Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013 Article ID 723976 13 pageshttpdxdoiorg1011552013723976

Research ArticleHeat Equations Associated with Weinstein Operator andApplications

Hatem Mejjaoli

Department of Mathematics College of Sciences Taibah University PO Box 30002 Al Madinah Al Munawarah Saudi Arabia

Correspondence should be addressed to HatemMejjaoli hatemmejjaoliipestrnutn

Received 31 May 2013 Accepted 3 August 2013

Academic Editor Dashan Fan

Copyright copy 2013 Hatem Mejjaoli This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We establish a characterization for the homogeneous Weinstein-Besov spaces via the Weinstein heat semigroup Next we obtainthe generalized Sobolev embedding theorems

Dedicated to Khalifa Trimeche

1 Introduction

We consider the Weinstein operator defined on R119889times ]0infin[

by

Δ 120573 =

119889+1

sum

119894=1

1205972

1205971199092

119894

+

2120573 + 1

119909119889+1

120597

120597119909119889+1

120573 gt minus

1

2

= Δ 119889 +L120573

(1)

where Δ 119889 is the Laplacian for the 119889-first variables and L120573 isthe Bessel operator for the last variable given by

L120573 =

1205972

1205971199092

119889+1

+

2120573 + 1

119909119889+1

120597

120597119909119889+1

120573 gt minus

1

2

(2)

For 119889 gt 2 the operator Δ 120573 is the Laplace-Beltrami operatoron the Riemannian space R119889

times ]0infin[ equipped with thefollowing metric

1198891199042= 119909

2(2120573+1)(119889minus1)

119889+1

119889+1

sum

119894=1

1198891199092

119894(3)

(cf [1 2])TheWeinstein operatorΔ 120573 has several applicationsin pure and applied mathematics especially in the fluidmechanics (cf [3])

The harmonic analysis associated with the Weinsteinoperator is studied by Ben Nahia and Ben Salem (cf [1 2])In particular the authors have introduced and studied thegeneralized Fourier transform associated with the Weinsteinoperator This transform is called the Weinstein transformWe note that for this transform we have studied the uncer-tainty principle (cf [4]) and the Gabor transform (cf [5])

In the present paper we intend to continue our studyof generalized spaces of type Sobolev associated with theWeinstein operator that started in [6]

In this paper we consider the Weinstein heat equation

120597119905119906 minus Δ 120573119906 = 119891 (119905 119909) (119905 119909) isin (0infin) timesR119889+1

+

119906|119905=0 = 119892

(4)

We study (4) to focus on the following problems

(1) Characterize the homogeneous Weinstein-Besovspaces via the Weinstein heat semigroup

(2) Prove the imbedding Sobolev theorems

I have studied the generalized Sobolev spaces in thecontext of differential-differences operators (cf [7ndash10])

The remaining part of the paper is organized as followsSection 2 is a summary of the main results in the harmonicanalysis associatedwith theWeinstein operators In Section 3

2 Journal of Function Spaces and Applications

we introduce and study the homogeneous Weinstein-Besovspaces the homogeneous Weinstein-Triebel-Lizorkin spacesand the homogeneous Weinstein-Riesz potential spaces InSection 4 we characterize the homogeneousWeinstein Besovspaces via the Weinstein heat semigroup Next we prove theSobolev embedding theorems

2 Preliminaries

In order to confirm the basic and standard notations webriefly overview the Weinstein operator and the relatedharmonic analysis Main references are [1 2]

21 Harmonic Analysis Associated with the Weinstein Opera-tor In this subsection we collect some notations and resultson theWeinstein kernel theWeinstein intertwining operatorand its dual the Weinstein transform and the Weinsteinconvolution

In the following

R119889+1

+= R

119889times [0infin[

119909 = (1199091 119909119889 119909119889+1) = (1199091015840 119909119889+1) isin R

119889+1

+

119909 = radic1199092

1+ sdot sdot sdot + 119909

2

119889+1

(5)

We denote by 119862lowast(R119889+1

) the space of continuous functionson R119889+1 even with respect to the last variable 119862119901

lowast(R119889+1

) thespace of functions of class 119862119901 on R119889+1 even with respectto the last variable Elowast(R

119889+1) the space of 119862infin-functions on

R119889+1 even with respect to the last variable Slowast(R119889+1

) theSchwartz space of rapidly decreasing functions onR119889+1 evenwith respect to the last variable 119863lowast(R

119889+1) the space of 119862infin-

functions on R119889+1 which are of compact support even withrespect to the last variable S1015840

lowast(R119889+1

) the space of temperatedistributions on R119889+1 even with respect to the last variableIt is the topological dual of Slowast(R

119889+1)

We consider the Weinstein operator Δ 120573 defined by

forall119909 = (1199091015840 119909119889+1) isin R

119889times ]0infin[

Δ 120573119891 (119909) = Δ 1199091015840119891 (1199091015840 119909119889+1) +L120573119909

119889+1

119891 (1199091015840 119909119889+1)

119891 isin 1198622

lowast(R

119889+1)

(6)

where Δ 1199091015840 is the Laplace operator on R119889 and L120573119909119889+1

is theBessel operator on ]0infin[ given by

L120573119909119889+1

=

1198892

1198891199092

119889+1

+

2120573 + 1

119909119889+1

119889

119889119909119889+1

120573 gt minus

1

2

(7)

TheWeinstein kernel Λ is given by

Λ (119909 119911) = 119890119894⟨11990910158401199111015840⟩119895120573 (119909119889+1119911119889+1)

forall (119909 119911) isin R119889+1

times C119889+1

(8)

where 119895120573(119909119889+1119911119889+1) is the normalized Bessel function TheWeinstein kernel satisfies the following properties

(i) For each 119911 isin R119889+1

+ we have

Δ 120573Λ (119909 119911) = minus1199112Λ (119909 119911) forall119909 isin R

119889+1

+ (9)

(ii) For all 119911 119905 isin C119889+1 we have

Λ (119911 119905) = Λ (119905 119911) Λ (119911 0) = 1

Λ (120582119911 119905) = Λ (119911 120582119905) forall120582 isin C(10)

(iii) For all ] isin N119889+1 119909 isin R119889+1 and 119911 isin C119889+1 we have1003816100381610038161003816119863

]119911Λ (119909 119911)

1003816100381610038161003816le 119909

|]| exp (119909 Im 119911) (11)

where 119863]119911= 120597

](120597119911

]1

1sdot sdot sdot 120597119911

]119889+1

119889+1) and |]| = ]1 + sdot sdot sdot + ]119889+1 In

particular1003816100381610038161003816Λ (119909 119910)

1003816100381610038161003816le 1 forall119909 119910 isin R

119889+1 (12)

The Weinstein intertwining operator is the operator R120573

defined on 119862lowast(R119889+1

) by

R120573119891 (1199091015840 119909119889+1)

=

2Γ (120573 + 1)

radic120587Γ (120573 + 12)

119909minus2120573

119889+1

timesint

119909119889+1

0

(1199092

119889+1minus 119905

2)

120573minus12

119891 (1199091015840 119905) 119889119905 119909119889+1 gt 0

119891 (1199091015840 0) 119909119889+1 = 0

(13)

R120573 is a topological isomorphism from Elowast(R119889+1

) ontoitself satisfying the following transmutation relation

Δ 120573 (R120573119891) = R120573 (Δ 119889+1119891) forall119891 isin Elowast (R119889+1

) (14)

where Δ 119889+1 = sum119889+1

119895=11205972

119895is the Laplacian on R119889+1

We denote by119871119901120573(R119889+1

+) the space ofmeasurable functions

on R119889+1

+such that

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)= (int

R119889+1+

1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119901119889120583120573 (119909))

1119901

lt infin

if 1 le 119901 lt infin

10038171003817100381710038171198911003817100381710038171003817119871infin120573(R119889+1+

)= ess sup

119909isinR119889+1+

1003816100381610038161003816119891 (119909)

1003816100381610038161003816lt infin

(15)

where 119889120583120573 is the measure on R119889+1

+given by

119889120583120573 (1199091015840 119909119889+1) = 119909

2120573+1

119889+1119889119909

1015840119889119909119889+1

(16)

TheWeinstein transform is given for 119891 in 1198711120573(R119889+1

+) by

F119882 (119891) (119910) = int

R119889+1+

119891 (119909)Λ (minus119909 119910) 119889120583120573 (119909) forall119910 isin R119889+1

+

(17)

Some basic properties of this transform are as follows

Journal of Function Spaces and Applications 3

(i) For 119891 in 1198711120573(R119889+1

+)

1003817100381710038171003817F119882 (119891)

1003817100381710038171003817119871infin120573(R119889+1+

)le100381710038171003817100381711989110038171003817100381710038171198711120573(R119889+1+

) (18)

(ii) For 119891 in Slowast(R119889+1

) we have

F119882 (Δ 120573119891) (119910) = minus10038171003817100381710038171199101003817100381710038171003817

2F119882 (119891) (119910) forall119910 isin R

119889+1

+ (19)

(iii) For all 119891 in 1198711120573(R119889+1

+) ifF119882(119891) belongs to 119871

1

120573(R119889+1

+)

then

119891 (119910) = 119862 (120573)int

R119889+1+

F119882 (119891) (119909) Λ (119909 119910) 119889120583120573 (119909) ae

(20)

where

119862 (120573) =

1

1205871198894120573+1198892

(Γ (120573 + 1))2 (21)

(iv) For 119891 isin Slowast(R119889+1

) if we define

F119882 (119891) (119910) = F119882 (119891) (minus119910) (22)

then

(F119882)minus1=

1

119862 (120573)

F119882 (23)

Proposition 1 (see [2]) (i) the Weinstein transform F119882 is atopological isomorphism fromSlowast(R

119889+1) onto itself and for all

f in Slowast(R119889+1

)

int

R119889+1+

1003816100381610038161003816119891 (119909)

1003816100381610038161003816

2119889120583120573 (119909) = 119862 (120573)int

R119889+1+

1003816100381610038161003816F119882 (119891) (120585)

1003816100381610038161003816

2119889120583120573 (120585)

(24)

(ii) In particular the renormalized Weinstein transform119891 rarr 119862(120573)

12F119882(119891) can be uniquely extended to an isometric

isomorphism from 1198712

120573(R119889+1

+) onto itself

In the Fourier analysis the translation operator is givenby 119891 997891rarr 119891(sdot + 119909)

In harmonic analysis associated for the operator Δ 120573 thegeneralized translation operator 120591119909 119909 isin R119889+1

+is defined by

forall119910 isin R119889+1

+

120591119909119891 (119910) =

Γ (120573 + 1)

radic120587Γ (120573 + 12)

times int

120587

0

119891(1199091015840+ 119910

1015840 radic119909

2

119889+1+ 119910

2

119889+1+ 2119909119889+1119910119889+1 cos 120579)

times (sin 120579)2120573119889120579(25)

where 119891 isin 119862lowast(R119889+1

)

By using the Weinstein kernel we can also define ageneralized translation For functions 119891 isin Slowast(R

119889+1) and

119910 isin R119889+1

+the generalized translation 120591119910119891 is defined by the

following relation

F119882 (120591119910119891) (119909) = Λ (119909 119910)F119882 (119891) (119909) (26)

By using the generalized translation we define the general-ized convolution product119891lowast

119882119892 of functions119891 119892 isin 1198711

120573(R119889+1

+)

as follows

119891lowast119882119892 (119909) = int

R119889+1+

120591119909119891 (minus1199101015840 119910119889+1) 119892 (119910) 119889120583120573 (119910) (27)

This convolution is commutative and associative and itsatisfies the following

(i) For all 119891 119892 isin 1198711

120573(R119889+1

+) 119891lowast119882 119892 belongs to 1198711

120573(R119889+1

+)

and

F119882 (119891lowast119882119892) = F119882 (119891)F119882 (119892) (28)

(ii) Let 1 le 119901 119902 119903 le infin such that 1119901 + 1119902 minus 1119903 = 1If 119891 isin 119871

119901

120573(R119889+1

+) and 119892 isin 119871

119902

120573(R119889+1

+) then 119891lowast119882119892 isin

119871119903

120573(R119889+1

+) and

1003817100381710038171003817119891lowast119882119892

1003817100381710038171003817119871119903120573(R119889+1+

)le10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171198921003817100381710038171003817119871119902

120573(R119889+1+

) (29)

We define the tempered distribution T119891 associated with119891 isin 119871

119901

120573(R119889+1

+) by

⟨T119891 120601⟩ = int

R119889+1+

119891 (119909) 120601 (119909) 119889120583120573 (119909) (30)

for 120601 isin Slowast(R119889+1

) and denote by ⟨119891 120601⟩120573 the integral in therighthand side

Definition 2 The Weinstein transform F119882(120591) of a distribu-tion 120591 isin S1015840

lowast(R119889+1

) is defined by

⟨F119882 (120591) 120601⟩ = ⟨120591F119882 (120601)⟩ (31)

for 120601 isin Slowast(R119889+1

)

In particular for 119891 isin 119871119901

120573(R119889+1

+) it follows that for 120601 isin

Slowast(R119889+1

)

⟨F119882 (119891) 120601⟩ = ⟨F119882 (T119891) 120601⟩ = ⟨T119891F119882 (120601)⟩

= ⟨119891F119882 (120601)⟩120573

(32)

Proposition 3 The Weinstein transform F119882 is a topologicalisomorphism from S1015840

lowast(R119889+1

) onto itself

Definition 4 The generalized convolution product of a dis-tribution 119878 in S1015840

lowast(R119889+1

) and a function 120601 in Slowast(R119889+1

) is thefunction 119878lowast119882120601 defined by

119878lowast119882 120601 (119909) = ⟨119878119910 120591minus119910120601 (119909)⟩ (33)

4 Journal of Function Spaces and Applications

Proposition 5 Let 119891 be in 119871119901

120573(R119889+1

+) 1 le 119901 le infin and 120601

in Slowast(R119889+1

) Then the distribution T119891lowast119882120601 is given by thefunction 119891lowast119882120601 and T119891lowast119882120601 belongs to 119871119901

120573(R119889+1

+) Moreover

for all 120595 isin Slowast(R119889+1

)

⟨T119891lowast119882120601 120595⟩ = ⟨

119891 120601lowast119882 ⟩120573 (34)

where (119909) = 120595(minus119909) and

F119882 (T119891lowast119882 120601) = F119882 (T119891)F119882 (120601) (35)

For each 119906 isin S1015840

lowast(R119889+1

) we define the distribution Δ 120573119906

by ⟨Δ 120573119906 120595⟩ = ⟨119906 Δ 120573120595⟩ and this distribution satisfies thefollowing property

F119882 (Δ 120573119906) = minus10038171003817100381710038171199101003817100381710038171003817

2F119882 (119906) (36)

In the following we denote T119891 given by (30) by 119891 forsimplicity

3 B119904120573

119901119902 F119904120573

119901119902(R119889+1

+) H119904

119901120573Spaces and

Basic Properties

31 HomogeneousWeinstein-Littlewood-Paley DecompositionOne of the main tools in this paper is the homogeneousLittlewood-Paley decomposition of distribution associatedwith theWeinstein operators into dyadic blocs of frequencies

Lemma6 One defines byC the ring of center 0 of small radius12 and great radius 2 There exists two radial functions 120595 and120593 the values of which are in the interval [0 1] belonging to119863lowast(R

119889+1) such that

supp120595 sub 119861 (0 1) supp120593 sub C

forall120585 isin R119889+1

120595 (120585) + sum

119895ge0

120593 (2minus119895120585) = 1

forall120585 isin C sum

119895isinZ

120593 (2minus119895120585) = 1

|119899 minus 119898| ge 2 997904rArr supp120593 (2minus119899sdot) cap supp120593 (2minus119898sdot) = 0

119895 ge 1 997904rArr supp120595 cap supp120593 (2minus119895sdot) = 0

(37)

Notations We denote by

Δ 119895119891 = Fminus1

119882(120593(

120585

2119895)F119882 (119891))

119878119895119891 = sum

119899le119895minus1

Δ 119899119891 forall119895 isin Z(38)

The distribution Δ 119895119891 is called the 119895th dyadic block of thehomogeneous Littlewood-Paley decomposition of 119891 associ-ated with the Weinstein operators

Throughout this paper we define 120601 and 120594 by 120601 = Fminus1

119882(120593)

and 120594 = Fminus1

119882(120595)

When dealing with the Littlewood-Paley decompositionit is convenient to introduce the functions and 120593 belongingto119863lowast(R

119889+1) such that equiv 1 on supp120595 and 120593 equiv 1 on supp120593

Remark 7 We remark that

F119882 (119878119895119891) (120585) = (

120585

2119895)F119882 (119878119895119891) (120585)

F119882 (Δ 119895119891) (120585) = 120593(

120585

2119895)F119882 (Δ 119895119891) (120585)

(39)

We put

120601 = F

minus1

119882(120593) 120594 = F

minus1

119882() (40)

Definition 8 One denotes by S1015840

ℎ120573lowast(R119889+1

) the space oftempered distribution such that

lim119895rarrminusinfin

119878119895119906 = 0 in S1015840

lowast(R

119889+1) (41)

Proposition 9 (Bernstein inequalities) For all 120583 isin N119889+1 and120590 isin R for all 119895 isin Z for all 1 le 119901 119902 le infin and for all119891 isin S1015840

lowast(R119889+1

) one has the following

(i) 10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902

120573(R119889+1+

)

le

10038171003817100381710038171003817

120601

10038171003817100381710038171003817119871119903120573(R119889+1+

)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

)2119895(119889+2120573+2)(1119901minus1119902)

with 1

119902

=

1

119901

+

1

119903

minus 1

(ii) 10038171003817100381710038171003817119878119895119891

10038171003817100381710038171003817119871119902

120573(R119889+1+

)

le10038171003817100381710038171205941003817100381710038171003817119871119903120573(R119889+1+

)

10038171003817100381710038171003817119878119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

)2119895(119889+2120573+2)(1119901minus1119902)

with 1

119902

=

1

119901

+

1

119903

minus 1

(iii)1003817100381710038171003817100381710038171003817

(radicminusΔ 120573)

120590

Δ 119895119891

1003817100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)

le

10038171003817100381710038171003817F

minus1

119882(10038171003817100381710038171205851003817100381710038171003817

120590120593)

100381710038171003817100381710038171198711120573(R119889+1+

)

times

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

)2119895120590

(42)

Proof Using Remark 7 we deduce from Proposition 5 that

119878119895119891 = 2119895(119889+2120573+2)

120594 (2119895sdot) lowast119882119878119895119891

Δ 119895119891 = 2119895(119889+2120573+2)

120601 (2119895sdot) lowast119882Δ 119895119891

(43)

Thus from the relation (29) we prove (i) (ii) and (iii)

32 Definitions In the following we define analogues of thehomogeneous Besov Triebel-Lizorkin and Riesz potentialspaces associated with the Weinstein operators on R119889+1

+and

obtain their basic propertiesFrom now we make the convention that for all non-

negative sequence 119886119902119902isinZ the notation (sum119902119886119903

119902)1119903 stands for

sup119902119886119902 in the case 119903 = infin

Journal of Function Spaces and Applications 5

Definition 10 Let 119904 isin R and 119901 119902 isin [1infin]The homogeneousWeinstein-Besov spaces B

119904120573

119901119902(R119889+1

+) are the spaces of distri-

bution in S1015840

ℎ120573lowast(R119889+1

) such that

10038171003817100381710038171198911003817100381710038171003817B119904120573

119901119902(R119889+1+

)= (sum

119895isinZ

(211990411989510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119902

)

1119902

lt infin (44)

Proposition 11 (see [6]) Let 119904 isin R and 119901 and 119902 two elementsof [1infin] the space B

119904120573

119901119902(R119889+1

+) is the set of 119891 isin S1015840

ℎ120573lowast(R119889+1

)

verifying

10038171003817100381710038171003817

119891

10038171003817100381710038171003817B119904120573

119901119902(R119889+1+

)= (int

infin

0

(119905minus1199041003817100381710038171003817119891lowast119882120601119905

1003817100381710038171003817119871119901

120573(R119889+1+

))

119902119889119905

119905

)

1119902

lt infin

(45)

where 120601119905(119909) = (11199052120573+2+119889

)120601(119909119905) for all 119905 isin (0infin) and 119909 isin

R119889+1

+

Definition 12 For 119904 isin R and 119901 119902 isin [1infin] one writes10038171003817100381710038171198911003817100381710038171003817119861119904120573

119901119902(R119889+1+

)=10038171003817100381710038171198780119891

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ (sum

119895ge1

(211990411989510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119902

)

1119902

(46)

The nonhomogeneous Besov space 119861119904120573119901119902(R119889+1

+) associated

with the Weinstein operators is defined by

119861119904120573

119901119902(R

119889+1

+) = 119891 isin S

1015840(R

119889)

10038171003817100381710038171198911003817100381710038171003817119861119904120573

119901119902(R119889+1+

)lt infin (47)

We give now another definition equivalent to the nonho-mogeneous Besov space 119861119904120573119901119902(R119889+1

+)

Proposition 13 Let 119904 isin R and119901 and 119902 two elements of [1infin]the space 119861119904120573119901119902(R119889+1

+) is the set of 119891 isin S1015840

(R119889) verifying

10038171003817100381710038171003817

119891

10038171003817100381710038171003817119861119904120573

119901119902(R119889+1+

)=1003817100381710038171003817119891lowast119882120595

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ (int

1

0

(119905minus1199041003817100381710038171003817119891lowast119882120601119905

1003817100381710038171003817119871119901

120573(R119889+1+

))

119902119889119905

119905

)

1119902

lt infin

(48)

Definition 14 Let 119904 isin R and 1 le 119901 119902 le infin the homogeneousWeinstein-Triebel-Lizorkin space F

119904120573

119901119902(R119889+1

+) is the space of

distribution in S1015840

ℎ120573lowast(R119889+1

) such that

10038171003817100381710038171198911003817100381710038171003817F119904120573

119901119902(R119889+1+

)=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

211990411989511990210038161003816100381610038161003816Δ 119895119891

10038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)

lt infin (49)

Definition 15 For 119904 isin R the operatorR119904

120573from S1015840

ℎ120573lowast(R119889+1

)

to S1015840

ℎ120573lowast(R119889+1

) is defined by

R119904

120573(119891) = F

minus1

119882(sdot

119904F119882119891) (50)

The operatorRminus119904

120573is called Weinstein-Riesz potential space

Definition 16 For 119904 isin R and 1 le 119901 le infin the homogeneousWeinstein-Riesz potential space H119904

119901120573(R119889+1

+) is defined

as the space Rminus119904

120573(119871

119901

120573(R119889+1

+)) equipped with the norm

119891H119904119901120573(R119889+1+

)= R119904

120573(119891)

119871119901

120573(R119889+1+

)

Proposition 17 Let 119904 isin R and 1 le 119901 119902 le infinThe operator Δ 120573 is a linear continuous operator from

B119904120573

119901119902(R119889+1

+) into B

119904minus2120573

119901119902 (R119889+1

+) and from H119904

119901120573(R119889+1

+) into

H119904minus2

119901120573(R119889+1

+)

Proof We obtain these results by the similar ideas used in thenonhomogeneous case (cf [6])

Proposition 18 Let 119904 119905 isin R and 1 le 119901 119902 le infin The operatorR119905

120573is a linear continuous injective operator from B

119904120573

119901119902(R119889+1

+)

onto B119904minus119905120573

119901119902 (R119889+1

+) and from H119904

119901120573(R119889+1

+) onto H119904minus119905

119901120573(R119889+1

+)

Proof We obtain these results by the similar ideas used in thenonhomogeneous case (cf [6])

33 Embeddings As in the Euclidean case (cf [11]) themono-tone character of 119897119902-spaces and the Minkowskis inequalityyield the following

Proposition 19 If 1 le 1199021 lt 1199022 le infin one has

B119904120573

1199011199021

(R119889+1

+) 997893rarr

B119904120573

1199011199022

(R119889+1

+) (1 le 119901 le infin 119904 isin R)

(51)

MoreoverB119904120573

1199011(R

119889+1

+) 997893rarr

H119904

119901120573(R

119889+1

+) 997893rarr

B119904120573

119901infin(R

119889+1

+)

(1 le 119901 le infin 119904 isin R)

(52)

If 1199040 = 1199041 one also has

(H1199040

119901120573(R

119889+1

+)

H1199041

119901120573(R

119889+1

+))

120579119902=

B119904120573

119901119902(R

119889+1

+)

(1 le 119901 119902 le infin 120579 isin (0 1))

(53)

where 119904 = (1 minus 120579)1199040 + 1205791199041

Proposition 20 One assumes that 119904 minus (119889 + 2120573 + 2)119901 = 1199041 minus

(119889 + 2120573 + 2)1199011 Then the following inclusion holds

B119904120573

119901119902(R

119889+1

+) 997893rarr

B1199041120573

11990111199021

(R119889+1

+)

(1 le 119901 le 1199011 le infin 1 le 119902 le 1199021 le infin 119904 1199041 isin R)

(54)

Proof In order to prove the inclusion we use the estimate

Δ 119895119891 = 2119895(119889+2120573+2)

120601 (2119895sdot) lowast119882Δ 119895119891 (55)

Proposition 9(i) gives that10038171003817100381710038171003817Δ 119895119891

100381710038171003817100381710038171198711199011

120573(R119889+1+

)=

100381710038171003817100381710038172119895(119889+2120573+2)

120601 (2119895sdot) lowast119882Δ 119895119891

100381710038171003817100381710038171198711199011

120573(R119889+1+

)

le 1198622119895(119889+2120573+2)(1119901minus1119901

1)10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

(56)

6 Journal of Function Spaces and Applications

By definition of the homogeneous Weinstein-Besov spaceswe therefore infer10038171003817100381710038171198911003817100381710038171003817B1199041120573

11990111199021(R119889+1+

)

= (

infin

sum

119895=minusinfin

(2119895119904110038171003817100381710038171003817Δ 119895119891

100381710038171003817100381710038171198711199011

120573(R119889+1+

))

1199021

)

11199021

le 119862(sum

119895isinZ

(211989511990412119895(119889+2120573+2)(1119901minus1119901

1)10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

1199021

)

11199021

le 119862(sum

119895isinZ

(211989511990410038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

1199021

)

11199021

le 11986210038171003817100381710038171198911003817100381710038171003817B119904120573

119901119902(R119889+1+

)

(57)

since 119902 le 1199021 This gives the inclusion

Proposition 21 (1) If 119906 belongs to B119904120573

119901119902(R119889+1

+) cap

B119905120573

119901119902(R119889+1

+)

then 119906 belongs to B120579119904+(1minus120579)119905120573

119901119902 (R119889+1

+) for all 120579 isin [0 1] and

119906B120579119904+(1minus120579)119905120573

119901119902(R119889+1+

)le 119906

120579

B119904120573

119901119902(R119889+1+

)119906

1minus120579

B119905120573

119901119902(R119889+1+

) (58)

(2) If 119906 belongs to B119904120573

119901infin(R119889+1

+) cap

B119905120573

119901infin(R119889+1

+) and 119904 lt 119905

then 119906 belongs to B120579119904+(1minus120579)119905120573

1199011(R119889+1

+) for all 120579 isin (0 1) and there

exists a positive constant 119862(119905 119904) such that

119906B120579119904+(1minus120579)119905120573

1199011(R119889+1+

)le 119862 (119905 119904) 119906

120579

B119904120573

119901infin(R119889+1+

)119906

1minus120579

B119905120573

119901infin(R119889+1+

) (59)

(3) If 119906 belongs to B119904120573

119901infin(R119889+1

+) cap

B119904+120576120573

119901infin (R119889+1

+) and 120576 gt

0 then 119906 belongs to B119904120573

1199011(R119889+1

+) and there exists a positive

constant 119862 such that

119906B119904120573

1199011(R119889+1+

)le

119862

120576

119906B119904120573

119901infin(R119889+1+

)log

2(119890 +

119906B119904+120576120573

119901infin(R119889+1+

)

119906B119904120573

119901infin(R119889+1+

)

)

(60)

Proof (1) is obvious from the Holderrsquos inequality As for (2)we write 119906

B120579119904+(1minus120579)119905120573

1199011(R119889+1+

)as

sum

119895le119873

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ sum

119895gt119873

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

(61)

where 119873 is chosen here after By the definition of thehomogeneous Weinstein-Besov norms we see that

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 2

119895(1minus120579)(119905minus119904)119906

B119904120573

119901infin(R119889+1+

)

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 2

minus119895120579(119905minus119904)119906

B119905120573

119901infin(R119889+1+

)

(62)

and thus 119906B120579119904+(1minus120579)119905120573

1199011(R119889+1+

)is dominated by

119906B119904120573

119901infin(R119889+1+

)sum

119895le119873

2119895(1minus120579)(119905minus119904)

+ 119906B119905120573

119901infin(R119889+1+

)sum

119895gt119873

2minus119895120579(119905minus119904)

le 119862119906B119904120573

119901infin(R119889+1+

)

2(119873+1)(1minus120579)(119905minus119904)

2(1minus120579)(119905minus119904)

minus 1

+ 119906B119905120573

119901infin(R119889+1+

)

2minus119873120579(119905minus119904)

1 minus 2minus120579(119905minus119904)

(63)

Hence in order to complete the proof of (2) it suffices tochoose119873 such that

119906B119905120573

119901infin(R119889)

119906B119904120573

119901infin(R119889)

le 2119873(119905minus119904)

lt 2

119906B119905120573

119901infin(R119889)

119906B119904120573

119901infin(R119889)

(64)

As for (3) it is easy to see that 119906B119904120573

1199011(R119889+1+

)is dominated as

sum

119895le119873minus1

211989511990410038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ sum

119895ge119873

211989511990410038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le (119873 + 1) 119906B119904120573

119901infin(R119889+1+

)+

2minus(119873minus1)120576

2120576minus 1

119906B119904+120576120573

119901infin(R119889+1+

)

(65)

Hence letting

119873 = 1 + [

[

1

120576

log2

119906B119904+120576120573

119901infin(R119889+1+

)

119906B119904120573

119901infin(R119889+1+

)

]

]

(66)

we can obtain the desired estimate

Proposition 22 Let 119902 isin (1infin) and let 119904 isin R such that0 lt 119904 lt (119889 + 2120573 + 2)119902 then one has

B119904120573

119902119902(R

119889+1

+) =

F119904120573

119902119902(R

119889+1

+) 997893rarr

F119904120573

119902infin(R

119889+1

+)

997893rarrF119904minus((119889+2120573+2)119902)120573

infininfin(R

119889+1

+)

(67)

H119904

119902120573(R

119889+1

+) =

F119904120573

1199022(R

119889+1

+) 997893rarr

F119904120573

119902infin(R

119889+1

+)

997893rarrF119904minus((119889+2120573+2)119902)120573

infininfin(R

119889+1

+)

(68)

Proof We obtain these results by the similar ideas used in thenonhomogeneous case (cf [6])

Theorem 23 Let 119886 119887 gt 0 and let 1199021 1199022 isin [1infin] Let120579 = 119886(119886 + 119887) isin (0 1) and let 1119901 = (1 minus 120579)1199021 +

1205791199022 Then there exists a constant 119862 such that for every119891 isin

F119886120573

1199021infin(R119889+1

+) cap

Fminus119887120573

1199022infin(R119889+1

+) then one has

1003816100381610038161003816119891 (119909)

1003816100381610038161003816le 119862(sup

119895isinZ

2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

1minus120579

(sup119895isinZ

2minus119887119895 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

120579

(69)

In particular one gets

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

F119886120573

1199021infin(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

120579

Fminus119887120573

1199022infin(R119889+1+

) (70)

Journal of Function Spaces and Applications 7

Proof Let 119891 be a Schwartz class we have

1003816100381610038161003816119891 (119909)

1003816100381610038161003816le sum

119895isinZ

10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816

le sum

119895isinZ

min(2minus119886119895sup119895isinZ

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816))

(71)

We define119873(119909) as the largest index such that

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816) le 2

minus119886119895sup119895isinZ

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816) (72)

and we write1003816100381610038161003816119891 (119909)

1003816100381610038161003816le sum

119895le119873(119909)

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

+ sum

119895gt119873(119909)

2minus119886119895sup

119895

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

le 119862(sup119895isinZ

2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

119887(119886+119887)

times (sup119895isinZ

2minus119887119895 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

119886(119886+119887)

(73)

Thus (69) is proved In order to obtain (70) it is enough toapply the Holder inequality in the expression above since wehave 120579 = 119886(119886+119887) isin (0 1) and let 1119901 = (1minus120579)1199021+1205791199022

Corollary 24 Let 119902 isin (1infin) and let 119904 isin R such that 0 lt 119904 lt

(119889 + 2120573 + 2)119902 then one has

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus(119902119901)

Bminus((2120573+2+119889)119902minus119904)120573

infininfin (R119889+1

+ )

10038171003817100381710038171198911003817100381710038171003817

119902119901

B119904120573

119902119902(R119889+1

+ )

(74)

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus(119902119901)

Bminus((2120573+2+119889)119902minus119904)120573

infininfin (R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

119902119901

H119904119902120573(R119889+1+

) (75)

where 119901 = 119902(2120573 + 2 + 119889)(2120573 + 2 + 119889 minus 119902119904)

Proof By choosing 119886 = 119904 gt 0 minus119887 = 119904 minus (119889 + 2120573 + 2)119902 lt 01199021 = 119902 and 1199022 = infin we deduce (74) from the relations (70)and (67) In the same way we deduce (75) from the relations(70) and (68)

4 Generalized Heat Equation

41 Characterization for the Weinstein-Besov Spaces TheWeinstein heat equation reads

120597119905119906 (119905 119909) minus Δ 120573119906 (119905 119909) = 119891 (119905 119909) (119905 119909) isin [0infin) timesR119889+1

+

119906|119905=0 = 119892

(76)

We introduce the Weinstein heat semigroup 119867120573(119905) for theWeinstein-Laplace operator

119867120573 (119905) 119892 (119909) =

int

R119889+1+

Γ120573 (119905 119909 119910) 119892 (119910) 119889120583120573 (119910) if 119905 gt 0

119892 (119909) if 119905 = 0(77)

where Γ120573 is the Weinstein heat kernel defined by

Γ120573 (119905 119909 119910) = 120591119909 (119864(120573)

119905) (119910) (78)

where

119864(120573)

119905(119910) =

2

1205871198892Γ (120573 + 1) (4119905)

120573+1+1198892119890minus11991024119905 (79)

Thus

119867120573 (119905) 119892 (119909) = 119892lowast119882119864(120573)

119905 (119909) (80)

In practice we use the integral formulation of (76)

119906 (119905 119909) = 119867120573 (119905) 119892 (119909) + 119866120573 (119891) (119905 119909)

= 119867120573 (119905) 119892 (119909) + int

119905

0

119867120573 (119905 minus 119904) 119891 (119904 119909) 119889119904

(81)

Remark 25 The function 119864(120573)119905

is the Gauss kernel associatedwith Weinstein operators This function satisfies

forall120585 isin R119889+1

+ F119882 (119864

(120573)

119905) (120585) = 119890

minus1199051205852

(82)

Proposition 26 Let 1 le 119901 le 119903 le infin and let 119891 isin 119871119901

120573(R119889+1

+)

Then the operator 119867120573(119905) maps 119871119901

120573(R119889+1

+) continuously to

119871119903

120573(R119889+1

+) and

10038171003817100381710038171003817119867120573 (119905) 119891

10038171003817100381710038171003817119871119903120573(R119889)

le 119862119905minus((119889+2120573+2)2)(1119901minus1119903)1003817

1003817100381710038171198911003817100381710038171003817119871119901

120573(R119889)

(83)

Moreover1003817100381710038171003817100381710038171003817

(minusΔ 120573)

1205752

119867120573 (119905) 119891

1003817100381710038171003817100381710038171003817119871119903120573(R119889)

le 119862119905minus1205752minus((119889+2120573+2)2)(1119901minus1119903)1003817

1003817100381710038171198911003817100381710038171003817119871119901

120573(R119889)

(84)

for all 120575 gt 0

Proof It follows from the relations (80) and (29) combinedwith scaling property of the kernel 119864(120573)

119905

In this section we prove estimates for the Weinstein heatsemigroupThese estimates are based on the following result

Lemma 27 Let C be an annulus Positive constants 119888 and 119862exist such that for any 119901 in [1infin] and any couple (119905 120582) ofpositive real numbers one has

suppF119882 (119906) sub 120582C 997904rArr

10038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le 119862119890minus1198881199051205822

119906119871119901

120573(R119889+1+

)

(85)

8 Journal of Function Spaces and Applications

Proof We again consider a function Θ in 119863(R119889+1

+ 0) the

value of which is identically 1 in neighborhood of annulusCWe can also assume without loss of generality that 120582 = 1 Wethen have

119867120573 (119905) 119906 = 119892 (119905 sdot) lowast119882119906 (86)where

119892 (119905 sdot) = Fminus1

119882(Θ (120585) 119890

minus1199051205852

) (87)

The lemma is proved provided that we can find positive realnumbers 119888 and 119862 such that

forall119905 gt 01003817100381710038171003817119892 (119905 sdot)

10038171003817100381710038171198711120573(R119889+1+

)le 119862119890

minus119888119905 (88)

To begin we perform integrations by parts in (87) We get1003816100381610038161003816119892 (119905 119909)

1003816100381610038161003816=

1

119888120573

(1 + 1199092)

minus(119889+[2120573]+3)

times int

R119889+1+

Λ (119909 120585) (119868119889 minus Δ 120573)

119889+[2120573]+3

times (Θ (120585) 119890minus1199051205852

) 119889120583120573 (120585)

(89)

Using Leibnizrsquos formula we obtain1003816100381610038161003816119892 (119905 119909)

1003816100381610038161003816le 119862(1 + 119909

2)

minus(119889+[2120573]+3)

119890minus119888119905 (90)

and (88) follows

For any interval 119868 of R (bounded or unbounded) wedefine themixed space-time 119871119901(119868 119871119902

120573(R119889+1

+)) Banach space of

(classes of) measurable functions 119906 119868 rarr 119871119902

120573(R119889+1

+) such

that 119906119871119901(119868119871

119902

120573(R119889+1+

))lt infin with

119906119871119901(119868119871119902

120573(R119889+1+

))= (int

119868

119906 (119905 sdot)119901

119871119902

120573(R119889+1+

)119889119905)

1119901

if 1 le 119901 119902 lt infin

119906119871infin(119868119871119902

120573(R119889+1+

))= ess sup

119905isin119868

119906 (119905 sdot)119871119902

120573(R119889+1+

) if 1 le 119902 lt infin

(91)Corollary 28 Let C be an annulus and 120582 a positive realnumber Let 1199060 (resp 119891 = 119891(119905 119909)) satisfy suppF119882(1199060) sub 120582C(resp suppF119882(119891(119905 sdot)) sub 120582C for all 119905 in [0 119879]) Consider 119906 asolution of

120597119905119906 minus Δ 120573119906 = 0 119906|119905=0 = 1199060 (92)and V a solution of

120597119905V minus Δ 120573V = 119891 (119905 sdot) V|119905=0 = 0 (93)There exist positive constants 119888 and 119862 depending only on Csuch that for any 1 le 119886 le 119887 le infin and 1 le 119901 le 119902 le infin we have

119906119871119902([0119879]119871119887120573(R119889+1+

))le 119862120582

(119889+2120573+2)(1119886minus1119887)120582minus21199021003817

1003817100381710038171199060

1003817100381710038171003817119871119886120573(R119889+1+

)

V119871119902([0119879]119871119887120573(R119889+1+

))le 119862120582

minus2(1+1119902minus1119901)120582(119889+2120573+2)(1119886minus1119887)

times10038171003817100381710038171198911003817100381710038171003817119871119901([0119879]119871119886

120573(R119889+1+

))

(94)

Proof It suffices to use the fact that

119906 (119905 sdot) = 119867120573 (119905) 1199060 V (119905 sdot) = int119905

0

119867120573 (119905 minus 119904) 119891 (119904 sdot) 119889119904

(95)

Combining Lemma 27 and Youngrsquos inequality (29) withscaling property of the kernel 119864(120573)

119905now yields the result

Theorem 29 Let 119904 be a positive real number and (119901 119903) isin

[1infin]2 A constant 119862 exists which satisfies the following

property For 119906 isin Bminus2119904120573

119901119903 (R119889+1

+) one has

119862minus1119906

Bminus2119904120573

119901119903(R119889+1+

)le

10038171003817100381710038171003817100381710038171003817

10038171003817100381710038171003817119905119904119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862119906Bminus2119904120573

119901119903(R119889+1+

)

(96)

To prove this result we need the following lemma

Lemma 30 There exist two positive constants 120581 and 119862

depending only on 120593 such that for all 1 le 119901 le infin 120591 ge 0 and119895 isin Z one has

10038171003817100381710038171003817Δ 119895 (119867120573 (120591) 119906)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862119890

minus1205812211989512059110038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

) (97)

Proof The result follows immediately by applying Lemma 27and because Δ 119895(119867120573(120591))119906 = (119867120573(120591)Δ 119895)119906

Proof of Theorem 29 Using Lemma 30 and considering thefact that the operator Δ 119895 commutes with the operator 119867120573(119905)

and the definition of the homogeneous Weinstein-Besov(semi) norm we get

10038171003817100381710038171003817119905119904119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862119906

Bminus2119904120573

119901119903(R119889+1+

)sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903119895

(98)

where (119888119903119895)119895isinZ denotes as in all this proof a generic elementof the unit sphere of 119897119903(Z) In the case when 119903 = infin therequired inequality comes immediately from the followingeasy result For any positive 119904 we have

sup119905gt0

sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

lt infin (99)

In the case 119903 lt infin using the Holder inequality with theweight 22119895119904119890minus1205811199052

2119895

(99) and the Fubini theorem we obtain

int

infin

0

11990511990311990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

119889119905

119905

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)int

infin

0

(sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

)

119903minus1

times (sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903

119903119895)

119889119905

119905

Journal of Function Spaces and Applications 9

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)int

infin

0

(sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903

119903119895)

119889119905

119905

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)sum

119895isinZ

119888119903

119903119895int

infin

0

(11990511990422119895119904119890minus12058111990522119895

)

119889119905

119905

le 119862Γ (119904) 119906119903

Bminus2119904120573

119901119903(R119889+1+

)

(100)

In order to prove the other inequality let us observe thatfor any 119904 greater than minus1 we have

Δ 119895119906 =

1

Γ (119904 + 1)

int

infin

0

119905119904(minusΔ 120573)

119904+1

119867120573 (119905) Δ 119895119906 119889119905 (101)

Then Lemma 30 Proposition 9 and the fact that the operatorΔ 119895 commutes with the operator119867120573(119905) lead to the following

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862int

infin

0

11990511990422119895(119904+1)

119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889119905

(102)

In the case 119903 = infin we simply write

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862(sup

119905gt0

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

times int

infin

0

22119895(119904+1)

119890minus12058111990522119895

119889119905

le 11986222119895119904(sup119905gt0

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

(103)

In the case 119903 lt infin Holderrsquos inequality with the weight 119890minus12058111990522119895

gives

(int

infin

0

119905119904119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889119905)

119903

le 1198622minus2119895(119903minus1)

int

infin

0

119905119903119904119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)119889119905

(104)

Thanks to (99) and Fubinirsquos theorem we infer from (102) that

sum

119895isinZ

2minus211989511990311990410038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)le 119862int

infin

0

11990511990311990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

119889119905

119905

(105)

The theorem is proved

Second Proof ofTheorem 29We only consider the case 1 le 119903 ltinfinThe case 119903 = infin can be shown similarlyWe first prove that

119862minus1119906

Bminus2119904120573

119901119903(R119889+1+

)le

10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

(106)

It is easy to see that

Δ 119895119906 = 120594119895lowast119882119864(120573)

2minus2119895lowast119882119906

(107)

where 120594119895 = Fminus1

119882(120593(2

minus119895120585)119890

2minus21198951205852

) and119864(120573)2minus2119895

is the Gauss kernelassociated with Weinstein operators By relation (29) we get

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le

10038171003817100381710038171003817120594119895

100381710038171003817100381710038171198711120573(R119889+1+

)

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

) (108)

As10038171003817100381710038171003817120594119895

100381710038171003817100381710038171198711120573(R119889+1+

)= int

R119889+1+

100381610038161003816100381610038161003816F

minus1

119882(120593 (120585) 119890

1205852

)

100381610038161003816100381610038161003816119889120583120573 (120585) lt infin (109)

we obtain10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

) (110)

Moreover simple calculations give that

119864(120573)

2minus2119895lowast119882119906 = 119867120573 (2

minus4119895minus 119905

2) (119864

(120573)

1199052lowast119882119906)

(111)

Thus from Proposition 26 it follows that100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1

+ )

(112)

for any 119905 isin [2minus119895minus1 2minus119895] which implies that

sum

119895isinZ

2minus211990411989511990310038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

le 119862 sum

119895isinZ

int

2minus119895

2minus119895minus1(119905

2119904100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862int

infin

0

(1199052119904100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862int

infin

0

(11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

(113)

where we have used the fact that 119864(120573)1199052lowast119882119906 = 119867120573(119905

2)119906

We now prove that10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862119906Bminus2119904120573

119901119903(R119889+1+

) (114)

Indeed one has

119864(120573)

2minus2119895lowast119882119906 = sum

119899isinZ

119864(120573)

2minus2119895lowast119882Δ 119899+119895119906 (115)

Arguing as above we have100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1

+ )

(116)

for any 119905 isin [2minus119895 21minus119895] Thus10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

= 2

100381710038171003817100381710038171003817100381710038171003817

1003817100381710038171003817100381710038171199052119904119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)

100381710038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862 sum

119895isinZ

int

21minus119895

2minus119895(2

minus2119895119904100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862 sum

119895isinZ

(2minus2119895119904

sum

119899isinZ

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

(117)

10 Journal of Function Spaces and Applications

On the other hand it is easy to see that100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 1198622

minus211989911990410038171003817100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

) (118)

for any 119904 gt 0 For 0 lt 1199041 lt 119904 lt 1199042 and by using theMinkowskiinequality we have

sum

119895isinZ

(2minus2119895119904

sum

119899isinZ

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

le sum

119895isinZ

(2minus2119895119904

0

sum

minusinfin

2minus21198991199041

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

+ sum

119895isinZ

(2minus2119895119904

sum

119899isinN

2minus21198991199042

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

le 119862(

0

sum

minusinfin

2minus2119899(119904

1minus119904)(sum

119895isinZ

(2minus2(119899+119895)11990311990410038171003817

100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

)

1119903

)

119903

+ 119862(sum

N

2minus2119899(119904

2minus119904)(sum

119895isinZ

(2minus2(119899+119895)11990311990410038171003817

100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

)

1119903

)

119903

le 119862 sum

119895isinZ

2minus211989511990311990410038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

(119)

The result is immediately from (117) and (119)

42 Embedding Sobolev Theorems

Theorem 31 Let 1 lt 119901 lt infin and let 0 lt 119904 lt (119889 + 2120573 +

2)119901There exists a positive constant119862 such that for all function119891 isin

H119904

119901120573(R119889+1

+) one has

10038171003817100381710038171198911003817100381710038171003817119871119902

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

H119904119901120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

120579

B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

) (120)

where 120579 = 119904119901(119889+2120573+2) and 119902 = 119901(119889+2120573+2)(119889+2120573+2minus119901119904)

Proof Bydensity we can suppose that119891belongs toSlowast(R119889+1

)It is easy to see that

119891 = int

infin

0

119867120573 (119905) Δ 120573119891119889119905(121)

and decompose the integral in two parts as follows

119891 = int

119860

0

119867120573 (119905) Δ 120573119891119889119905 + int

infin

119860

119867120573 (119905) Δ 120573119891119889119905(122)

where 119860 is a constant to be fixed laterOn the other hand byTheorem 29 we obtain10038171003817100381710038171003817119867120573 (119905) Δ 120573119891

10038171003817100381710038171003817119871infin120573(R119889+1+

)

le

119862

1199051minus(12)(119904minus(119889+2120573+2)119901)

10038171003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(123)

Therefore after integrating we get

int

infin

119860

10038171003817100381710038171003817119867120573 (119905) Δ 120573119891

10038171003817100381710038171003817119871infin120573(R119889+1+

)119889119905

le 119860(12)(119904minus(119889+2120573+2)119901)1003817

1003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(124)

On the other hand denoting 119892 = (minusΔ 120573)1199042119891 we have

119867120573 (119905) Δ 120573119891 =

1

(minus119905)1minus1199042

119867120573 (119905) (minus119905Δ 120573)

1minus1199042

119892 (125)

We proceed as in [8] we prove that1003816100381610038161003816100381610038161003816

119867120573 (119905) (minus119905Δ 120573)

1minus1199042

119892 (119909)

1003816100381610038161003816100381610038161003816

le 119862 (119904)119872120573 (119892) (119909) (126)

where119872120573(119892) is a maximal function of 119892 associated with theWeinstein operators (cf [12])

This leads to100381610038161003816100381610038161003816100381610038161003816

int

119860

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

100381610038161003816100381610038161003816100381610038161003816

le 1198621198601199042119872120573 (119892) (119909) (127)

In conclusion we get10038161003816100381610038161003816100381610038161003816

int

infin

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

10038161003816100381610038161003816100381610038161003816

le 119862 (1198601199042119872120573 (119892) (119909)

+119860(12)(119904minus(119889+2120573+2)119901)1003817

1003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

))

(128)

and the choice of 119860 such that

119860(119889+2120573+2)2119901

119872120573 (119892) (119909) =10038171003817100381710038171198911003817100381710038171003817B119904minus(119889+2120573+2)119901120573

infininfin (R119889+1+

)(129)

ensures that10038161003816100381610038161003816100381610038161003816

int

infin

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

10038161003816100381610038161003816100381610038161003816

le 119862(119872120573(119892)(119909))

1minus(119901119904(119889+2120573+2))10038171003817100381710038171198911003817100381710038171003817

119901119904(119889+2120573+2)

B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(130)

Finally taking the 119871119902120573norm with 119902 = 119901(119889 + 2120573 + 2)(119889 + 2120573 +

2minus119901119904) ends the proof thanks to the fact themaximal function119872120573 is bounded of 119871119902

120573(R119889+1

+) into itself for 119902 gt 1

Theorem 32 Let 1 lt 119901 lt 119902 lt infin For all function 119891 such that119891 isin

H1199041

119901120573(R119889+1

+)⋂

Bminus120573120573

infininfin(R119889+1

+) one has

10038171003817100381710038171198911003817100381710038171003817H119904119901120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

120579

H1199041

119901120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573120573

infininfin(R119889+1

+) (131)

where 120579 = 119901119902 119904 = 1205791199041 minus (1 minus 120579)120573 with 120573 gt 0 minus120573 lt 119904 lt 1199041

Proof It suffices to prove that1003817100381710038171003817100381710038171003817

(minusΔ 120573)

(119904minus1199041)2

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le 11986210038171003817100381710038171198911003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(132)

Journal of Function Spaces and Applications 11

Indeed we use the following identity (which may be easilyproven by taking the Weinstein transform in 119909 of both sides)

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

infin

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905 (133)

with 120575 = 1199041 minus 119904 gt 0We decompose the integral in two parts as follows

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

119879

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

+

1

Γ (1205752)

int

infin

119879

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

(134)

where 119879 is a constant to be fixed laterWe proceed as in [8] we obtain

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119872120573 (119891) (119909) (135)

On the other hand we use Theorem 29 and the fact that 119891belongs to Bminus120573minus119904

1120573

infininfin(R119889+1

+) to deduce that

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119905

(minus120573minus1199041)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

) (136)

Thus by applying the preceding estimates on the right part of(134) we obtain

1003816100381610038161003816100381610038161003816

(minusΔ 120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621

Γ (1205752)

1198791205752119872120573 (119891) (119909)

+

1198622

Γ (1205752)

119879(120575minus120573minus119904

1)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

(137)

We fix now

119879 = (

10038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

119872120573 (119891) (119909)

)

2(120573+1199041)

(138)

We obtain

1003816100381610038161003816100381610038161003816

(minusΔ120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621 + 1198622

Γ (1205752)

(119872120573(119891)(119909))

12057910038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(139)

Thus we deduce that1003817100381710038171003817100381710038171003817

(minusΔ120573)

minus1205752

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le

1198621 + 1198622

Γ (1205752)

10038171003817100381710038171003817119872120573 (119891)

10038171003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(140)

To conclude we used the fact that the maximal function119872120573

is bounded of 119871119902120573(R119889+1

+) into itself for 119902 gt 1

43 Estimates in Generalized Besov Spaces For any interval 119868ofR (bounded or unbounded) and a normed space 119865(R119889+1

+)

we define the mixed space-time 119871119901(119868 119865(R119889+1

+)) space of

(classes of) measurable functions 119906 119868 rarr 119865(R119889+1

+) such that

||119906||119871119901(119868119865(R119889+1+

)) lt infin with

119906119871119901(119868119865(R119889+1+

)) = (int

119868

119906 (119905 sdot)119901

119865(R119889+1+

)119889119905)

1119901

if 1 le 119901 lt infin

119906119871infin(119868119865(R119889+1+

)) = ess sup119905isin119868

119906 (119905 sdot)119865(R119889+1+

)

(141)

For any interval 119868 of R (bounded or unbounded) anda Banach space 119883 we define the mixed space-time 119862(119868119883)space of continuous functions 119868 rarr 119883 When 119868 is bounded119862(119868119883) is a Banach space with the norm of 119871infin(119868 119883)

Theorem 33 Let 119904 isin R and 1 le 119901 119902 119903 le infin Let 119879 gt 0 119892 isin

B119904120573

119901119903(R119889+1

+) and119891 in 119871119902((0 119879) B

119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76)

has a unique solution

119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903(R

119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R

119889+1

+))

(142)

and there exists a constant 119862 such that for all 1199021 isin [119902infin] onehas

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862(10038171003817100381710038171198921003817100381710038171003817B119904120573

119901119903(R119889+1+

)+10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)B

119904minus2+(2119902)120573

119901119903(R119889+1+

)))

(143)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] B119904120573

119901119903(R119889+1

+))

Proof Since 119892 and 119891 are temperate distributions (76) has aunique solution 119906 in S1015840

((0 119879) timesR119889+1

+) which satisfies

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(144)

Next we notice that applying Δ 119895 to (76) and using formula(81) yield

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(145)

Therefore

10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(146)

12 Journal of Function Spaces and Applications

By virtue of Lemma 30 we thus have for some 120581 gt 0

10038171003817100381710038171003817Δ 119895119906(119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

119890minus12058122119895(119905minus120591) 10038171003817

100381710038171003817Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(147)

Applying convolution inequalities we get

10038171003817100381710038171003817Δ 119895119906

100381710038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862[

[

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

)

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

]

]

(148)

with 11199022 = 1+ 11199021 minus1119902 Finally taking the 119897119903(Z) norm we

conclude that (with the usual convention if 119903 = infin)

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

1199031199021

)(211989511990410038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

]

]

1119903

+ 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

1199031199022

)

times (2119895(119904minus2+2119902)10038171003817

100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902(0119879119871

119901

120573(R119889+1+

))

119903

]

]

1119903

(149)

which insures that 119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903 (R119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R119889+1

+)) and yields the desired inequality

Since 119906 belongs to 119862([0 119879]B119904120573

119901119903(R119889+1

+)) in the case where

119903 is finite may be easily deduced from the density ofSlowast(R

119889+1)⋂

B119904120573

119901119903(R119889+1

+) in B

119904120573

119901119903(R)

Theorem 34 Let 119904 isin R 119879 gt 0 and 1 le 119901 119902 119903 le infinOne supposes that 119892 isin 119861

119904120573

119901119903(R119889+1

+) and 119891 isin 119871

119902((0 119879)

119861119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76) has a unique solution 119906 belong-

ing to

119871119902((0 119879) 119861

119904+(2119902)120573

119901119903(R

119889+1

+))⋂119871

infin((0 119879) 119861

119904120573

119901119903(R

119889+1

+))

(150)

and there exists a constant 119862 such that for all 119902 le 1199021 le infin

1199061198711199021 ((0119879)119861

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862 [ (1 + 11987911199021)10038171003817100381710038171198921003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (1 + 1198791+1119902

1minus1119902

)10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)119861

119904minus2+(2119902)120573

119901119903(R119889+1+

))]

(151)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] 119861119904120573119901119903(R119889+1

+))

Proof Since 119892 119891 are tempered (76) has a unique solution 119906in S1015840

((0 119879) timesR119889+1

+) satisfying

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(152)

Hence applying Δ 119895 119895 ge 0 to (81) we see that

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(153)

and thus by Lemma 30 we can deduce that10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+int

119905

0

119890minus12058122119895(119905minus120591)10038171003817

100381710038171003817Δ 119895119891(120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(154)

Then it follows from convolution inequalities thatΔ 1198951199061198711199021 ((0119879)119871

119901

120573(R119889+1+

))is dominated by

(

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

(155)

with 11199022 = 1 + 11199021 minus 1119902 Moreover similarly as above wecan obtain that

1003817100381710038171003817Δminus1119906(119905 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)le1003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

1003817100381710038171003817Δminus1119891 (120591 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(156)

and thus if 1 le 119902 le 1199021 le infin1003817100381710038171003817Δminus1119906

10038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862(119879111990211003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)+ 119879

111990221003817100381710038171003817Δminus1119891

1003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

)))

(157)

Journal of Function Spaces and Applications 13

Finally taking the 119897119903-normwith respect to 119895 in (155) and (157)with the usual convention if 119903 = infin we can deduce the desiredestimate

Acknowledgment

Theauthor gratefully acknowledges theDeanship of ScientificResearch at the University of Taibah The author is deeplyindebted to the referee for providing constructive commentsand help in improving the contents of this paper

References

[1] Z Ben Nahia and N Ben Salem ldquoSpherical harmonics andapplications associated with the Weinstein operatorrdquo in Pro-ceedings of the International Conference on PotentialTheory heldin Kouty Czech Republic (ICPT rsquo94) pp 235ndash241 1996

[2] Z Ben Nahia and N Ben Salem ldquoOn a mean value propertyassociated with the Weinstein operatorrdquo in Proceedings of theInternational Conference on Potential Theory held in KoutyCzech Republic (ICPT rsquo94) pp 243ndash253 1996

[3] M Brelot ldquoEquation de Weinstein et potentiels de MarcelRieszrdquo in Seminaire de Theorie de Potentiel Paris No 3 vol 681of Lecture Notes in Mathematics pp 18ndash38 Springer BerlinGermany 1978

[4] H Mejjaoli and M Salhi ldquoUncertainty principles for theweinstein transformrdquo Czechoslovak Mathematical Journal vol61 no 4 pp 941ndash974 2011

[5] H Mejjaoli and A Ould Ahmed Salem ldquoWeinstein Gabortransform and applicationsrdquo Advanced Studies in Pure Mathe-matics vol 2 no 3 pp 203ndash210 2012

[6] H Mejjaoli ldquoBesov spaces associated withthe Weinstein opera-tor and applicationsrdquo In press

[7] T Kawazoe and H Mejjaoli ldquoGeneralized Besov spaces andtheir applicationsrdquo Tokyo Journal of Mathematics vol 35 no 2pp 297ndash320 2012

[8] H Mejjaoli ldquoLittlewood-Paley decomposition associated withthe Dunkl operators and paraproduct operatorsrdquo Journal ofInequalities in Pure and Applied Mathematics vol 9 no 4 pp1ndash25 2008

[9] H Mejjaoli and N Sraeib ldquoGeneralized sobolev spaces inquantum calculus and applicationsrdquo Journal of Inequalities andSpecial Functions vol 1 no 4 pp 43ndash64 2012

[10] H Mejjaoli ldquoGeneralized homogeneous Besov spaces and theirapplicationsrdquo Serdica Mathematical Journal vol 38 no 4 pp575ndash614 2012

[11] H Triebel Interpolation Theory Functions Spaces DifferentialOperators North-Holland AmsterdamThe Netherlands 1978

[12] V S Guliev ldquoOn maximal function and fractional integralassociated with the Bessel differential operatorrdquo MathematicalInequalities and Applications vol 6 no 2 pp 317ndash330 2003

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Stochastic AnalysisInternational Journal of

2 Journal of Function Spaces and Applications

we introduce and study the homogeneous Weinstein-Besovspaces the homogeneous Weinstein-Triebel-Lizorkin spacesand the homogeneous Weinstein-Riesz potential spaces InSection 4 we characterize the homogeneousWeinstein Besovspaces via the Weinstein heat semigroup Next we prove theSobolev embedding theorems

2 Preliminaries

In order to confirm the basic and standard notations webriefly overview the Weinstein operator and the relatedharmonic analysis Main references are [1 2]

21 Harmonic Analysis Associated with the Weinstein Opera-tor In this subsection we collect some notations and resultson theWeinstein kernel theWeinstein intertwining operatorand its dual the Weinstein transform and the Weinsteinconvolution

In the following

R119889+1

+= R

119889times [0infin[

119909 = (1199091 119909119889 119909119889+1) = (1199091015840 119909119889+1) isin R

119889+1

+

119909 = radic1199092

1+ sdot sdot sdot + 119909

2

119889+1

(5)

We denote by 119862lowast(R119889+1

) the space of continuous functionson R119889+1 even with respect to the last variable 119862119901

lowast(R119889+1

) thespace of functions of class 119862119901 on R119889+1 even with respectto the last variable Elowast(R

119889+1) the space of 119862infin-functions on

R119889+1 even with respect to the last variable Slowast(R119889+1

) theSchwartz space of rapidly decreasing functions onR119889+1 evenwith respect to the last variable 119863lowast(R

119889+1) the space of 119862infin-

functions on R119889+1 which are of compact support even withrespect to the last variable S1015840

lowast(R119889+1

) the space of temperatedistributions on R119889+1 even with respect to the last variableIt is the topological dual of Slowast(R

119889+1)

We consider the Weinstein operator Δ 120573 defined by

forall119909 = (1199091015840 119909119889+1) isin R

119889times ]0infin[

Δ 120573119891 (119909) = Δ 1199091015840119891 (1199091015840 119909119889+1) +L120573119909

119889+1

119891 (1199091015840 119909119889+1)

119891 isin 1198622

lowast(R

119889+1)

(6)

where Δ 1199091015840 is the Laplace operator on R119889 and L120573119909119889+1

is theBessel operator on ]0infin[ given by

L120573119909119889+1

=

1198892

1198891199092

119889+1

+

2120573 + 1

119909119889+1

119889

119889119909119889+1

120573 gt minus

1

2

(7)

TheWeinstein kernel Λ is given by

Λ (119909 119911) = 119890119894⟨11990910158401199111015840⟩119895120573 (119909119889+1119911119889+1)

forall (119909 119911) isin R119889+1

times C119889+1

(8)

where 119895120573(119909119889+1119911119889+1) is the normalized Bessel function TheWeinstein kernel satisfies the following properties

(i) For each 119911 isin R119889+1

+ we have

Δ 120573Λ (119909 119911) = minus1199112Λ (119909 119911) forall119909 isin R

119889+1

+ (9)

(ii) For all 119911 119905 isin C119889+1 we have

Λ (119911 119905) = Λ (119905 119911) Λ (119911 0) = 1

Λ (120582119911 119905) = Λ (119911 120582119905) forall120582 isin C(10)

(iii) For all ] isin N119889+1 119909 isin R119889+1 and 119911 isin C119889+1 we have1003816100381610038161003816119863

]119911Λ (119909 119911)

1003816100381610038161003816le 119909

|]| exp (119909 Im 119911) (11)

where 119863]119911= 120597

](120597119911

]1

1sdot sdot sdot 120597119911

]119889+1

119889+1) and |]| = ]1 + sdot sdot sdot + ]119889+1 In

particular1003816100381610038161003816Λ (119909 119910)

1003816100381610038161003816le 1 forall119909 119910 isin R

119889+1 (12)

The Weinstein intertwining operator is the operator R120573

defined on 119862lowast(R119889+1

) by

R120573119891 (1199091015840 119909119889+1)

=

2Γ (120573 + 1)

radic120587Γ (120573 + 12)

119909minus2120573

119889+1

timesint

119909119889+1

0

(1199092

119889+1minus 119905

2)

120573minus12

119891 (1199091015840 119905) 119889119905 119909119889+1 gt 0

119891 (1199091015840 0) 119909119889+1 = 0

(13)

R120573 is a topological isomorphism from Elowast(R119889+1

) ontoitself satisfying the following transmutation relation

Δ 120573 (R120573119891) = R120573 (Δ 119889+1119891) forall119891 isin Elowast (R119889+1

) (14)

where Δ 119889+1 = sum119889+1

119895=11205972

119895is the Laplacian on R119889+1

We denote by119871119901120573(R119889+1

+) the space ofmeasurable functions

on R119889+1

+such that

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)= (int

R119889+1+

1003816100381610038161003816119891 (119909)

1003816100381610038161003816

119901119889120583120573 (119909))

1119901

lt infin

if 1 le 119901 lt infin

10038171003817100381710038171198911003817100381710038171003817119871infin120573(R119889+1+

)= ess sup

119909isinR119889+1+

1003816100381610038161003816119891 (119909)

1003816100381610038161003816lt infin

(15)

where 119889120583120573 is the measure on R119889+1

+given by

119889120583120573 (1199091015840 119909119889+1) = 119909

2120573+1

119889+1119889119909

1015840119889119909119889+1

(16)

TheWeinstein transform is given for 119891 in 1198711120573(R119889+1

+) by

F119882 (119891) (119910) = int

R119889+1+

119891 (119909)Λ (minus119909 119910) 119889120583120573 (119909) forall119910 isin R119889+1

+

(17)

Some basic properties of this transform are as follows

Journal of Function Spaces and Applications 3

(i) For 119891 in 1198711120573(R119889+1

+)

1003817100381710038171003817F119882 (119891)

1003817100381710038171003817119871infin120573(R119889+1+

)le100381710038171003817100381711989110038171003817100381710038171198711120573(R119889+1+

) (18)

(ii) For 119891 in Slowast(R119889+1

) we have

F119882 (Δ 120573119891) (119910) = minus10038171003817100381710038171199101003817100381710038171003817

2F119882 (119891) (119910) forall119910 isin R

119889+1

+ (19)

(iii) For all 119891 in 1198711120573(R119889+1

+) ifF119882(119891) belongs to 119871

1

120573(R119889+1

+)

then

119891 (119910) = 119862 (120573)int

R119889+1+

F119882 (119891) (119909) Λ (119909 119910) 119889120583120573 (119909) ae

(20)

where

119862 (120573) =

1

1205871198894120573+1198892

(Γ (120573 + 1))2 (21)

(iv) For 119891 isin Slowast(R119889+1

) if we define

F119882 (119891) (119910) = F119882 (119891) (minus119910) (22)

then

(F119882)minus1=

1

119862 (120573)

F119882 (23)

Proposition 1 (see [2]) (i) the Weinstein transform F119882 is atopological isomorphism fromSlowast(R

119889+1) onto itself and for all

f in Slowast(R119889+1

)

int

R119889+1+

1003816100381610038161003816119891 (119909)

1003816100381610038161003816

2119889120583120573 (119909) = 119862 (120573)int

R119889+1+

1003816100381610038161003816F119882 (119891) (120585)

1003816100381610038161003816

2119889120583120573 (120585)

(24)

(ii) In particular the renormalized Weinstein transform119891 rarr 119862(120573)

12F119882(119891) can be uniquely extended to an isometric

isomorphism from 1198712

120573(R119889+1

+) onto itself

In the Fourier analysis the translation operator is givenby 119891 997891rarr 119891(sdot + 119909)

In harmonic analysis associated for the operator Δ 120573 thegeneralized translation operator 120591119909 119909 isin R119889+1

+is defined by

forall119910 isin R119889+1

+

120591119909119891 (119910) =

Γ (120573 + 1)

radic120587Γ (120573 + 12)

times int

120587

0

119891(1199091015840+ 119910

1015840 radic119909

2

119889+1+ 119910

2

119889+1+ 2119909119889+1119910119889+1 cos 120579)

times (sin 120579)2120573119889120579(25)

where 119891 isin 119862lowast(R119889+1

)

By using the Weinstein kernel we can also define ageneralized translation For functions 119891 isin Slowast(R

119889+1) and

119910 isin R119889+1

+the generalized translation 120591119910119891 is defined by the

following relation

F119882 (120591119910119891) (119909) = Λ (119909 119910)F119882 (119891) (119909) (26)

By using the generalized translation we define the general-ized convolution product119891lowast

119882119892 of functions119891 119892 isin 1198711

120573(R119889+1

+)

as follows

119891lowast119882119892 (119909) = int

R119889+1+

120591119909119891 (minus1199101015840 119910119889+1) 119892 (119910) 119889120583120573 (119910) (27)

This convolution is commutative and associative and itsatisfies the following

(i) For all 119891 119892 isin 1198711

120573(R119889+1

+) 119891lowast119882 119892 belongs to 1198711

120573(R119889+1

+)

and

F119882 (119891lowast119882119892) = F119882 (119891)F119882 (119892) (28)

(ii) Let 1 le 119901 119902 119903 le infin such that 1119901 + 1119902 minus 1119903 = 1If 119891 isin 119871

119901

120573(R119889+1

+) and 119892 isin 119871

119902

120573(R119889+1

+) then 119891lowast119882119892 isin

119871119903

120573(R119889+1

+) and

1003817100381710038171003817119891lowast119882119892

1003817100381710038171003817119871119903120573(R119889+1+

)le10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171198921003817100381710038171003817119871119902

120573(R119889+1+

) (29)

We define the tempered distribution T119891 associated with119891 isin 119871

119901

120573(R119889+1

+) by

⟨T119891 120601⟩ = int

R119889+1+

119891 (119909) 120601 (119909) 119889120583120573 (119909) (30)

for 120601 isin Slowast(R119889+1

) and denote by ⟨119891 120601⟩120573 the integral in therighthand side

Definition 2 The Weinstein transform F119882(120591) of a distribu-tion 120591 isin S1015840

lowast(R119889+1

) is defined by

⟨F119882 (120591) 120601⟩ = ⟨120591F119882 (120601)⟩ (31)

for 120601 isin Slowast(R119889+1

)

In particular for 119891 isin 119871119901

120573(R119889+1

+) it follows that for 120601 isin

Slowast(R119889+1

)

⟨F119882 (119891) 120601⟩ = ⟨F119882 (T119891) 120601⟩ = ⟨T119891F119882 (120601)⟩

= ⟨119891F119882 (120601)⟩120573

(32)

Proposition 3 The Weinstein transform F119882 is a topologicalisomorphism from S1015840

lowast(R119889+1

) onto itself

Definition 4 The generalized convolution product of a dis-tribution 119878 in S1015840

lowast(R119889+1

) and a function 120601 in Slowast(R119889+1

) is thefunction 119878lowast119882120601 defined by

119878lowast119882 120601 (119909) = ⟨119878119910 120591minus119910120601 (119909)⟩ (33)

4 Journal of Function Spaces and Applications

Proposition 5 Let 119891 be in 119871119901

120573(R119889+1

+) 1 le 119901 le infin and 120601

in Slowast(R119889+1

) Then the distribution T119891lowast119882120601 is given by thefunction 119891lowast119882120601 and T119891lowast119882120601 belongs to 119871119901

120573(R119889+1

+) Moreover

for all 120595 isin Slowast(R119889+1

)

⟨T119891lowast119882120601 120595⟩ = ⟨

119891 120601lowast119882 ⟩120573 (34)

where (119909) = 120595(minus119909) and

F119882 (T119891lowast119882 120601) = F119882 (T119891)F119882 (120601) (35)

For each 119906 isin S1015840

lowast(R119889+1

) we define the distribution Δ 120573119906

by ⟨Δ 120573119906 120595⟩ = ⟨119906 Δ 120573120595⟩ and this distribution satisfies thefollowing property

F119882 (Δ 120573119906) = minus10038171003817100381710038171199101003817100381710038171003817

2F119882 (119906) (36)

In the following we denote T119891 given by (30) by 119891 forsimplicity

3 B119904120573

119901119902 F119904120573

119901119902(R119889+1

+) H119904

119901120573Spaces and

Basic Properties

31 HomogeneousWeinstein-Littlewood-Paley DecompositionOne of the main tools in this paper is the homogeneousLittlewood-Paley decomposition of distribution associatedwith theWeinstein operators into dyadic blocs of frequencies

Lemma6 One defines byC the ring of center 0 of small radius12 and great radius 2 There exists two radial functions 120595 and120593 the values of which are in the interval [0 1] belonging to119863lowast(R

119889+1) such that

supp120595 sub 119861 (0 1) supp120593 sub C

forall120585 isin R119889+1

120595 (120585) + sum

119895ge0

120593 (2minus119895120585) = 1

forall120585 isin C sum

119895isinZ

120593 (2minus119895120585) = 1

|119899 minus 119898| ge 2 997904rArr supp120593 (2minus119899sdot) cap supp120593 (2minus119898sdot) = 0

119895 ge 1 997904rArr supp120595 cap supp120593 (2minus119895sdot) = 0

(37)

Notations We denote by

Δ 119895119891 = Fminus1

119882(120593(

120585

2119895)F119882 (119891))

119878119895119891 = sum

119899le119895minus1

Δ 119899119891 forall119895 isin Z(38)

The distribution Δ 119895119891 is called the 119895th dyadic block of thehomogeneous Littlewood-Paley decomposition of 119891 associ-ated with the Weinstein operators

Throughout this paper we define 120601 and 120594 by 120601 = Fminus1

119882(120593)

and 120594 = Fminus1

119882(120595)

When dealing with the Littlewood-Paley decompositionit is convenient to introduce the functions and 120593 belongingto119863lowast(R

119889+1) such that equiv 1 on supp120595 and 120593 equiv 1 on supp120593

Remark 7 We remark that

F119882 (119878119895119891) (120585) = (

120585

2119895)F119882 (119878119895119891) (120585)

F119882 (Δ 119895119891) (120585) = 120593(

120585

2119895)F119882 (Δ 119895119891) (120585)

(39)

We put

120601 = F

minus1

119882(120593) 120594 = F

minus1

119882() (40)

Definition 8 One denotes by S1015840

ℎ120573lowast(R119889+1

) the space oftempered distribution such that

lim119895rarrminusinfin

119878119895119906 = 0 in S1015840

lowast(R

119889+1) (41)

Proposition 9 (Bernstein inequalities) For all 120583 isin N119889+1 and120590 isin R for all 119895 isin Z for all 1 le 119901 119902 le infin and for all119891 isin S1015840

lowast(R119889+1

) one has the following

(i) 10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902

120573(R119889+1+

)

le

10038171003817100381710038171003817

120601

10038171003817100381710038171003817119871119903120573(R119889+1+

)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

)2119895(119889+2120573+2)(1119901minus1119902)

with 1

119902

=

1

119901

+

1

119903

minus 1

(ii) 10038171003817100381710038171003817119878119895119891

10038171003817100381710038171003817119871119902

120573(R119889+1+

)

le10038171003817100381710038171205941003817100381710038171003817119871119903120573(R119889+1+

)

10038171003817100381710038171003817119878119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

)2119895(119889+2120573+2)(1119901minus1119902)

with 1

119902

=

1

119901

+

1

119903

minus 1

(iii)1003817100381710038171003817100381710038171003817

(radicminusΔ 120573)

120590

Δ 119895119891

1003817100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)

le

10038171003817100381710038171003817F

minus1

119882(10038171003817100381710038171205851003817100381710038171003817

120590120593)

100381710038171003817100381710038171198711120573(R119889+1+

)

times

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

)2119895120590

(42)

Proof Using Remark 7 we deduce from Proposition 5 that

119878119895119891 = 2119895(119889+2120573+2)

120594 (2119895sdot) lowast119882119878119895119891

Δ 119895119891 = 2119895(119889+2120573+2)

120601 (2119895sdot) lowast119882Δ 119895119891

(43)

Thus from the relation (29) we prove (i) (ii) and (iii)

32 Definitions In the following we define analogues of thehomogeneous Besov Triebel-Lizorkin and Riesz potentialspaces associated with the Weinstein operators on R119889+1

+and

obtain their basic propertiesFrom now we make the convention that for all non-

negative sequence 119886119902119902isinZ the notation (sum119902119886119903

119902)1119903 stands for

sup119902119886119902 in the case 119903 = infin

Journal of Function Spaces and Applications 5

Definition 10 Let 119904 isin R and 119901 119902 isin [1infin]The homogeneousWeinstein-Besov spaces B

119904120573

119901119902(R119889+1

+) are the spaces of distri-

bution in S1015840

ℎ120573lowast(R119889+1

) such that

10038171003817100381710038171198911003817100381710038171003817B119904120573

119901119902(R119889+1+

)= (sum

119895isinZ

(211990411989510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119902

)

1119902

lt infin (44)

Proposition 11 (see [6]) Let 119904 isin R and 119901 and 119902 two elementsof [1infin] the space B

119904120573

119901119902(R119889+1

+) is the set of 119891 isin S1015840

ℎ120573lowast(R119889+1

)

verifying

10038171003817100381710038171003817

119891

10038171003817100381710038171003817B119904120573

119901119902(R119889+1+

)= (int

infin

0

(119905minus1199041003817100381710038171003817119891lowast119882120601119905

1003817100381710038171003817119871119901

120573(R119889+1+

))

119902119889119905

119905

)

1119902

lt infin

(45)

where 120601119905(119909) = (11199052120573+2+119889

)120601(119909119905) for all 119905 isin (0infin) and 119909 isin

R119889+1

+

Definition 12 For 119904 isin R and 119901 119902 isin [1infin] one writes10038171003817100381710038171198911003817100381710038171003817119861119904120573

119901119902(R119889+1+

)=10038171003817100381710038171198780119891

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ (sum

119895ge1

(211990411989510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119902

)

1119902

(46)

The nonhomogeneous Besov space 119861119904120573119901119902(R119889+1

+) associated

with the Weinstein operators is defined by

119861119904120573

119901119902(R

119889+1

+) = 119891 isin S

1015840(R

119889)

10038171003817100381710038171198911003817100381710038171003817119861119904120573

119901119902(R119889+1+

)lt infin (47)

We give now another definition equivalent to the nonho-mogeneous Besov space 119861119904120573119901119902(R119889+1

+)

Proposition 13 Let 119904 isin R and119901 and 119902 two elements of [1infin]the space 119861119904120573119901119902(R119889+1

+) is the set of 119891 isin S1015840

(R119889) verifying

10038171003817100381710038171003817

119891

10038171003817100381710038171003817119861119904120573

119901119902(R119889+1+

)=1003817100381710038171003817119891lowast119882120595

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ (int

1

0

(119905minus1199041003817100381710038171003817119891lowast119882120601119905

1003817100381710038171003817119871119901

120573(R119889+1+

))

119902119889119905

119905

)

1119902

lt infin

(48)

Definition 14 Let 119904 isin R and 1 le 119901 119902 le infin the homogeneousWeinstein-Triebel-Lizorkin space F

119904120573

119901119902(R119889+1

+) is the space of

distribution in S1015840

ℎ120573lowast(R119889+1

) such that

10038171003817100381710038171198911003817100381710038171003817F119904120573

119901119902(R119889+1+

)=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

211990411989511990210038161003816100381610038161003816Δ 119895119891

10038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)

lt infin (49)

Definition 15 For 119904 isin R the operatorR119904

120573from S1015840

ℎ120573lowast(R119889+1

)

to S1015840

ℎ120573lowast(R119889+1

) is defined by

R119904

120573(119891) = F

minus1

119882(sdot

119904F119882119891) (50)

The operatorRminus119904

120573is called Weinstein-Riesz potential space

Definition 16 For 119904 isin R and 1 le 119901 le infin the homogeneousWeinstein-Riesz potential space H119904

119901120573(R119889+1

+) is defined

as the space Rminus119904

120573(119871

119901

120573(R119889+1

+)) equipped with the norm

119891H119904119901120573(R119889+1+

)= R119904

120573(119891)

119871119901

120573(R119889+1+

)

Proposition 17 Let 119904 isin R and 1 le 119901 119902 le infinThe operator Δ 120573 is a linear continuous operator from

B119904120573

119901119902(R119889+1

+) into B

119904minus2120573

119901119902 (R119889+1

+) and from H119904

119901120573(R119889+1

+) into

H119904minus2

119901120573(R119889+1

+)

Proof We obtain these results by the similar ideas used in thenonhomogeneous case (cf [6])

Proposition 18 Let 119904 119905 isin R and 1 le 119901 119902 le infin The operatorR119905

120573is a linear continuous injective operator from B

119904120573

119901119902(R119889+1

+)

onto B119904minus119905120573

119901119902 (R119889+1

+) and from H119904

119901120573(R119889+1

+) onto H119904minus119905

119901120573(R119889+1

+)

Proof We obtain these results by the similar ideas used in thenonhomogeneous case (cf [6])

33 Embeddings As in the Euclidean case (cf [11]) themono-tone character of 119897119902-spaces and the Minkowskis inequalityyield the following

Proposition 19 If 1 le 1199021 lt 1199022 le infin one has

B119904120573

1199011199021

(R119889+1

+) 997893rarr

B119904120573

1199011199022

(R119889+1

+) (1 le 119901 le infin 119904 isin R)

(51)

MoreoverB119904120573

1199011(R

119889+1

+) 997893rarr

H119904

119901120573(R

119889+1

+) 997893rarr

B119904120573

119901infin(R

119889+1

+)

(1 le 119901 le infin 119904 isin R)

(52)

If 1199040 = 1199041 one also has

(H1199040

119901120573(R

119889+1

+)

H1199041

119901120573(R

119889+1

+))

120579119902=

B119904120573

119901119902(R

119889+1

+)

(1 le 119901 119902 le infin 120579 isin (0 1))

(53)

where 119904 = (1 minus 120579)1199040 + 1205791199041

Proposition 20 One assumes that 119904 minus (119889 + 2120573 + 2)119901 = 1199041 minus

(119889 + 2120573 + 2)1199011 Then the following inclusion holds

B119904120573

119901119902(R

119889+1

+) 997893rarr

B1199041120573

11990111199021

(R119889+1

+)

(1 le 119901 le 1199011 le infin 1 le 119902 le 1199021 le infin 119904 1199041 isin R)

(54)

Proof In order to prove the inclusion we use the estimate

Δ 119895119891 = 2119895(119889+2120573+2)

120601 (2119895sdot) lowast119882Δ 119895119891 (55)

Proposition 9(i) gives that10038171003817100381710038171003817Δ 119895119891

100381710038171003817100381710038171198711199011

120573(R119889+1+

)=

100381710038171003817100381710038172119895(119889+2120573+2)

120601 (2119895sdot) lowast119882Δ 119895119891

100381710038171003817100381710038171198711199011

120573(R119889+1+

)

le 1198622119895(119889+2120573+2)(1119901minus1119901

1)10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

(56)

6 Journal of Function Spaces and Applications

By definition of the homogeneous Weinstein-Besov spaceswe therefore infer10038171003817100381710038171198911003817100381710038171003817B1199041120573

11990111199021(R119889+1+

)

= (

infin

sum

119895=minusinfin

(2119895119904110038171003817100381710038171003817Δ 119895119891

100381710038171003817100381710038171198711199011

120573(R119889+1+

))

1199021

)

11199021

le 119862(sum

119895isinZ

(211989511990412119895(119889+2120573+2)(1119901minus1119901

1)10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

1199021

)

11199021

le 119862(sum

119895isinZ

(211989511990410038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

1199021

)

11199021

le 11986210038171003817100381710038171198911003817100381710038171003817B119904120573

119901119902(R119889+1+

)

(57)

since 119902 le 1199021 This gives the inclusion

Proposition 21 (1) If 119906 belongs to B119904120573

119901119902(R119889+1

+) cap

B119905120573

119901119902(R119889+1

+)

then 119906 belongs to B120579119904+(1minus120579)119905120573

119901119902 (R119889+1

+) for all 120579 isin [0 1] and

119906B120579119904+(1minus120579)119905120573

119901119902(R119889+1+

)le 119906

120579

B119904120573

119901119902(R119889+1+

)119906

1minus120579

B119905120573

119901119902(R119889+1+

) (58)

(2) If 119906 belongs to B119904120573

119901infin(R119889+1

+) cap

B119905120573

119901infin(R119889+1

+) and 119904 lt 119905

then 119906 belongs to B120579119904+(1minus120579)119905120573

1199011(R119889+1

+) for all 120579 isin (0 1) and there

exists a positive constant 119862(119905 119904) such that

119906B120579119904+(1minus120579)119905120573

1199011(R119889+1+

)le 119862 (119905 119904) 119906

120579

B119904120573

119901infin(R119889+1+

)119906

1minus120579

B119905120573

119901infin(R119889+1+

) (59)

(3) If 119906 belongs to B119904120573

119901infin(R119889+1

+) cap

B119904+120576120573

119901infin (R119889+1

+) and 120576 gt

0 then 119906 belongs to B119904120573

1199011(R119889+1

+) and there exists a positive

constant 119862 such that

119906B119904120573

1199011(R119889+1+

)le

119862

120576

119906B119904120573

119901infin(R119889+1+

)log

2(119890 +

119906B119904+120576120573

119901infin(R119889+1+

)

119906B119904120573

119901infin(R119889+1+

)

)

(60)

Proof (1) is obvious from the Holderrsquos inequality As for (2)we write 119906

B120579119904+(1minus120579)119905120573

1199011(R119889+1+

)as

sum

119895le119873

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ sum

119895gt119873

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

(61)

where 119873 is chosen here after By the definition of thehomogeneous Weinstein-Besov norms we see that

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 2

119895(1minus120579)(119905minus119904)119906

B119904120573

119901infin(R119889+1+

)

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 2

minus119895120579(119905minus119904)119906

B119905120573

119901infin(R119889+1+

)

(62)

and thus 119906B120579119904+(1minus120579)119905120573

1199011(R119889+1+

)is dominated by

119906B119904120573

119901infin(R119889+1+

)sum

119895le119873

2119895(1minus120579)(119905minus119904)

+ 119906B119905120573

119901infin(R119889+1+

)sum

119895gt119873

2minus119895120579(119905minus119904)

le 119862119906B119904120573

119901infin(R119889+1+

)

2(119873+1)(1minus120579)(119905minus119904)

2(1minus120579)(119905minus119904)

minus 1

+ 119906B119905120573

119901infin(R119889+1+

)

2minus119873120579(119905minus119904)

1 minus 2minus120579(119905minus119904)

(63)

Hence in order to complete the proof of (2) it suffices tochoose119873 such that

119906B119905120573

119901infin(R119889)

119906B119904120573

119901infin(R119889)

le 2119873(119905minus119904)

lt 2

119906B119905120573

119901infin(R119889)

119906B119904120573

119901infin(R119889)

(64)

As for (3) it is easy to see that 119906B119904120573

1199011(R119889+1+

)is dominated as

sum

119895le119873minus1

211989511990410038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ sum

119895ge119873

211989511990410038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le (119873 + 1) 119906B119904120573

119901infin(R119889+1+

)+

2minus(119873minus1)120576

2120576minus 1

119906B119904+120576120573

119901infin(R119889+1+

)

(65)

Hence letting

119873 = 1 + [

[

1

120576

log2

119906B119904+120576120573

119901infin(R119889+1+

)

119906B119904120573

119901infin(R119889+1+

)

]

]

(66)

we can obtain the desired estimate

Proposition 22 Let 119902 isin (1infin) and let 119904 isin R such that0 lt 119904 lt (119889 + 2120573 + 2)119902 then one has

B119904120573

119902119902(R

119889+1

+) =

F119904120573

119902119902(R

119889+1

+) 997893rarr

F119904120573

119902infin(R

119889+1

+)

997893rarrF119904minus((119889+2120573+2)119902)120573

infininfin(R

119889+1

+)

(67)

H119904

119902120573(R

119889+1

+) =

F119904120573

1199022(R

119889+1

+) 997893rarr

F119904120573

119902infin(R

119889+1

+)

997893rarrF119904minus((119889+2120573+2)119902)120573

infininfin(R

119889+1

+)

(68)

Proof We obtain these results by the similar ideas used in thenonhomogeneous case (cf [6])

Theorem 23 Let 119886 119887 gt 0 and let 1199021 1199022 isin [1infin] Let120579 = 119886(119886 + 119887) isin (0 1) and let 1119901 = (1 minus 120579)1199021 +

1205791199022 Then there exists a constant 119862 such that for every119891 isin

F119886120573

1199021infin(R119889+1

+) cap

Fminus119887120573

1199022infin(R119889+1

+) then one has

1003816100381610038161003816119891 (119909)

1003816100381610038161003816le 119862(sup

119895isinZ

2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

1minus120579

(sup119895isinZ

2minus119887119895 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

120579

(69)

In particular one gets

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

F119886120573

1199021infin(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

120579

Fminus119887120573

1199022infin(R119889+1+

) (70)

Journal of Function Spaces and Applications 7

Proof Let 119891 be a Schwartz class we have

1003816100381610038161003816119891 (119909)

1003816100381610038161003816le sum

119895isinZ

10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816

le sum

119895isinZ

min(2minus119886119895sup119895isinZ

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816))

(71)

We define119873(119909) as the largest index such that

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816) le 2

minus119886119895sup119895isinZ

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816) (72)

and we write1003816100381610038161003816119891 (119909)

1003816100381610038161003816le sum

119895le119873(119909)

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

+ sum

119895gt119873(119909)

2minus119886119895sup

119895

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

le 119862(sup119895isinZ

2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

119887(119886+119887)

times (sup119895isinZ

2minus119887119895 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

119886(119886+119887)

(73)

Thus (69) is proved In order to obtain (70) it is enough toapply the Holder inequality in the expression above since wehave 120579 = 119886(119886+119887) isin (0 1) and let 1119901 = (1minus120579)1199021+1205791199022

Corollary 24 Let 119902 isin (1infin) and let 119904 isin R such that 0 lt 119904 lt

(119889 + 2120573 + 2)119902 then one has

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus(119902119901)

Bminus((2120573+2+119889)119902minus119904)120573

infininfin (R119889+1

+ )

10038171003817100381710038171198911003817100381710038171003817

119902119901

B119904120573

119902119902(R119889+1

+ )

(74)

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus(119902119901)

Bminus((2120573+2+119889)119902minus119904)120573

infininfin (R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

119902119901

H119904119902120573(R119889+1+

) (75)

where 119901 = 119902(2120573 + 2 + 119889)(2120573 + 2 + 119889 minus 119902119904)

Proof By choosing 119886 = 119904 gt 0 minus119887 = 119904 minus (119889 + 2120573 + 2)119902 lt 01199021 = 119902 and 1199022 = infin we deduce (74) from the relations (70)and (67) In the same way we deduce (75) from the relations(70) and (68)

4 Generalized Heat Equation

41 Characterization for the Weinstein-Besov Spaces TheWeinstein heat equation reads

120597119905119906 (119905 119909) minus Δ 120573119906 (119905 119909) = 119891 (119905 119909) (119905 119909) isin [0infin) timesR119889+1

+

119906|119905=0 = 119892

(76)

We introduce the Weinstein heat semigroup 119867120573(119905) for theWeinstein-Laplace operator

119867120573 (119905) 119892 (119909) =

int

R119889+1+

Γ120573 (119905 119909 119910) 119892 (119910) 119889120583120573 (119910) if 119905 gt 0

119892 (119909) if 119905 = 0(77)

where Γ120573 is the Weinstein heat kernel defined by

Γ120573 (119905 119909 119910) = 120591119909 (119864(120573)

119905) (119910) (78)

where

119864(120573)

119905(119910) =

2

1205871198892Γ (120573 + 1) (4119905)

120573+1+1198892119890minus11991024119905 (79)

Thus

119867120573 (119905) 119892 (119909) = 119892lowast119882119864(120573)

119905 (119909) (80)

In practice we use the integral formulation of (76)

119906 (119905 119909) = 119867120573 (119905) 119892 (119909) + 119866120573 (119891) (119905 119909)

= 119867120573 (119905) 119892 (119909) + int

119905

0

119867120573 (119905 minus 119904) 119891 (119904 119909) 119889119904

(81)

Remark 25 The function 119864(120573)119905

is the Gauss kernel associatedwith Weinstein operators This function satisfies

forall120585 isin R119889+1

+ F119882 (119864

(120573)

119905) (120585) = 119890

minus1199051205852

(82)

Proposition 26 Let 1 le 119901 le 119903 le infin and let 119891 isin 119871119901

120573(R119889+1

+)

Then the operator 119867120573(119905) maps 119871119901

120573(R119889+1

+) continuously to

119871119903

120573(R119889+1

+) and

10038171003817100381710038171003817119867120573 (119905) 119891

10038171003817100381710038171003817119871119903120573(R119889)

le 119862119905minus((119889+2120573+2)2)(1119901minus1119903)1003817

1003817100381710038171198911003817100381710038171003817119871119901

120573(R119889)

(83)

Moreover1003817100381710038171003817100381710038171003817

(minusΔ 120573)

1205752

119867120573 (119905) 119891

1003817100381710038171003817100381710038171003817119871119903120573(R119889)

le 119862119905minus1205752minus((119889+2120573+2)2)(1119901minus1119903)1003817

1003817100381710038171198911003817100381710038171003817119871119901

120573(R119889)

(84)

for all 120575 gt 0

Proof It follows from the relations (80) and (29) combinedwith scaling property of the kernel 119864(120573)

119905

In this section we prove estimates for the Weinstein heatsemigroupThese estimates are based on the following result

Lemma 27 Let C be an annulus Positive constants 119888 and 119862exist such that for any 119901 in [1infin] and any couple (119905 120582) ofpositive real numbers one has

suppF119882 (119906) sub 120582C 997904rArr

10038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le 119862119890minus1198881199051205822

119906119871119901

120573(R119889+1+

)

(85)

8 Journal of Function Spaces and Applications

Proof We again consider a function Θ in 119863(R119889+1

+ 0) the

value of which is identically 1 in neighborhood of annulusCWe can also assume without loss of generality that 120582 = 1 Wethen have

119867120573 (119905) 119906 = 119892 (119905 sdot) lowast119882119906 (86)where

119892 (119905 sdot) = Fminus1

119882(Θ (120585) 119890

minus1199051205852

) (87)

The lemma is proved provided that we can find positive realnumbers 119888 and 119862 such that

forall119905 gt 01003817100381710038171003817119892 (119905 sdot)

10038171003817100381710038171198711120573(R119889+1+

)le 119862119890

minus119888119905 (88)

To begin we perform integrations by parts in (87) We get1003816100381610038161003816119892 (119905 119909)

1003816100381610038161003816=

1

119888120573

(1 + 1199092)

minus(119889+[2120573]+3)

times int

R119889+1+

Λ (119909 120585) (119868119889 minus Δ 120573)

119889+[2120573]+3

times (Θ (120585) 119890minus1199051205852

) 119889120583120573 (120585)

(89)

Using Leibnizrsquos formula we obtain1003816100381610038161003816119892 (119905 119909)

1003816100381610038161003816le 119862(1 + 119909

2)

minus(119889+[2120573]+3)

119890minus119888119905 (90)

and (88) follows

For any interval 119868 of R (bounded or unbounded) wedefine themixed space-time 119871119901(119868 119871119902

120573(R119889+1

+)) Banach space of

(classes of) measurable functions 119906 119868 rarr 119871119902

120573(R119889+1

+) such

that 119906119871119901(119868119871

119902

120573(R119889+1+

))lt infin with

119906119871119901(119868119871119902

120573(R119889+1+

))= (int

119868

119906 (119905 sdot)119901

119871119902

120573(R119889+1+

)119889119905)

1119901

if 1 le 119901 119902 lt infin

119906119871infin(119868119871119902

120573(R119889+1+

))= ess sup

119905isin119868

119906 (119905 sdot)119871119902

120573(R119889+1+

) if 1 le 119902 lt infin

(91)Corollary 28 Let C be an annulus and 120582 a positive realnumber Let 1199060 (resp 119891 = 119891(119905 119909)) satisfy suppF119882(1199060) sub 120582C(resp suppF119882(119891(119905 sdot)) sub 120582C for all 119905 in [0 119879]) Consider 119906 asolution of

120597119905119906 minus Δ 120573119906 = 0 119906|119905=0 = 1199060 (92)and V a solution of

120597119905V minus Δ 120573V = 119891 (119905 sdot) V|119905=0 = 0 (93)There exist positive constants 119888 and 119862 depending only on Csuch that for any 1 le 119886 le 119887 le infin and 1 le 119901 le 119902 le infin we have

119906119871119902([0119879]119871119887120573(R119889+1+

))le 119862120582

(119889+2120573+2)(1119886minus1119887)120582minus21199021003817

1003817100381710038171199060

1003817100381710038171003817119871119886120573(R119889+1+

)

V119871119902([0119879]119871119887120573(R119889+1+

))le 119862120582

minus2(1+1119902minus1119901)120582(119889+2120573+2)(1119886minus1119887)

times10038171003817100381710038171198911003817100381710038171003817119871119901([0119879]119871119886

120573(R119889+1+

))

(94)

Proof It suffices to use the fact that

119906 (119905 sdot) = 119867120573 (119905) 1199060 V (119905 sdot) = int119905

0

119867120573 (119905 minus 119904) 119891 (119904 sdot) 119889119904

(95)

Combining Lemma 27 and Youngrsquos inequality (29) withscaling property of the kernel 119864(120573)

119905now yields the result

Theorem 29 Let 119904 be a positive real number and (119901 119903) isin

[1infin]2 A constant 119862 exists which satisfies the following

property For 119906 isin Bminus2119904120573

119901119903 (R119889+1

+) one has

119862minus1119906

Bminus2119904120573

119901119903(R119889+1+

)le

10038171003817100381710038171003817100381710038171003817

10038171003817100381710038171003817119905119904119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862119906Bminus2119904120573

119901119903(R119889+1+

)

(96)

To prove this result we need the following lemma

Lemma 30 There exist two positive constants 120581 and 119862

depending only on 120593 such that for all 1 le 119901 le infin 120591 ge 0 and119895 isin Z one has

10038171003817100381710038171003817Δ 119895 (119867120573 (120591) 119906)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862119890

minus1205812211989512059110038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

) (97)

Proof The result follows immediately by applying Lemma 27and because Δ 119895(119867120573(120591))119906 = (119867120573(120591)Δ 119895)119906

Proof of Theorem 29 Using Lemma 30 and considering thefact that the operator Δ 119895 commutes with the operator 119867120573(119905)

and the definition of the homogeneous Weinstein-Besov(semi) norm we get

10038171003817100381710038171003817119905119904119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862119906

Bminus2119904120573

119901119903(R119889+1+

)sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903119895

(98)

where (119888119903119895)119895isinZ denotes as in all this proof a generic elementof the unit sphere of 119897119903(Z) In the case when 119903 = infin therequired inequality comes immediately from the followingeasy result For any positive 119904 we have

sup119905gt0

sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

lt infin (99)

In the case 119903 lt infin using the Holder inequality with theweight 22119895119904119890minus1205811199052

2119895

(99) and the Fubini theorem we obtain

int

infin

0

11990511990311990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

119889119905

119905

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)int

infin

0

(sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

)

119903minus1

times (sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903

119903119895)

119889119905

119905

Journal of Function Spaces and Applications 9

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)int

infin

0

(sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903

119903119895)

119889119905

119905

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)sum

119895isinZ

119888119903

119903119895int

infin

0

(11990511990422119895119904119890minus12058111990522119895

)

119889119905

119905

le 119862Γ (119904) 119906119903

Bminus2119904120573

119901119903(R119889+1+

)

(100)

In order to prove the other inequality let us observe thatfor any 119904 greater than minus1 we have

Δ 119895119906 =

1

Γ (119904 + 1)

int

infin

0

119905119904(minusΔ 120573)

119904+1

119867120573 (119905) Δ 119895119906 119889119905 (101)

Then Lemma 30 Proposition 9 and the fact that the operatorΔ 119895 commutes with the operator119867120573(119905) lead to the following

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862int

infin

0

11990511990422119895(119904+1)

119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889119905

(102)

In the case 119903 = infin we simply write

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862(sup

119905gt0

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

times int

infin

0

22119895(119904+1)

119890minus12058111990522119895

119889119905

le 11986222119895119904(sup119905gt0

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

(103)

In the case 119903 lt infin Holderrsquos inequality with the weight 119890minus12058111990522119895

gives

(int

infin

0

119905119904119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889119905)

119903

le 1198622minus2119895(119903minus1)

int

infin

0

119905119903119904119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)119889119905

(104)

Thanks to (99) and Fubinirsquos theorem we infer from (102) that

sum

119895isinZ

2minus211989511990311990410038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)le 119862int

infin

0

11990511990311990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

119889119905

119905

(105)

The theorem is proved

Second Proof ofTheorem 29We only consider the case 1 le 119903 ltinfinThe case 119903 = infin can be shown similarlyWe first prove that

119862minus1119906

Bminus2119904120573

119901119903(R119889+1+

)le

10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

(106)

It is easy to see that

Δ 119895119906 = 120594119895lowast119882119864(120573)

2minus2119895lowast119882119906

(107)

where 120594119895 = Fminus1

119882(120593(2

minus119895120585)119890

2minus21198951205852

) and119864(120573)2minus2119895

is the Gauss kernelassociated with Weinstein operators By relation (29) we get

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le

10038171003817100381710038171003817120594119895

100381710038171003817100381710038171198711120573(R119889+1+

)

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

) (108)

As10038171003817100381710038171003817120594119895

100381710038171003817100381710038171198711120573(R119889+1+

)= int

R119889+1+

100381610038161003816100381610038161003816F

minus1

119882(120593 (120585) 119890

1205852

)

100381610038161003816100381610038161003816119889120583120573 (120585) lt infin (109)

we obtain10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

) (110)

Moreover simple calculations give that

119864(120573)

2minus2119895lowast119882119906 = 119867120573 (2

minus4119895minus 119905

2) (119864

(120573)

1199052lowast119882119906)

(111)

Thus from Proposition 26 it follows that100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1

+ )

(112)

for any 119905 isin [2minus119895minus1 2minus119895] which implies that

sum

119895isinZ

2minus211990411989511990310038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

le 119862 sum

119895isinZ

int

2minus119895

2minus119895minus1(119905

2119904100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862int

infin

0

(1199052119904100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862int

infin

0

(11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

(113)

where we have used the fact that 119864(120573)1199052lowast119882119906 = 119867120573(119905

2)119906

We now prove that10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862119906Bminus2119904120573

119901119903(R119889+1+

) (114)

Indeed one has

119864(120573)

2minus2119895lowast119882119906 = sum

119899isinZ

119864(120573)

2minus2119895lowast119882Δ 119899+119895119906 (115)

Arguing as above we have100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1

+ )

(116)

for any 119905 isin [2minus119895 21minus119895] Thus10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

= 2

100381710038171003817100381710038171003817100381710038171003817

1003817100381710038171003817100381710038171199052119904119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)

100381710038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862 sum

119895isinZ

int

21minus119895

2minus119895(2

minus2119895119904100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862 sum

119895isinZ

(2minus2119895119904

sum

119899isinZ

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

(117)

10 Journal of Function Spaces and Applications

On the other hand it is easy to see that100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 1198622

minus211989911990410038171003817100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

) (118)

for any 119904 gt 0 For 0 lt 1199041 lt 119904 lt 1199042 and by using theMinkowskiinequality we have

sum

119895isinZ

(2minus2119895119904

sum

119899isinZ

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

le sum

119895isinZ

(2minus2119895119904

0

sum

minusinfin

2minus21198991199041

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

+ sum

119895isinZ

(2minus2119895119904

sum

119899isinN

2minus21198991199042

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

le 119862(

0

sum

minusinfin

2minus2119899(119904

1minus119904)(sum

119895isinZ

(2minus2(119899+119895)11990311990410038171003817

100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

)

1119903

)

119903

+ 119862(sum

N

2minus2119899(119904

2minus119904)(sum

119895isinZ

(2minus2(119899+119895)11990311990410038171003817

100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

)

1119903

)

119903

le 119862 sum

119895isinZ

2minus211989511990311990410038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

(119)

The result is immediately from (117) and (119)

42 Embedding Sobolev Theorems

Theorem 31 Let 1 lt 119901 lt infin and let 0 lt 119904 lt (119889 + 2120573 +

2)119901There exists a positive constant119862 such that for all function119891 isin

H119904

119901120573(R119889+1

+) one has

10038171003817100381710038171198911003817100381710038171003817119871119902

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

H119904119901120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

120579

B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

) (120)

where 120579 = 119904119901(119889+2120573+2) and 119902 = 119901(119889+2120573+2)(119889+2120573+2minus119901119904)

Proof Bydensity we can suppose that119891belongs toSlowast(R119889+1

)It is easy to see that

119891 = int

infin

0

119867120573 (119905) Δ 120573119891119889119905(121)

and decompose the integral in two parts as follows

119891 = int

119860

0

119867120573 (119905) Δ 120573119891119889119905 + int

infin

119860

119867120573 (119905) Δ 120573119891119889119905(122)

where 119860 is a constant to be fixed laterOn the other hand byTheorem 29 we obtain10038171003817100381710038171003817119867120573 (119905) Δ 120573119891

10038171003817100381710038171003817119871infin120573(R119889+1+

)

le

119862

1199051minus(12)(119904minus(119889+2120573+2)119901)

10038171003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(123)

Therefore after integrating we get

int

infin

119860

10038171003817100381710038171003817119867120573 (119905) Δ 120573119891

10038171003817100381710038171003817119871infin120573(R119889+1+

)119889119905

le 119860(12)(119904minus(119889+2120573+2)119901)1003817

1003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(124)

On the other hand denoting 119892 = (minusΔ 120573)1199042119891 we have

119867120573 (119905) Δ 120573119891 =

1

(minus119905)1minus1199042

119867120573 (119905) (minus119905Δ 120573)

1minus1199042

119892 (125)

We proceed as in [8] we prove that1003816100381610038161003816100381610038161003816

119867120573 (119905) (minus119905Δ 120573)

1minus1199042

119892 (119909)

1003816100381610038161003816100381610038161003816

le 119862 (119904)119872120573 (119892) (119909) (126)

where119872120573(119892) is a maximal function of 119892 associated with theWeinstein operators (cf [12])

This leads to100381610038161003816100381610038161003816100381610038161003816

int

119860

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

100381610038161003816100381610038161003816100381610038161003816

le 1198621198601199042119872120573 (119892) (119909) (127)

In conclusion we get10038161003816100381610038161003816100381610038161003816

int

infin

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

10038161003816100381610038161003816100381610038161003816

le 119862 (1198601199042119872120573 (119892) (119909)

+119860(12)(119904minus(119889+2120573+2)119901)1003817

1003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

))

(128)

and the choice of 119860 such that

119860(119889+2120573+2)2119901

119872120573 (119892) (119909) =10038171003817100381710038171198911003817100381710038171003817B119904minus(119889+2120573+2)119901120573

infininfin (R119889+1+

)(129)

ensures that10038161003816100381610038161003816100381610038161003816

int

infin

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

10038161003816100381610038161003816100381610038161003816

le 119862(119872120573(119892)(119909))

1minus(119901119904(119889+2120573+2))10038171003817100381710038171198911003817100381710038171003817

119901119904(119889+2120573+2)

B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(130)

Finally taking the 119871119902120573norm with 119902 = 119901(119889 + 2120573 + 2)(119889 + 2120573 +

2minus119901119904) ends the proof thanks to the fact themaximal function119872120573 is bounded of 119871119902

120573(R119889+1

+) into itself for 119902 gt 1

Theorem 32 Let 1 lt 119901 lt 119902 lt infin For all function 119891 such that119891 isin

H1199041

119901120573(R119889+1

+)⋂

Bminus120573120573

infininfin(R119889+1

+) one has

10038171003817100381710038171198911003817100381710038171003817H119904119901120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

120579

H1199041

119901120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573120573

infininfin(R119889+1

+) (131)

where 120579 = 119901119902 119904 = 1205791199041 minus (1 minus 120579)120573 with 120573 gt 0 minus120573 lt 119904 lt 1199041

Proof It suffices to prove that1003817100381710038171003817100381710038171003817

(minusΔ 120573)

(119904minus1199041)2

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le 11986210038171003817100381710038171198911003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(132)

Journal of Function Spaces and Applications 11

Indeed we use the following identity (which may be easilyproven by taking the Weinstein transform in 119909 of both sides)

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

infin

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905 (133)

with 120575 = 1199041 minus 119904 gt 0We decompose the integral in two parts as follows

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

119879

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

+

1

Γ (1205752)

int

infin

119879

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

(134)

where 119879 is a constant to be fixed laterWe proceed as in [8] we obtain

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119872120573 (119891) (119909) (135)

On the other hand we use Theorem 29 and the fact that 119891belongs to Bminus120573minus119904

1120573

infininfin(R119889+1

+) to deduce that

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119905

(minus120573minus1199041)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

) (136)

Thus by applying the preceding estimates on the right part of(134) we obtain

1003816100381610038161003816100381610038161003816

(minusΔ 120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621

Γ (1205752)

1198791205752119872120573 (119891) (119909)

+

1198622

Γ (1205752)

119879(120575minus120573minus119904

1)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

(137)

We fix now

119879 = (

10038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

119872120573 (119891) (119909)

)

2(120573+1199041)

(138)

We obtain

1003816100381610038161003816100381610038161003816

(minusΔ120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621 + 1198622

Γ (1205752)

(119872120573(119891)(119909))

12057910038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(139)

Thus we deduce that1003817100381710038171003817100381710038171003817

(minusΔ120573)

minus1205752

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le

1198621 + 1198622

Γ (1205752)

10038171003817100381710038171003817119872120573 (119891)

10038171003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(140)

To conclude we used the fact that the maximal function119872120573

is bounded of 119871119902120573(R119889+1

+) into itself for 119902 gt 1

43 Estimates in Generalized Besov Spaces For any interval 119868ofR (bounded or unbounded) and a normed space 119865(R119889+1

+)

we define the mixed space-time 119871119901(119868 119865(R119889+1

+)) space of

(classes of) measurable functions 119906 119868 rarr 119865(R119889+1

+) such that

||119906||119871119901(119868119865(R119889+1+

)) lt infin with

119906119871119901(119868119865(R119889+1+

)) = (int

119868

119906 (119905 sdot)119901

119865(R119889+1+

)119889119905)

1119901

if 1 le 119901 lt infin

119906119871infin(119868119865(R119889+1+

)) = ess sup119905isin119868

119906 (119905 sdot)119865(R119889+1+

)

(141)

For any interval 119868 of R (bounded or unbounded) anda Banach space 119883 we define the mixed space-time 119862(119868119883)space of continuous functions 119868 rarr 119883 When 119868 is bounded119862(119868119883) is a Banach space with the norm of 119871infin(119868 119883)

Theorem 33 Let 119904 isin R and 1 le 119901 119902 119903 le infin Let 119879 gt 0 119892 isin

B119904120573

119901119903(R119889+1

+) and119891 in 119871119902((0 119879) B

119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76)

has a unique solution

119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903(R

119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R

119889+1

+))

(142)

and there exists a constant 119862 such that for all 1199021 isin [119902infin] onehas

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862(10038171003817100381710038171198921003817100381710038171003817B119904120573

119901119903(R119889+1+

)+10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)B

119904minus2+(2119902)120573

119901119903(R119889+1+

)))

(143)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] B119904120573

119901119903(R119889+1

+))

Proof Since 119892 and 119891 are temperate distributions (76) has aunique solution 119906 in S1015840

((0 119879) timesR119889+1

+) which satisfies

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(144)

Next we notice that applying Δ 119895 to (76) and using formula(81) yield

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(145)

Therefore

10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(146)

12 Journal of Function Spaces and Applications

By virtue of Lemma 30 we thus have for some 120581 gt 0

10038171003817100381710038171003817Δ 119895119906(119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

119890minus12058122119895(119905minus120591) 10038171003817

100381710038171003817Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(147)

Applying convolution inequalities we get

10038171003817100381710038171003817Δ 119895119906

100381710038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862[

[

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

)

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

]

]

(148)

with 11199022 = 1+ 11199021 minus1119902 Finally taking the 119897119903(Z) norm we

conclude that (with the usual convention if 119903 = infin)

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

1199031199021

)(211989511990410038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

]

]

1119903

+ 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

1199031199022

)

times (2119895(119904minus2+2119902)10038171003817

100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902(0119879119871

119901

120573(R119889+1+

))

119903

]

]

1119903

(149)

which insures that 119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903 (R119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R119889+1

+)) and yields the desired inequality

Since 119906 belongs to 119862([0 119879]B119904120573

119901119903(R119889+1

+)) in the case where

119903 is finite may be easily deduced from the density ofSlowast(R

119889+1)⋂

B119904120573

119901119903(R119889+1

+) in B

119904120573

119901119903(R)

Theorem 34 Let 119904 isin R 119879 gt 0 and 1 le 119901 119902 119903 le infinOne supposes that 119892 isin 119861

119904120573

119901119903(R119889+1

+) and 119891 isin 119871

119902((0 119879)

119861119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76) has a unique solution 119906 belong-

ing to

119871119902((0 119879) 119861

119904+(2119902)120573

119901119903(R

119889+1

+))⋂119871

infin((0 119879) 119861

119904120573

119901119903(R

119889+1

+))

(150)

and there exists a constant 119862 such that for all 119902 le 1199021 le infin

1199061198711199021 ((0119879)119861

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862 [ (1 + 11987911199021)10038171003817100381710038171198921003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (1 + 1198791+1119902

1minus1119902

)10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)119861

119904minus2+(2119902)120573

119901119903(R119889+1+

))]

(151)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] 119861119904120573119901119903(R119889+1

+))

Proof Since 119892 119891 are tempered (76) has a unique solution 119906in S1015840

((0 119879) timesR119889+1

+) satisfying

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(152)

Hence applying Δ 119895 119895 ge 0 to (81) we see that

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(153)

and thus by Lemma 30 we can deduce that10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+int

119905

0

119890minus12058122119895(119905minus120591)10038171003817

100381710038171003817Δ 119895119891(120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(154)

Then it follows from convolution inequalities thatΔ 1198951199061198711199021 ((0119879)119871

119901

120573(R119889+1+

))is dominated by

(

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

(155)

with 11199022 = 1 + 11199021 minus 1119902 Moreover similarly as above wecan obtain that

1003817100381710038171003817Δminus1119906(119905 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)le1003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

1003817100381710038171003817Δminus1119891 (120591 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(156)

and thus if 1 le 119902 le 1199021 le infin1003817100381710038171003817Δminus1119906

10038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862(119879111990211003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)+ 119879

111990221003817100381710038171003817Δminus1119891

1003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

)))

(157)

Journal of Function Spaces and Applications 13

Finally taking the 119897119903-normwith respect to 119895 in (155) and (157)with the usual convention if 119903 = infin we can deduce the desiredestimate

Acknowledgment

Theauthor gratefully acknowledges theDeanship of ScientificResearch at the University of Taibah The author is deeplyindebted to the referee for providing constructive commentsand help in improving the contents of this paper

References

[1] Z Ben Nahia and N Ben Salem ldquoSpherical harmonics andapplications associated with the Weinstein operatorrdquo in Pro-ceedings of the International Conference on PotentialTheory heldin Kouty Czech Republic (ICPT rsquo94) pp 235ndash241 1996

[2] Z Ben Nahia and N Ben Salem ldquoOn a mean value propertyassociated with the Weinstein operatorrdquo in Proceedings of theInternational Conference on Potential Theory held in KoutyCzech Republic (ICPT rsquo94) pp 243ndash253 1996

[3] M Brelot ldquoEquation de Weinstein et potentiels de MarcelRieszrdquo in Seminaire de Theorie de Potentiel Paris No 3 vol 681of Lecture Notes in Mathematics pp 18ndash38 Springer BerlinGermany 1978

[4] H Mejjaoli and M Salhi ldquoUncertainty principles for theweinstein transformrdquo Czechoslovak Mathematical Journal vol61 no 4 pp 941ndash974 2011

[5] H Mejjaoli and A Ould Ahmed Salem ldquoWeinstein Gabortransform and applicationsrdquo Advanced Studies in Pure Mathe-matics vol 2 no 3 pp 203ndash210 2012

[6] H Mejjaoli ldquoBesov spaces associated withthe Weinstein opera-tor and applicationsrdquo In press

[7] T Kawazoe and H Mejjaoli ldquoGeneralized Besov spaces andtheir applicationsrdquo Tokyo Journal of Mathematics vol 35 no 2pp 297ndash320 2012

[8] H Mejjaoli ldquoLittlewood-Paley decomposition associated withthe Dunkl operators and paraproduct operatorsrdquo Journal ofInequalities in Pure and Applied Mathematics vol 9 no 4 pp1ndash25 2008

[9] H Mejjaoli and N Sraeib ldquoGeneralized sobolev spaces inquantum calculus and applicationsrdquo Journal of Inequalities andSpecial Functions vol 1 no 4 pp 43ndash64 2012

[10] H Mejjaoli ldquoGeneralized homogeneous Besov spaces and theirapplicationsrdquo Serdica Mathematical Journal vol 38 no 4 pp575ndash614 2012

[11] H Triebel Interpolation Theory Functions Spaces DifferentialOperators North-Holland AmsterdamThe Netherlands 1978

[12] V S Guliev ldquoOn maximal function and fractional integralassociated with the Bessel differential operatorrdquo MathematicalInequalities and Applications vol 6 no 2 pp 317ndash330 2003

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Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 3

(i) For 119891 in 1198711120573(R119889+1

+)

1003817100381710038171003817F119882 (119891)

1003817100381710038171003817119871infin120573(R119889+1+

)le100381710038171003817100381711989110038171003817100381710038171198711120573(R119889+1+

) (18)

(ii) For 119891 in Slowast(R119889+1

) we have

F119882 (Δ 120573119891) (119910) = minus10038171003817100381710038171199101003817100381710038171003817

2F119882 (119891) (119910) forall119910 isin R

119889+1

+ (19)

(iii) For all 119891 in 1198711120573(R119889+1

+) ifF119882(119891) belongs to 119871

1

120573(R119889+1

+)

then

119891 (119910) = 119862 (120573)int

R119889+1+

F119882 (119891) (119909) Λ (119909 119910) 119889120583120573 (119909) ae

(20)

where

119862 (120573) =

1

1205871198894120573+1198892

(Γ (120573 + 1))2 (21)

(iv) For 119891 isin Slowast(R119889+1

) if we define

F119882 (119891) (119910) = F119882 (119891) (minus119910) (22)

then

(F119882)minus1=

1

119862 (120573)

F119882 (23)

Proposition 1 (see [2]) (i) the Weinstein transform F119882 is atopological isomorphism fromSlowast(R

119889+1) onto itself and for all

f in Slowast(R119889+1

)

int

R119889+1+

1003816100381610038161003816119891 (119909)

1003816100381610038161003816

2119889120583120573 (119909) = 119862 (120573)int

R119889+1+

1003816100381610038161003816F119882 (119891) (120585)

1003816100381610038161003816

2119889120583120573 (120585)

(24)

(ii) In particular the renormalized Weinstein transform119891 rarr 119862(120573)

12F119882(119891) can be uniquely extended to an isometric

isomorphism from 1198712

120573(R119889+1

+) onto itself

In the Fourier analysis the translation operator is givenby 119891 997891rarr 119891(sdot + 119909)

In harmonic analysis associated for the operator Δ 120573 thegeneralized translation operator 120591119909 119909 isin R119889+1

+is defined by

forall119910 isin R119889+1

+

120591119909119891 (119910) =

Γ (120573 + 1)

radic120587Γ (120573 + 12)

times int

120587

0

119891(1199091015840+ 119910

1015840 radic119909

2

119889+1+ 119910

2

119889+1+ 2119909119889+1119910119889+1 cos 120579)

times (sin 120579)2120573119889120579(25)

where 119891 isin 119862lowast(R119889+1

)

By using the Weinstein kernel we can also define ageneralized translation For functions 119891 isin Slowast(R

119889+1) and

119910 isin R119889+1

+the generalized translation 120591119910119891 is defined by the

following relation

F119882 (120591119910119891) (119909) = Λ (119909 119910)F119882 (119891) (119909) (26)

By using the generalized translation we define the general-ized convolution product119891lowast

119882119892 of functions119891 119892 isin 1198711

120573(R119889+1

+)

as follows

119891lowast119882119892 (119909) = int

R119889+1+

120591119909119891 (minus1199101015840 119910119889+1) 119892 (119910) 119889120583120573 (119910) (27)

This convolution is commutative and associative and itsatisfies the following

(i) For all 119891 119892 isin 1198711

120573(R119889+1

+) 119891lowast119882 119892 belongs to 1198711

120573(R119889+1

+)

and

F119882 (119891lowast119882119892) = F119882 (119891)F119882 (119892) (28)

(ii) Let 1 le 119901 119902 119903 le infin such that 1119901 + 1119902 minus 1119903 = 1If 119891 isin 119871

119901

120573(R119889+1

+) and 119892 isin 119871

119902

120573(R119889+1

+) then 119891lowast119882119892 isin

119871119903

120573(R119889+1

+) and

1003817100381710038171003817119891lowast119882119892

1003817100381710038171003817119871119903120573(R119889+1+

)le10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171198921003817100381710038171003817119871119902

120573(R119889+1+

) (29)

We define the tempered distribution T119891 associated with119891 isin 119871

119901

120573(R119889+1

+) by

⟨T119891 120601⟩ = int

R119889+1+

119891 (119909) 120601 (119909) 119889120583120573 (119909) (30)

for 120601 isin Slowast(R119889+1

) and denote by ⟨119891 120601⟩120573 the integral in therighthand side

Definition 2 The Weinstein transform F119882(120591) of a distribu-tion 120591 isin S1015840

lowast(R119889+1

) is defined by

⟨F119882 (120591) 120601⟩ = ⟨120591F119882 (120601)⟩ (31)

for 120601 isin Slowast(R119889+1

)

In particular for 119891 isin 119871119901

120573(R119889+1

+) it follows that for 120601 isin

Slowast(R119889+1

)

⟨F119882 (119891) 120601⟩ = ⟨F119882 (T119891) 120601⟩ = ⟨T119891F119882 (120601)⟩

= ⟨119891F119882 (120601)⟩120573

(32)

Proposition 3 The Weinstein transform F119882 is a topologicalisomorphism from S1015840

lowast(R119889+1

) onto itself

Definition 4 The generalized convolution product of a dis-tribution 119878 in S1015840

lowast(R119889+1

) and a function 120601 in Slowast(R119889+1

) is thefunction 119878lowast119882120601 defined by

119878lowast119882 120601 (119909) = ⟨119878119910 120591minus119910120601 (119909)⟩ (33)

4 Journal of Function Spaces and Applications

Proposition 5 Let 119891 be in 119871119901

120573(R119889+1

+) 1 le 119901 le infin and 120601

in Slowast(R119889+1

) Then the distribution T119891lowast119882120601 is given by thefunction 119891lowast119882120601 and T119891lowast119882120601 belongs to 119871119901

120573(R119889+1

+) Moreover

for all 120595 isin Slowast(R119889+1

)

⟨T119891lowast119882120601 120595⟩ = ⟨

119891 120601lowast119882 ⟩120573 (34)

where (119909) = 120595(minus119909) and

F119882 (T119891lowast119882 120601) = F119882 (T119891)F119882 (120601) (35)

For each 119906 isin S1015840

lowast(R119889+1

) we define the distribution Δ 120573119906

by ⟨Δ 120573119906 120595⟩ = ⟨119906 Δ 120573120595⟩ and this distribution satisfies thefollowing property

F119882 (Δ 120573119906) = minus10038171003817100381710038171199101003817100381710038171003817

2F119882 (119906) (36)

In the following we denote T119891 given by (30) by 119891 forsimplicity

3 B119904120573

119901119902 F119904120573

119901119902(R119889+1

+) H119904

119901120573Spaces and

Basic Properties

31 HomogeneousWeinstein-Littlewood-Paley DecompositionOne of the main tools in this paper is the homogeneousLittlewood-Paley decomposition of distribution associatedwith theWeinstein operators into dyadic blocs of frequencies

Lemma6 One defines byC the ring of center 0 of small radius12 and great radius 2 There exists two radial functions 120595 and120593 the values of which are in the interval [0 1] belonging to119863lowast(R

119889+1) such that

supp120595 sub 119861 (0 1) supp120593 sub C

forall120585 isin R119889+1

120595 (120585) + sum

119895ge0

120593 (2minus119895120585) = 1

forall120585 isin C sum

119895isinZ

120593 (2minus119895120585) = 1

|119899 minus 119898| ge 2 997904rArr supp120593 (2minus119899sdot) cap supp120593 (2minus119898sdot) = 0

119895 ge 1 997904rArr supp120595 cap supp120593 (2minus119895sdot) = 0

(37)

Notations We denote by

Δ 119895119891 = Fminus1

119882(120593(

120585

2119895)F119882 (119891))

119878119895119891 = sum

119899le119895minus1

Δ 119899119891 forall119895 isin Z(38)

The distribution Δ 119895119891 is called the 119895th dyadic block of thehomogeneous Littlewood-Paley decomposition of 119891 associ-ated with the Weinstein operators

Throughout this paper we define 120601 and 120594 by 120601 = Fminus1

119882(120593)

and 120594 = Fminus1

119882(120595)

When dealing with the Littlewood-Paley decompositionit is convenient to introduce the functions and 120593 belongingto119863lowast(R

119889+1) such that equiv 1 on supp120595 and 120593 equiv 1 on supp120593

Remark 7 We remark that

F119882 (119878119895119891) (120585) = (

120585

2119895)F119882 (119878119895119891) (120585)

F119882 (Δ 119895119891) (120585) = 120593(

120585

2119895)F119882 (Δ 119895119891) (120585)

(39)

We put

120601 = F

minus1

119882(120593) 120594 = F

minus1

119882() (40)

Definition 8 One denotes by S1015840

ℎ120573lowast(R119889+1

) the space oftempered distribution such that

lim119895rarrminusinfin

119878119895119906 = 0 in S1015840

lowast(R

119889+1) (41)

Proposition 9 (Bernstein inequalities) For all 120583 isin N119889+1 and120590 isin R for all 119895 isin Z for all 1 le 119901 119902 le infin and for all119891 isin S1015840

lowast(R119889+1

) one has the following

(i) 10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902

120573(R119889+1+

)

le

10038171003817100381710038171003817

120601

10038171003817100381710038171003817119871119903120573(R119889+1+

)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

)2119895(119889+2120573+2)(1119901minus1119902)

with 1

119902

=

1

119901

+

1

119903

minus 1

(ii) 10038171003817100381710038171003817119878119895119891

10038171003817100381710038171003817119871119902

120573(R119889+1+

)

le10038171003817100381710038171205941003817100381710038171003817119871119903120573(R119889+1+

)

10038171003817100381710038171003817119878119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

)2119895(119889+2120573+2)(1119901minus1119902)

with 1

119902

=

1

119901

+

1

119903

minus 1

(iii)1003817100381710038171003817100381710038171003817

(radicminusΔ 120573)

120590

Δ 119895119891

1003817100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)

le

10038171003817100381710038171003817F

minus1

119882(10038171003817100381710038171205851003817100381710038171003817

120590120593)

100381710038171003817100381710038171198711120573(R119889+1+

)

times

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

)2119895120590

(42)

Proof Using Remark 7 we deduce from Proposition 5 that

119878119895119891 = 2119895(119889+2120573+2)

120594 (2119895sdot) lowast119882119878119895119891

Δ 119895119891 = 2119895(119889+2120573+2)

120601 (2119895sdot) lowast119882Δ 119895119891

(43)

Thus from the relation (29) we prove (i) (ii) and (iii)

32 Definitions In the following we define analogues of thehomogeneous Besov Triebel-Lizorkin and Riesz potentialspaces associated with the Weinstein operators on R119889+1

+and

obtain their basic propertiesFrom now we make the convention that for all non-

negative sequence 119886119902119902isinZ the notation (sum119902119886119903

119902)1119903 stands for

sup119902119886119902 in the case 119903 = infin

Journal of Function Spaces and Applications 5

Definition 10 Let 119904 isin R and 119901 119902 isin [1infin]The homogeneousWeinstein-Besov spaces B

119904120573

119901119902(R119889+1

+) are the spaces of distri-

bution in S1015840

ℎ120573lowast(R119889+1

) such that

10038171003817100381710038171198911003817100381710038171003817B119904120573

119901119902(R119889+1+

)= (sum

119895isinZ

(211990411989510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119902

)

1119902

lt infin (44)

Proposition 11 (see [6]) Let 119904 isin R and 119901 and 119902 two elementsof [1infin] the space B

119904120573

119901119902(R119889+1

+) is the set of 119891 isin S1015840

ℎ120573lowast(R119889+1

)

verifying

10038171003817100381710038171003817

119891

10038171003817100381710038171003817B119904120573

119901119902(R119889+1+

)= (int

infin

0

(119905minus1199041003817100381710038171003817119891lowast119882120601119905

1003817100381710038171003817119871119901

120573(R119889+1+

))

119902119889119905

119905

)

1119902

lt infin

(45)

where 120601119905(119909) = (11199052120573+2+119889

)120601(119909119905) for all 119905 isin (0infin) and 119909 isin

R119889+1

+

Definition 12 For 119904 isin R and 119901 119902 isin [1infin] one writes10038171003817100381710038171198911003817100381710038171003817119861119904120573

119901119902(R119889+1+

)=10038171003817100381710038171198780119891

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ (sum

119895ge1

(211990411989510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119902

)

1119902

(46)

The nonhomogeneous Besov space 119861119904120573119901119902(R119889+1

+) associated

with the Weinstein operators is defined by

119861119904120573

119901119902(R

119889+1

+) = 119891 isin S

1015840(R

119889)

10038171003817100381710038171198911003817100381710038171003817119861119904120573

119901119902(R119889+1+

)lt infin (47)

We give now another definition equivalent to the nonho-mogeneous Besov space 119861119904120573119901119902(R119889+1

+)

Proposition 13 Let 119904 isin R and119901 and 119902 two elements of [1infin]the space 119861119904120573119901119902(R119889+1

+) is the set of 119891 isin S1015840

(R119889) verifying

10038171003817100381710038171003817

119891

10038171003817100381710038171003817119861119904120573

119901119902(R119889+1+

)=1003817100381710038171003817119891lowast119882120595

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ (int

1

0

(119905minus1199041003817100381710038171003817119891lowast119882120601119905

1003817100381710038171003817119871119901

120573(R119889+1+

))

119902119889119905

119905

)

1119902

lt infin

(48)

Definition 14 Let 119904 isin R and 1 le 119901 119902 le infin the homogeneousWeinstein-Triebel-Lizorkin space F

119904120573

119901119902(R119889+1

+) is the space of

distribution in S1015840

ℎ120573lowast(R119889+1

) such that

10038171003817100381710038171198911003817100381710038171003817F119904120573

119901119902(R119889+1+

)=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

211990411989511990210038161003816100381610038161003816Δ 119895119891

10038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)

lt infin (49)

Definition 15 For 119904 isin R the operatorR119904

120573from S1015840

ℎ120573lowast(R119889+1

)

to S1015840

ℎ120573lowast(R119889+1

) is defined by

R119904

120573(119891) = F

minus1

119882(sdot

119904F119882119891) (50)

The operatorRminus119904

120573is called Weinstein-Riesz potential space

Definition 16 For 119904 isin R and 1 le 119901 le infin the homogeneousWeinstein-Riesz potential space H119904

119901120573(R119889+1

+) is defined

as the space Rminus119904

120573(119871

119901

120573(R119889+1

+)) equipped with the norm

119891H119904119901120573(R119889+1+

)= R119904

120573(119891)

119871119901

120573(R119889+1+

)

Proposition 17 Let 119904 isin R and 1 le 119901 119902 le infinThe operator Δ 120573 is a linear continuous operator from

B119904120573

119901119902(R119889+1

+) into B

119904minus2120573

119901119902 (R119889+1

+) and from H119904

119901120573(R119889+1

+) into

H119904minus2

119901120573(R119889+1

+)

Proof We obtain these results by the similar ideas used in thenonhomogeneous case (cf [6])

Proposition 18 Let 119904 119905 isin R and 1 le 119901 119902 le infin The operatorR119905

120573is a linear continuous injective operator from B

119904120573

119901119902(R119889+1

+)

onto B119904minus119905120573

119901119902 (R119889+1

+) and from H119904

119901120573(R119889+1

+) onto H119904minus119905

119901120573(R119889+1

+)

Proof We obtain these results by the similar ideas used in thenonhomogeneous case (cf [6])

33 Embeddings As in the Euclidean case (cf [11]) themono-tone character of 119897119902-spaces and the Minkowskis inequalityyield the following

Proposition 19 If 1 le 1199021 lt 1199022 le infin one has

B119904120573

1199011199021

(R119889+1

+) 997893rarr

B119904120573

1199011199022

(R119889+1

+) (1 le 119901 le infin 119904 isin R)

(51)

MoreoverB119904120573

1199011(R

119889+1

+) 997893rarr

H119904

119901120573(R

119889+1

+) 997893rarr

B119904120573

119901infin(R

119889+1

+)

(1 le 119901 le infin 119904 isin R)

(52)

If 1199040 = 1199041 one also has

(H1199040

119901120573(R

119889+1

+)

H1199041

119901120573(R

119889+1

+))

120579119902=

B119904120573

119901119902(R

119889+1

+)

(1 le 119901 119902 le infin 120579 isin (0 1))

(53)

where 119904 = (1 minus 120579)1199040 + 1205791199041

Proposition 20 One assumes that 119904 minus (119889 + 2120573 + 2)119901 = 1199041 minus

(119889 + 2120573 + 2)1199011 Then the following inclusion holds

B119904120573

119901119902(R

119889+1

+) 997893rarr

B1199041120573

11990111199021

(R119889+1

+)

(1 le 119901 le 1199011 le infin 1 le 119902 le 1199021 le infin 119904 1199041 isin R)

(54)

Proof In order to prove the inclusion we use the estimate

Δ 119895119891 = 2119895(119889+2120573+2)

120601 (2119895sdot) lowast119882Δ 119895119891 (55)

Proposition 9(i) gives that10038171003817100381710038171003817Δ 119895119891

100381710038171003817100381710038171198711199011

120573(R119889+1+

)=

100381710038171003817100381710038172119895(119889+2120573+2)

120601 (2119895sdot) lowast119882Δ 119895119891

100381710038171003817100381710038171198711199011

120573(R119889+1+

)

le 1198622119895(119889+2120573+2)(1119901minus1119901

1)10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

(56)

6 Journal of Function Spaces and Applications

By definition of the homogeneous Weinstein-Besov spaceswe therefore infer10038171003817100381710038171198911003817100381710038171003817B1199041120573

11990111199021(R119889+1+

)

= (

infin

sum

119895=minusinfin

(2119895119904110038171003817100381710038171003817Δ 119895119891

100381710038171003817100381710038171198711199011

120573(R119889+1+

))

1199021

)

11199021

le 119862(sum

119895isinZ

(211989511990412119895(119889+2120573+2)(1119901minus1119901

1)10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

1199021

)

11199021

le 119862(sum

119895isinZ

(211989511990410038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

1199021

)

11199021

le 11986210038171003817100381710038171198911003817100381710038171003817B119904120573

119901119902(R119889+1+

)

(57)

since 119902 le 1199021 This gives the inclusion

Proposition 21 (1) If 119906 belongs to B119904120573

119901119902(R119889+1

+) cap

B119905120573

119901119902(R119889+1

+)

then 119906 belongs to B120579119904+(1minus120579)119905120573

119901119902 (R119889+1

+) for all 120579 isin [0 1] and

119906B120579119904+(1minus120579)119905120573

119901119902(R119889+1+

)le 119906

120579

B119904120573

119901119902(R119889+1+

)119906

1minus120579

B119905120573

119901119902(R119889+1+

) (58)

(2) If 119906 belongs to B119904120573

119901infin(R119889+1

+) cap

B119905120573

119901infin(R119889+1

+) and 119904 lt 119905

then 119906 belongs to B120579119904+(1minus120579)119905120573

1199011(R119889+1

+) for all 120579 isin (0 1) and there

exists a positive constant 119862(119905 119904) such that

119906B120579119904+(1minus120579)119905120573

1199011(R119889+1+

)le 119862 (119905 119904) 119906

120579

B119904120573

119901infin(R119889+1+

)119906

1minus120579

B119905120573

119901infin(R119889+1+

) (59)

(3) If 119906 belongs to B119904120573

119901infin(R119889+1

+) cap

B119904+120576120573

119901infin (R119889+1

+) and 120576 gt

0 then 119906 belongs to B119904120573

1199011(R119889+1

+) and there exists a positive

constant 119862 such that

119906B119904120573

1199011(R119889+1+

)le

119862

120576

119906B119904120573

119901infin(R119889+1+

)log

2(119890 +

119906B119904+120576120573

119901infin(R119889+1+

)

119906B119904120573

119901infin(R119889+1+

)

)

(60)

Proof (1) is obvious from the Holderrsquos inequality As for (2)we write 119906

B120579119904+(1minus120579)119905120573

1199011(R119889+1+

)as

sum

119895le119873

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ sum

119895gt119873

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

(61)

where 119873 is chosen here after By the definition of thehomogeneous Weinstein-Besov norms we see that

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 2

119895(1minus120579)(119905minus119904)119906

B119904120573

119901infin(R119889+1+

)

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 2

minus119895120579(119905minus119904)119906

B119905120573

119901infin(R119889+1+

)

(62)

and thus 119906B120579119904+(1minus120579)119905120573

1199011(R119889+1+

)is dominated by

119906B119904120573

119901infin(R119889+1+

)sum

119895le119873

2119895(1minus120579)(119905minus119904)

+ 119906B119905120573

119901infin(R119889+1+

)sum

119895gt119873

2minus119895120579(119905minus119904)

le 119862119906B119904120573

119901infin(R119889+1+

)

2(119873+1)(1minus120579)(119905minus119904)

2(1minus120579)(119905minus119904)

minus 1

+ 119906B119905120573

119901infin(R119889+1+

)

2minus119873120579(119905minus119904)

1 minus 2minus120579(119905minus119904)

(63)

Hence in order to complete the proof of (2) it suffices tochoose119873 such that

119906B119905120573

119901infin(R119889)

119906B119904120573

119901infin(R119889)

le 2119873(119905minus119904)

lt 2

119906B119905120573

119901infin(R119889)

119906B119904120573

119901infin(R119889)

(64)

As for (3) it is easy to see that 119906B119904120573

1199011(R119889+1+

)is dominated as

sum

119895le119873minus1

211989511990410038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ sum

119895ge119873

211989511990410038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le (119873 + 1) 119906B119904120573

119901infin(R119889+1+

)+

2minus(119873minus1)120576

2120576minus 1

119906B119904+120576120573

119901infin(R119889+1+

)

(65)

Hence letting

119873 = 1 + [

[

1

120576

log2

119906B119904+120576120573

119901infin(R119889+1+

)

119906B119904120573

119901infin(R119889+1+

)

]

]

(66)

we can obtain the desired estimate

Proposition 22 Let 119902 isin (1infin) and let 119904 isin R such that0 lt 119904 lt (119889 + 2120573 + 2)119902 then one has

B119904120573

119902119902(R

119889+1

+) =

F119904120573

119902119902(R

119889+1

+) 997893rarr

F119904120573

119902infin(R

119889+1

+)

997893rarrF119904minus((119889+2120573+2)119902)120573

infininfin(R

119889+1

+)

(67)

H119904

119902120573(R

119889+1

+) =

F119904120573

1199022(R

119889+1

+) 997893rarr

F119904120573

119902infin(R

119889+1

+)

997893rarrF119904minus((119889+2120573+2)119902)120573

infininfin(R

119889+1

+)

(68)

Proof We obtain these results by the similar ideas used in thenonhomogeneous case (cf [6])

Theorem 23 Let 119886 119887 gt 0 and let 1199021 1199022 isin [1infin] Let120579 = 119886(119886 + 119887) isin (0 1) and let 1119901 = (1 minus 120579)1199021 +

1205791199022 Then there exists a constant 119862 such that for every119891 isin

F119886120573

1199021infin(R119889+1

+) cap

Fminus119887120573

1199022infin(R119889+1

+) then one has

1003816100381610038161003816119891 (119909)

1003816100381610038161003816le 119862(sup

119895isinZ

2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

1minus120579

(sup119895isinZ

2minus119887119895 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

120579

(69)

In particular one gets

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

F119886120573

1199021infin(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

120579

Fminus119887120573

1199022infin(R119889+1+

) (70)

Journal of Function Spaces and Applications 7

Proof Let 119891 be a Schwartz class we have

1003816100381610038161003816119891 (119909)

1003816100381610038161003816le sum

119895isinZ

10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816

le sum

119895isinZ

min(2minus119886119895sup119895isinZ

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816))

(71)

We define119873(119909) as the largest index such that

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816) le 2

minus119886119895sup119895isinZ

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816) (72)

and we write1003816100381610038161003816119891 (119909)

1003816100381610038161003816le sum

119895le119873(119909)

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

+ sum

119895gt119873(119909)

2minus119886119895sup

119895

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

le 119862(sup119895isinZ

2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

119887(119886+119887)

times (sup119895isinZ

2minus119887119895 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

119886(119886+119887)

(73)

Thus (69) is proved In order to obtain (70) it is enough toapply the Holder inequality in the expression above since wehave 120579 = 119886(119886+119887) isin (0 1) and let 1119901 = (1minus120579)1199021+1205791199022

Corollary 24 Let 119902 isin (1infin) and let 119904 isin R such that 0 lt 119904 lt

(119889 + 2120573 + 2)119902 then one has

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus(119902119901)

Bminus((2120573+2+119889)119902minus119904)120573

infininfin (R119889+1

+ )

10038171003817100381710038171198911003817100381710038171003817

119902119901

B119904120573

119902119902(R119889+1

+ )

(74)

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus(119902119901)

Bminus((2120573+2+119889)119902minus119904)120573

infininfin (R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

119902119901

H119904119902120573(R119889+1+

) (75)

where 119901 = 119902(2120573 + 2 + 119889)(2120573 + 2 + 119889 minus 119902119904)

Proof By choosing 119886 = 119904 gt 0 minus119887 = 119904 minus (119889 + 2120573 + 2)119902 lt 01199021 = 119902 and 1199022 = infin we deduce (74) from the relations (70)and (67) In the same way we deduce (75) from the relations(70) and (68)

4 Generalized Heat Equation

41 Characterization for the Weinstein-Besov Spaces TheWeinstein heat equation reads

120597119905119906 (119905 119909) minus Δ 120573119906 (119905 119909) = 119891 (119905 119909) (119905 119909) isin [0infin) timesR119889+1

+

119906|119905=0 = 119892

(76)

We introduce the Weinstein heat semigroup 119867120573(119905) for theWeinstein-Laplace operator

119867120573 (119905) 119892 (119909) =

int

R119889+1+

Γ120573 (119905 119909 119910) 119892 (119910) 119889120583120573 (119910) if 119905 gt 0

119892 (119909) if 119905 = 0(77)

where Γ120573 is the Weinstein heat kernel defined by

Γ120573 (119905 119909 119910) = 120591119909 (119864(120573)

119905) (119910) (78)

where

119864(120573)

119905(119910) =

2

1205871198892Γ (120573 + 1) (4119905)

120573+1+1198892119890minus11991024119905 (79)

Thus

119867120573 (119905) 119892 (119909) = 119892lowast119882119864(120573)

119905 (119909) (80)

In practice we use the integral formulation of (76)

119906 (119905 119909) = 119867120573 (119905) 119892 (119909) + 119866120573 (119891) (119905 119909)

= 119867120573 (119905) 119892 (119909) + int

119905

0

119867120573 (119905 minus 119904) 119891 (119904 119909) 119889119904

(81)

Remark 25 The function 119864(120573)119905

is the Gauss kernel associatedwith Weinstein operators This function satisfies

forall120585 isin R119889+1

+ F119882 (119864

(120573)

119905) (120585) = 119890

minus1199051205852

(82)

Proposition 26 Let 1 le 119901 le 119903 le infin and let 119891 isin 119871119901

120573(R119889+1

+)

Then the operator 119867120573(119905) maps 119871119901

120573(R119889+1

+) continuously to

119871119903

120573(R119889+1

+) and

10038171003817100381710038171003817119867120573 (119905) 119891

10038171003817100381710038171003817119871119903120573(R119889)

le 119862119905minus((119889+2120573+2)2)(1119901minus1119903)1003817

1003817100381710038171198911003817100381710038171003817119871119901

120573(R119889)

(83)

Moreover1003817100381710038171003817100381710038171003817

(minusΔ 120573)

1205752

119867120573 (119905) 119891

1003817100381710038171003817100381710038171003817119871119903120573(R119889)

le 119862119905minus1205752minus((119889+2120573+2)2)(1119901minus1119903)1003817

1003817100381710038171198911003817100381710038171003817119871119901

120573(R119889)

(84)

for all 120575 gt 0

Proof It follows from the relations (80) and (29) combinedwith scaling property of the kernel 119864(120573)

119905

In this section we prove estimates for the Weinstein heatsemigroupThese estimates are based on the following result

Lemma 27 Let C be an annulus Positive constants 119888 and 119862exist such that for any 119901 in [1infin] and any couple (119905 120582) ofpositive real numbers one has

suppF119882 (119906) sub 120582C 997904rArr

10038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le 119862119890minus1198881199051205822

119906119871119901

120573(R119889+1+

)

(85)

8 Journal of Function Spaces and Applications

Proof We again consider a function Θ in 119863(R119889+1

+ 0) the

value of which is identically 1 in neighborhood of annulusCWe can also assume without loss of generality that 120582 = 1 Wethen have

119867120573 (119905) 119906 = 119892 (119905 sdot) lowast119882119906 (86)where

119892 (119905 sdot) = Fminus1

119882(Θ (120585) 119890

minus1199051205852

) (87)

The lemma is proved provided that we can find positive realnumbers 119888 and 119862 such that

forall119905 gt 01003817100381710038171003817119892 (119905 sdot)

10038171003817100381710038171198711120573(R119889+1+

)le 119862119890

minus119888119905 (88)

To begin we perform integrations by parts in (87) We get1003816100381610038161003816119892 (119905 119909)

1003816100381610038161003816=

1

119888120573

(1 + 1199092)

minus(119889+[2120573]+3)

times int

R119889+1+

Λ (119909 120585) (119868119889 minus Δ 120573)

119889+[2120573]+3

times (Θ (120585) 119890minus1199051205852

) 119889120583120573 (120585)

(89)

Using Leibnizrsquos formula we obtain1003816100381610038161003816119892 (119905 119909)

1003816100381610038161003816le 119862(1 + 119909

2)

minus(119889+[2120573]+3)

119890minus119888119905 (90)

and (88) follows

For any interval 119868 of R (bounded or unbounded) wedefine themixed space-time 119871119901(119868 119871119902

120573(R119889+1

+)) Banach space of

(classes of) measurable functions 119906 119868 rarr 119871119902

120573(R119889+1

+) such

that 119906119871119901(119868119871

119902

120573(R119889+1+

))lt infin with

119906119871119901(119868119871119902

120573(R119889+1+

))= (int

119868

119906 (119905 sdot)119901

119871119902

120573(R119889+1+

)119889119905)

1119901

if 1 le 119901 119902 lt infin

119906119871infin(119868119871119902

120573(R119889+1+

))= ess sup

119905isin119868

119906 (119905 sdot)119871119902

120573(R119889+1+

) if 1 le 119902 lt infin

(91)Corollary 28 Let C be an annulus and 120582 a positive realnumber Let 1199060 (resp 119891 = 119891(119905 119909)) satisfy suppF119882(1199060) sub 120582C(resp suppF119882(119891(119905 sdot)) sub 120582C for all 119905 in [0 119879]) Consider 119906 asolution of

120597119905119906 minus Δ 120573119906 = 0 119906|119905=0 = 1199060 (92)and V a solution of

120597119905V minus Δ 120573V = 119891 (119905 sdot) V|119905=0 = 0 (93)There exist positive constants 119888 and 119862 depending only on Csuch that for any 1 le 119886 le 119887 le infin and 1 le 119901 le 119902 le infin we have

119906119871119902([0119879]119871119887120573(R119889+1+

))le 119862120582

(119889+2120573+2)(1119886minus1119887)120582minus21199021003817

1003817100381710038171199060

1003817100381710038171003817119871119886120573(R119889+1+

)

V119871119902([0119879]119871119887120573(R119889+1+

))le 119862120582

minus2(1+1119902minus1119901)120582(119889+2120573+2)(1119886minus1119887)

times10038171003817100381710038171198911003817100381710038171003817119871119901([0119879]119871119886

120573(R119889+1+

))

(94)

Proof It suffices to use the fact that

119906 (119905 sdot) = 119867120573 (119905) 1199060 V (119905 sdot) = int119905

0

119867120573 (119905 minus 119904) 119891 (119904 sdot) 119889119904

(95)

Combining Lemma 27 and Youngrsquos inequality (29) withscaling property of the kernel 119864(120573)

119905now yields the result

Theorem 29 Let 119904 be a positive real number and (119901 119903) isin

[1infin]2 A constant 119862 exists which satisfies the following

property For 119906 isin Bminus2119904120573

119901119903 (R119889+1

+) one has

119862minus1119906

Bminus2119904120573

119901119903(R119889+1+

)le

10038171003817100381710038171003817100381710038171003817

10038171003817100381710038171003817119905119904119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862119906Bminus2119904120573

119901119903(R119889+1+

)

(96)

To prove this result we need the following lemma

Lemma 30 There exist two positive constants 120581 and 119862

depending only on 120593 such that for all 1 le 119901 le infin 120591 ge 0 and119895 isin Z one has

10038171003817100381710038171003817Δ 119895 (119867120573 (120591) 119906)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862119890

minus1205812211989512059110038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

) (97)

Proof The result follows immediately by applying Lemma 27and because Δ 119895(119867120573(120591))119906 = (119867120573(120591)Δ 119895)119906

Proof of Theorem 29 Using Lemma 30 and considering thefact that the operator Δ 119895 commutes with the operator 119867120573(119905)

and the definition of the homogeneous Weinstein-Besov(semi) norm we get

10038171003817100381710038171003817119905119904119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862119906

Bminus2119904120573

119901119903(R119889+1+

)sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903119895

(98)

where (119888119903119895)119895isinZ denotes as in all this proof a generic elementof the unit sphere of 119897119903(Z) In the case when 119903 = infin therequired inequality comes immediately from the followingeasy result For any positive 119904 we have

sup119905gt0

sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

lt infin (99)

In the case 119903 lt infin using the Holder inequality with theweight 22119895119904119890minus1205811199052

2119895

(99) and the Fubini theorem we obtain

int

infin

0

11990511990311990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

119889119905

119905

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)int

infin

0

(sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

)

119903minus1

times (sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903

119903119895)

119889119905

119905

Journal of Function Spaces and Applications 9

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)int

infin

0

(sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903

119903119895)

119889119905

119905

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)sum

119895isinZ

119888119903

119903119895int

infin

0

(11990511990422119895119904119890minus12058111990522119895

)

119889119905

119905

le 119862Γ (119904) 119906119903

Bminus2119904120573

119901119903(R119889+1+

)

(100)

In order to prove the other inequality let us observe thatfor any 119904 greater than minus1 we have

Δ 119895119906 =

1

Γ (119904 + 1)

int

infin

0

119905119904(minusΔ 120573)

119904+1

119867120573 (119905) Δ 119895119906 119889119905 (101)

Then Lemma 30 Proposition 9 and the fact that the operatorΔ 119895 commutes with the operator119867120573(119905) lead to the following

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862int

infin

0

11990511990422119895(119904+1)

119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889119905

(102)

In the case 119903 = infin we simply write

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862(sup

119905gt0

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

times int

infin

0

22119895(119904+1)

119890minus12058111990522119895

119889119905

le 11986222119895119904(sup119905gt0

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

(103)

In the case 119903 lt infin Holderrsquos inequality with the weight 119890minus12058111990522119895

gives

(int

infin

0

119905119904119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889119905)

119903

le 1198622minus2119895(119903minus1)

int

infin

0

119905119903119904119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)119889119905

(104)

Thanks to (99) and Fubinirsquos theorem we infer from (102) that

sum

119895isinZ

2minus211989511990311990410038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)le 119862int

infin

0

11990511990311990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

119889119905

119905

(105)

The theorem is proved

Second Proof ofTheorem 29We only consider the case 1 le 119903 ltinfinThe case 119903 = infin can be shown similarlyWe first prove that

119862minus1119906

Bminus2119904120573

119901119903(R119889+1+

)le

10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

(106)

It is easy to see that

Δ 119895119906 = 120594119895lowast119882119864(120573)

2minus2119895lowast119882119906

(107)

where 120594119895 = Fminus1

119882(120593(2

minus119895120585)119890

2minus21198951205852

) and119864(120573)2minus2119895

is the Gauss kernelassociated with Weinstein operators By relation (29) we get

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le

10038171003817100381710038171003817120594119895

100381710038171003817100381710038171198711120573(R119889+1+

)

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

) (108)

As10038171003817100381710038171003817120594119895

100381710038171003817100381710038171198711120573(R119889+1+

)= int

R119889+1+

100381610038161003816100381610038161003816F

minus1

119882(120593 (120585) 119890

1205852

)

100381610038161003816100381610038161003816119889120583120573 (120585) lt infin (109)

we obtain10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

) (110)

Moreover simple calculations give that

119864(120573)

2minus2119895lowast119882119906 = 119867120573 (2

minus4119895minus 119905

2) (119864

(120573)

1199052lowast119882119906)

(111)

Thus from Proposition 26 it follows that100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1

+ )

(112)

for any 119905 isin [2minus119895minus1 2minus119895] which implies that

sum

119895isinZ

2minus211990411989511990310038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

le 119862 sum

119895isinZ

int

2minus119895

2minus119895minus1(119905

2119904100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862int

infin

0

(1199052119904100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862int

infin

0

(11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

(113)

where we have used the fact that 119864(120573)1199052lowast119882119906 = 119867120573(119905

2)119906

We now prove that10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862119906Bminus2119904120573

119901119903(R119889+1+

) (114)

Indeed one has

119864(120573)

2minus2119895lowast119882119906 = sum

119899isinZ

119864(120573)

2minus2119895lowast119882Δ 119899+119895119906 (115)

Arguing as above we have100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1

+ )

(116)

for any 119905 isin [2minus119895 21minus119895] Thus10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

= 2

100381710038171003817100381710038171003817100381710038171003817

1003817100381710038171003817100381710038171199052119904119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)

100381710038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862 sum

119895isinZ

int

21minus119895

2minus119895(2

minus2119895119904100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862 sum

119895isinZ

(2minus2119895119904

sum

119899isinZ

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

(117)

10 Journal of Function Spaces and Applications

On the other hand it is easy to see that100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 1198622

minus211989911990410038171003817100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

) (118)

for any 119904 gt 0 For 0 lt 1199041 lt 119904 lt 1199042 and by using theMinkowskiinequality we have

sum

119895isinZ

(2minus2119895119904

sum

119899isinZ

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

le sum

119895isinZ

(2minus2119895119904

0

sum

minusinfin

2minus21198991199041

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

+ sum

119895isinZ

(2minus2119895119904

sum

119899isinN

2minus21198991199042

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

le 119862(

0

sum

minusinfin

2minus2119899(119904

1minus119904)(sum

119895isinZ

(2minus2(119899+119895)11990311990410038171003817

100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

)

1119903

)

119903

+ 119862(sum

N

2minus2119899(119904

2minus119904)(sum

119895isinZ

(2minus2(119899+119895)11990311990410038171003817

100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

)

1119903

)

119903

le 119862 sum

119895isinZ

2minus211989511990311990410038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

(119)

The result is immediately from (117) and (119)

42 Embedding Sobolev Theorems

Theorem 31 Let 1 lt 119901 lt infin and let 0 lt 119904 lt (119889 + 2120573 +

2)119901There exists a positive constant119862 such that for all function119891 isin

H119904

119901120573(R119889+1

+) one has

10038171003817100381710038171198911003817100381710038171003817119871119902

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

H119904119901120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

120579

B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

) (120)

where 120579 = 119904119901(119889+2120573+2) and 119902 = 119901(119889+2120573+2)(119889+2120573+2minus119901119904)

Proof Bydensity we can suppose that119891belongs toSlowast(R119889+1

)It is easy to see that

119891 = int

infin

0

119867120573 (119905) Δ 120573119891119889119905(121)

and decompose the integral in two parts as follows

119891 = int

119860

0

119867120573 (119905) Δ 120573119891119889119905 + int

infin

119860

119867120573 (119905) Δ 120573119891119889119905(122)

where 119860 is a constant to be fixed laterOn the other hand byTheorem 29 we obtain10038171003817100381710038171003817119867120573 (119905) Δ 120573119891

10038171003817100381710038171003817119871infin120573(R119889+1+

)

le

119862

1199051minus(12)(119904minus(119889+2120573+2)119901)

10038171003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(123)

Therefore after integrating we get

int

infin

119860

10038171003817100381710038171003817119867120573 (119905) Δ 120573119891

10038171003817100381710038171003817119871infin120573(R119889+1+

)119889119905

le 119860(12)(119904minus(119889+2120573+2)119901)1003817

1003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(124)

On the other hand denoting 119892 = (minusΔ 120573)1199042119891 we have

119867120573 (119905) Δ 120573119891 =

1

(minus119905)1minus1199042

119867120573 (119905) (minus119905Δ 120573)

1minus1199042

119892 (125)

We proceed as in [8] we prove that1003816100381610038161003816100381610038161003816

119867120573 (119905) (minus119905Δ 120573)

1minus1199042

119892 (119909)

1003816100381610038161003816100381610038161003816

le 119862 (119904)119872120573 (119892) (119909) (126)

where119872120573(119892) is a maximal function of 119892 associated with theWeinstein operators (cf [12])

This leads to100381610038161003816100381610038161003816100381610038161003816

int

119860

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

100381610038161003816100381610038161003816100381610038161003816

le 1198621198601199042119872120573 (119892) (119909) (127)

In conclusion we get10038161003816100381610038161003816100381610038161003816

int

infin

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

10038161003816100381610038161003816100381610038161003816

le 119862 (1198601199042119872120573 (119892) (119909)

+119860(12)(119904minus(119889+2120573+2)119901)1003817

1003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

))

(128)

and the choice of 119860 such that

119860(119889+2120573+2)2119901

119872120573 (119892) (119909) =10038171003817100381710038171198911003817100381710038171003817B119904minus(119889+2120573+2)119901120573

infininfin (R119889+1+

)(129)

ensures that10038161003816100381610038161003816100381610038161003816

int

infin

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

10038161003816100381610038161003816100381610038161003816

le 119862(119872120573(119892)(119909))

1minus(119901119904(119889+2120573+2))10038171003817100381710038171198911003817100381710038171003817

119901119904(119889+2120573+2)

B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(130)

Finally taking the 119871119902120573norm with 119902 = 119901(119889 + 2120573 + 2)(119889 + 2120573 +

2minus119901119904) ends the proof thanks to the fact themaximal function119872120573 is bounded of 119871119902

120573(R119889+1

+) into itself for 119902 gt 1

Theorem 32 Let 1 lt 119901 lt 119902 lt infin For all function 119891 such that119891 isin

H1199041

119901120573(R119889+1

+)⋂

Bminus120573120573

infininfin(R119889+1

+) one has

10038171003817100381710038171198911003817100381710038171003817H119904119901120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

120579

H1199041

119901120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573120573

infininfin(R119889+1

+) (131)

where 120579 = 119901119902 119904 = 1205791199041 minus (1 minus 120579)120573 with 120573 gt 0 minus120573 lt 119904 lt 1199041

Proof It suffices to prove that1003817100381710038171003817100381710038171003817

(minusΔ 120573)

(119904minus1199041)2

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le 11986210038171003817100381710038171198911003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(132)

Journal of Function Spaces and Applications 11

Indeed we use the following identity (which may be easilyproven by taking the Weinstein transform in 119909 of both sides)

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

infin

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905 (133)

with 120575 = 1199041 minus 119904 gt 0We decompose the integral in two parts as follows

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

119879

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

+

1

Γ (1205752)

int

infin

119879

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

(134)

where 119879 is a constant to be fixed laterWe proceed as in [8] we obtain

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119872120573 (119891) (119909) (135)

On the other hand we use Theorem 29 and the fact that 119891belongs to Bminus120573minus119904

1120573

infininfin(R119889+1

+) to deduce that

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119905

(minus120573minus1199041)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

) (136)

Thus by applying the preceding estimates on the right part of(134) we obtain

1003816100381610038161003816100381610038161003816

(minusΔ 120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621

Γ (1205752)

1198791205752119872120573 (119891) (119909)

+

1198622

Γ (1205752)

119879(120575minus120573minus119904

1)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

(137)

We fix now

119879 = (

10038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

119872120573 (119891) (119909)

)

2(120573+1199041)

(138)

We obtain

1003816100381610038161003816100381610038161003816

(minusΔ120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621 + 1198622

Γ (1205752)

(119872120573(119891)(119909))

12057910038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(139)

Thus we deduce that1003817100381710038171003817100381710038171003817

(minusΔ120573)

minus1205752

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le

1198621 + 1198622

Γ (1205752)

10038171003817100381710038171003817119872120573 (119891)

10038171003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(140)

To conclude we used the fact that the maximal function119872120573

is bounded of 119871119902120573(R119889+1

+) into itself for 119902 gt 1

43 Estimates in Generalized Besov Spaces For any interval 119868ofR (bounded or unbounded) and a normed space 119865(R119889+1

+)

we define the mixed space-time 119871119901(119868 119865(R119889+1

+)) space of

(classes of) measurable functions 119906 119868 rarr 119865(R119889+1

+) such that

||119906||119871119901(119868119865(R119889+1+

)) lt infin with

119906119871119901(119868119865(R119889+1+

)) = (int

119868

119906 (119905 sdot)119901

119865(R119889+1+

)119889119905)

1119901

if 1 le 119901 lt infin

119906119871infin(119868119865(R119889+1+

)) = ess sup119905isin119868

119906 (119905 sdot)119865(R119889+1+

)

(141)

For any interval 119868 of R (bounded or unbounded) anda Banach space 119883 we define the mixed space-time 119862(119868119883)space of continuous functions 119868 rarr 119883 When 119868 is bounded119862(119868119883) is a Banach space with the norm of 119871infin(119868 119883)

Theorem 33 Let 119904 isin R and 1 le 119901 119902 119903 le infin Let 119879 gt 0 119892 isin

B119904120573

119901119903(R119889+1

+) and119891 in 119871119902((0 119879) B

119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76)

has a unique solution

119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903(R

119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R

119889+1

+))

(142)

and there exists a constant 119862 such that for all 1199021 isin [119902infin] onehas

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862(10038171003817100381710038171198921003817100381710038171003817B119904120573

119901119903(R119889+1+

)+10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)B

119904minus2+(2119902)120573

119901119903(R119889+1+

)))

(143)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] B119904120573

119901119903(R119889+1

+))

Proof Since 119892 and 119891 are temperate distributions (76) has aunique solution 119906 in S1015840

((0 119879) timesR119889+1

+) which satisfies

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(144)

Next we notice that applying Δ 119895 to (76) and using formula(81) yield

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(145)

Therefore

10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(146)

12 Journal of Function Spaces and Applications

By virtue of Lemma 30 we thus have for some 120581 gt 0

10038171003817100381710038171003817Δ 119895119906(119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

119890minus12058122119895(119905minus120591) 10038171003817

100381710038171003817Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(147)

Applying convolution inequalities we get

10038171003817100381710038171003817Δ 119895119906

100381710038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862[

[

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

)

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

]

]

(148)

with 11199022 = 1+ 11199021 minus1119902 Finally taking the 119897119903(Z) norm we

conclude that (with the usual convention if 119903 = infin)

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

1199031199021

)(211989511990410038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

]

]

1119903

+ 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

1199031199022

)

times (2119895(119904minus2+2119902)10038171003817

100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902(0119879119871

119901

120573(R119889+1+

))

119903

]

]

1119903

(149)

which insures that 119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903 (R119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R119889+1

+)) and yields the desired inequality

Since 119906 belongs to 119862([0 119879]B119904120573

119901119903(R119889+1

+)) in the case where

119903 is finite may be easily deduced from the density ofSlowast(R

119889+1)⋂

B119904120573

119901119903(R119889+1

+) in B

119904120573

119901119903(R)

Theorem 34 Let 119904 isin R 119879 gt 0 and 1 le 119901 119902 119903 le infinOne supposes that 119892 isin 119861

119904120573

119901119903(R119889+1

+) and 119891 isin 119871

119902((0 119879)

119861119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76) has a unique solution 119906 belong-

ing to

119871119902((0 119879) 119861

119904+(2119902)120573

119901119903(R

119889+1

+))⋂119871

infin((0 119879) 119861

119904120573

119901119903(R

119889+1

+))

(150)

and there exists a constant 119862 such that for all 119902 le 1199021 le infin

1199061198711199021 ((0119879)119861

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862 [ (1 + 11987911199021)10038171003817100381710038171198921003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (1 + 1198791+1119902

1minus1119902

)10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)119861

119904minus2+(2119902)120573

119901119903(R119889+1+

))]

(151)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] 119861119904120573119901119903(R119889+1

+))

Proof Since 119892 119891 are tempered (76) has a unique solution 119906in S1015840

((0 119879) timesR119889+1

+) satisfying

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(152)

Hence applying Δ 119895 119895 ge 0 to (81) we see that

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(153)

and thus by Lemma 30 we can deduce that10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+int

119905

0

119890minus12058122119895(119905minus120591)10038171003817

100381710038171003817Δ 119895119891(120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(154)

Then it follows from convolution inequalities thatΔ 1198951199061198711199021 ((0119879)119871

119901

120573(R119889+1+

))is dominated by

(

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

(155)

with 11199022 = 1 + 11199021 minus 1119902 Moreover similarly as above wecan obtain that

1003817100381710038171003817Δminus1119906(119905 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)le1003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

1003817100381710038171003817Δminus1119891 (120591 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(156)

and thus if 1 le 119902 le 1199021 le infin1003817100381710038171003817Δminus1119906

10038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862(119879111990211003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)+ 119879

111990221003817100381710038171003817Δminus1119891

1003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

)))

(157)

Journal of Function Spaces and Applications 13

Finally taking the 119897119903-normwith respect to 119895 in (155) and (157)with the usual convention if 119903 = infin we can deduce the desiredestimate

Acknowledgment

Theauthor gratefully acknowledges theDeanship of ScientificResearch at the University of Taibah The author is deeplyindebted to the referee for providing constructive commentsand help in improving the contents of this paper

References

[1] Z Ben Nahia and N Ben Salem ldquoSpherical harmonics andapplications associated with the Weinstein operatorrdquo in Pro-ceedings of the International Conference on PotentialTheory heldin Kouty Czech Republic (ICPT rsquo94) pp 235ndash241 1996

[2] Z Ben Nahia and N Ben Salem ldquoOn a mean value propertyassociated with the Weinstein operatorrdquo in Proceedings of theInternational Conference on Potential Theory held in KoutyCzech Republic (ICPT rsquo94) pp 243ndash253 1996

[3] M Brelot ldquoEquation de Weinstein et potentiels de MarcelRieszrdquo in Seminaire de Theorie de Potentiel Paris No 3 vol 681of Lecture Notes in Mathematics pp 18ndash38 Springer BerlinGermany 1978

[4] H Mejjaoli and M Salhi ldquoUncertainty principles for theweinstein transformrdquo Czechoslovak Mathematical Journal vol61 no 4 pp 941ndash974 2011

[5] H Mejjaoli and A Ould Ahmed Salem ldquoWeinstein Gabortransform and applicationsrdquo Advanced Studies in Pure Mathe-matics vol 2 no 3 pp 203ndash210 2012

[6] H Mejjaoli ldquoBesov spaces associated withthe Weinstein opera-tor and applicationsrdquo In press

[7] T Kawazoe and H Mejjaoli ldquoGeneralized Besov spaces andtheir applicationsrdquo Tokyo Journal of Mathematics vol 35 no 2pp 297ndash320 2012

[8] H Mejjaoli ldquoLittlewood-Paley decomposition associated withthe Dunkl operators and paraproduct operatorsrdquo Journal ofInequalities in Pure and Applied Mathematics vol 9 no 4 pp1ndash25 2008

[9] H Mejjaoli and N Sraeib ldquoGeneralized sobolev spaces inquantum calculus and applicationsrdquo Journal of Inequalities andSpecial Functions vol 1 no 4 pp 43ndash64 2012

[10] H Mejjaoli ldquoGeneralized homogeneous Besov spaces and theirapplicationsrdquo Serdica Mathematical Journal vol 38 no 4 pp575ndash614 2012

[11] H Triebel Interpolation Theory Functions Spaces DifferentialOperators North-Holland AmsterdamThe Netherlands 1978

[12] V S Guliev ldquoOn maximal function and fractional integralassociated with the Bessel differential operatorrdquo MathematicalInequalities and Applications vol 6 no 2 pp 317ndash330 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Function Spaces and Applications

Proposition 5 Let 119891 be in 119871119901

120573(R119889+1

+) 1 le 119901 le infin and 120601

in Slowast(R119889+1

) Then the distribution T119891lowast119882120601 is given by thefunction 119891lowast119882120601 and T119891lowast119882120601 belongs to 119871119901

120573(R119889+1

+) Moreover

for all 120595 isin Slowast(R119889+1

)

⟨T119891lowast119882120601 120595⟩ = ⟨

119891 120601lowast119882 ⟩120573 (34)

where (119909) = 120595(minus119909) and

F119882 (T119891lowast119882 120601) = F119882 (T119891)F119882 (120601) (35)

For each 119906 isin S1015840

lowast(R119889+1

) we define the distribution Δ 120573119906

by ⟨Δ 120573119906 120595⟩ = ⟨119906 Δ 120573120595⟩ and this distribution satisfies thefollowing property

F119882 (Δ 120573119906) = minus10038171003817100381710038171199101003817100381710038171003817

2F119882 (119906) (36)

In the following we denote T119891 given by (30) by 119891 forsimplicity

3 B119904120573

119901119902 F119904120573

119901119902(R119889+1

+) H119904

119901120573Spaces and

Basic Properties

31 HomogeneousWeinstein-Littlewood-Paley DecompositionOne of the main tools in this paper is the homogeneousLittlewood-Paley decomposition of distribution associatedwith theWeinstein operators into dyadic blocs of frequencies

Lemma6 One defines byC the ring of center 0 of small radius12 and great radius 2 There exists two radial functions 120595 and120593 the values of which are in the interval [0 1] belonging to119863lowast(R

119889+1) such that

supp120595 sub 119861 (0 1) supp120593 sub C

forall120585 isin R119889+1

120595 (120585) + sum

119895ge0

120593 (2minus119895120585) = 1

forall120585 isin C sum

119895isinZ

120593 (2minus119895120585) = 1

|119899 minus 119898| ge 2 997904rArr supp120593 (2minus119899sdot) cap supp120593 (2minus119898sdot) = 0

119895 ge 1 997904rArr supp120595 cap supp120593 (2minus119895sdot) = 0

(37)

Notations We denote by

Δ 119895119891 = Fminus1

119882(120593(

120585

2119895)F119882 (119891))

119878119895119891 = sum

119899le119895minus1

Δ 119899119891 forall119895 isin Z(38)

The distribution Δ 119895119891 is called the 119895th dyadic block of thehomogeneous Littlewood-Paley decomposition of 119891 associ-ated with the Weinstein operators

Throughout this paper we define 120601 and 120594 by 120601 = Fminus1

119882(120593)

and 120594 = Fminus1

119882(120595)

When dealing with the Littlewood-Paley decompositionit is convenient to introduce the functions and 120593 belongingto119863lowast(R

119889+1) such that equiv 1 on supp120595 and 120593 equiv 1 on supp120593

Remark 7 We remark that

F119882 (119878119895119891) (120585) = (

120585

2119895)F119882 (119878119895119891) (120585)

F119882 (Δ 119895119891) (120585) = 120593(

120585

2119895)F119882 (Δ 119895119891) (120585)

(39)

We put

120601 = F

minus1

119882(120593) 120594 = F

minus1

119882() (40)

Definition 8 One denotes by S1015840

ℎ120573lowast(R119889+1

) the space oftempered distribution such that

lim119895rarrminusinfin

119878119895119906 = 0 in S1015840

lowast(R

119889+1) (41)

Proposition 9 (Bernstein inequalities) For all 120583 isin N119889+1 and120590 isin R for all 119895 isin Z for all 1 le 119901 119902 le infin and for all119891 isin S1015840

lowast(R119889+1

) one has the following

(i) 10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902

120573(R119889+1+

)

le

10038171003817100381710038171003817

120601

10038171003817100381710038171003817119871119903120573(R119889+1+

)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

)2119895(119889+2120573+2)(1119901minus1119902)

with 1

119902

=

1

119901

+

1

119903

minus 1

(ii) 10038171003817100381710038171003817119878119895119891

10038171003817100381710038171003817119871119902

120573(R119889+1+

)

le10038171003817100381710038171205941003817100381710038171003817119871119903120573(R119889+1+

)

10038171003817100381710038171003817119878119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

)2119895(119889+2120573+2)(1119901minus1119902)

with 1

119902

=

1

119901

+

1

119903

minus 1

(iii)1003817100381710038171003817100381710038171003817

(radicminusΔ 120573)

120590

Δ 119895119891

1003817100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)

le

10038171003817100381710038171003817F

minus1

119882(10038171003817100381710038171205851003817100381710038171003817

120590120593)

100381710038171003817100381710038171198711120573(R119889+1+

)

times

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

)2119895120590

(42)

Proof Using Remark 7 we deduce from Proposition 5 that

119878119895119891 = 2119895(119889+2120573+2)

120594 (2119895sdot) lowast119882119878119895119891

Δ 119895119891 = 2119895(119889+2120573+2)

120601 (2119895sdot) lowast119882Δ 119895119891

(43)

Thus from the relation (29) we prove (i) (ii) and (iii)

32 Definitions In the following we define analogues of thehomogeneous Besov Triebel-Lizorkin and Riesz potentialspaces associated with the Weinstein operators on R119889+1

+and

obtain their basic propertiesFrom now we make the convention that for all non-

negative sequence 119886119902119902isinZ the notation (sum119902119886119903

119902)1119903 stands for

sup119902119886119902 in the case 119903 = infin

Journal of Function Spaces and Applications 5

Definition 10 Let 119904 isin R and 119901 119902 isin [1infin]The homogeneousWeinstein-Besov spaces B

119904120573

119901119902(R119889+1

+) are the spaces of distri-

bution in S1015840

ℎ120573lowast(R119889+1

) such that

10038171003817100381710038171198911003817100381710038171003817B119904120573

119901119902(R119889+1+

)= (sum

119895isinZ

(211990411989510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119902

)

1119902

lt infin (44)

Proposition 11 (see [6]) Let 119904 isin R and 119901 and 119902 two elementsof [1infin] the space B

119904120573

119901119902(R119889+1

+) is the set of 119891 isin S1015840

ℎ120573lowast(R119889+1

)

verifying

10038171003817100381710038171003817

119891

10038171003817100381710038171003817B119904120573

119901119902(R119889+1+

)= (int

infin

0

(119905minus1199041003817100381710038171003817119891lowast119882120601119905

1003817100381710038171003817119871119901

120573(R119889+1+

))

119902119889119905

119905

)

1119902

lt infin

(45)

where 120601119905(119909) = (11199052120573+2+119889

)120601(119909119905) for all 119905 isin (0infin) and 119909 isin

R119889+1

+

Definition 12 For 119904 isin R and 119901 119902 isin [1infin] one writes10038171003817100381710038171198911003817100381710038171003817119861119904120573

119901119902(R119889+1+

)=10038171003817100381710038171198780119891

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ (sum

119895ge1

(211990411989510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119902

)

1119902

(46)

The nonhomogeneous Besov space 119861119904120573119901119902(R119889+1

+) associated

with the Weinstein operators is defined by

119861119904120573

119901119902(R

119889+1

+) = 119891 isin S

1015840(R

119889)

10038171003817100381710038171198911003817100381710038171003817119861119904120573

119901119902(R119889+1+

)lt infin (47)

We give now another definition equivalent to the nonho-mogeneous Besov space 119861119904120573119901119902(R119889+1

+)

Proposition 13 Let 119904 isin R and119901 and 119902 two elements of [1infin]the space 119861119904120573119901119902(R119889+1

+) is the set of 119891 isin S1015840

(R119889) verifying

10038171003817100381710038171003817

119891

10038171003817100381710038171003817119861119904120573

119901119902(R119889+1+

)=1003817100381710038171003817119891lowast119882120595

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ (int

1

0

(119905minus1199041003817100381710038171003817119891lowast119882120601119905

1003817100381710038171003817119871119901

120573(R119889+1+

))

119902119889119905

119905

)

1119902

lt infin

(48)

Definition 14 Let 119904 isin R and 1 le 119901 119902 le infin the homogeneousWeinstein-Triebel-Lizorkin space F

119904120573

119901119902(R119889+1

+) is the space of

distribution in S1015840

ℎ120573lowast(R119889+1

) such that

10038171003817100381710038171198911003817100381710038171003817F119904120573

119901119902(R119889+1+

)=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

211990411989511990210038161003816100381610038161003816Δ 119895119891

10038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)

lt infin (49)

Definition 15 For 119904 isin R the operatorR119904

120573from S1015840

ℎ120573lowast(R119889+1

)

to S1015840

ℎ120573lowast(R119889+1

) is defined by

R119904

120573(119891) = F

minus1

119882(sdot

119904F119882119891) (50)

The operatorRminus119904

120573is called Weinstein-Riesz potential space

Definition 16 For 119904 isin R and 1 le 119901 le infin the homogeneousWeinstein-Riesz potential space H119904

119901120573(R119889+1

+) is defined

as the space Rminus119904

120573(119871

119901

120573(R119889+1

+)) equipped with the norm

119891H119904119901120573(R119889+1+

)= R119904

120573(119891)

119871119901

120573(R119889+1+

)

Proposition 17 Let 119904 isin R and 1 le 119901 119902 le infinThe operator Δ 120573 is a linear continuous operator from

B119904120573

119901119902(R119889+1

+) into B

119904minus2120573

119901119902 (R119889+1

+) and from H119904

119901120573(R119889+1

+) into

H119904minus2

119901120573(R119889+1

+)

Proof We obtain these results by the similar ideas used in thenonhomogeneous case (cf [6])

Proposition 18 Let 119904 119905 isin R and 1 le 119901 119902 le infin The operatorR119905

120573is a linear continuous injective operator from B

119904120573

119901119902(R119889+1

+)

onto B119904minus119905120573

119901119902 (R119889+1

+) and from H119904

119901120573(R119889+1

+) onto H119904minus119905

119901120573(R119889+1

+)

Proof We obtain these results by the similar ideas used in thenonhomogeneous case (cf [6])

33 Embeddings As in the Euclidean case (cf [11]) themono-tone character of 119897119902-spaces and the Minkowskis inequalityyield the following

Proposition 19 If 1 le 1199021 lt 1199022 le infin one has

B119904120573

1199011199021

(R119889+1

+) 997893rarr

B119904120573

1199011199022

(R119889+1

+) (1 le 119901 le infin 119904 isin R)

(51)

MoreoverB119904120573

1199011(R

119889+1

+) 997893rarr

H119904

119901120573(R

119889+1

+) 997893rarr

B119904120573

119901infin(R

119889+1

+)

(1 le 119901 le infin 119904 isin R)

(52)

If 1199040 = 1199041 one also has

(H1199040

119901120573(R

119889+1

+)

H1199041

119901120573(R

119889+1

+))

120579119902=

B119904120573

119901119902(R

119889+1

+)

(1 le 119901 119902 le infin 120579 isin (0 1))

(53)

where 119904 = (1 minus 120579)1199040 + 1205791199041

Proposition 20 One assumes that 119904 minus (119889 + 2120573 + 2)119901 = 1199041 minus

(119889 + 2120573 + 2)1199011 Then the following inclusion holds

B119904120573

119901119902(R

119889+1

+) 997893rarr

B1199041120573

11990111199021

(R119889+1

+)

(1 le 119901 le 1199011 le infin 1 le 119902 le 1199021 le infin 119904 1199041 isin R)

(54)

Proof In order to prove the inclusion we use the estimate

Δ 119895119891 = 2119895(119889+2120573+2)

120601 (2119895sdot) lowast119882Δ 119895119891 (55)

Proposition 9(i) gives that10038171003817100381710038171003817Δ 119895119891

100381710038171003817100381710038171198711199011

120573(R119889+1+

)=

100381710038171003817100381710038172119895(119889+2120573+2)

120601 (2119895sdot) lowast119882Δ 119895119891

100381710038171003817100381710038171198711199011

120573(R119889+1+

)

le 1198622119895(119889+2120573+2)(1119901minus1119901

1)10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

(56)

6 Journal of Function Spaces and Applications

By definition of the homogeneous Weinstein-Besov spaceswe therefore infer10038171003817100381710038171198911003817100381710038171003817B1199041120573

11990111199021(R119889+1+

)

= (

infin

sum

119895=minusinfin

(2119895119904110038171003817100381710038171003817Δ 119895119891

100381710038171003817100381710038171198711199011

120573(R119889+1+

))

1199021

)

11199021

le 119862(sum

119895isinZ

(211989511990412119895(119889+2120573+2)(1119901minus1119901

1)10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

1199021

)

11199021

le 119862(sum

119895isinZ

(211989511990410038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

1199021

)

11199021

le 11986210038171003817100381710038171198911003817100381710038171003817B119904120573

119901119902(R119889+1+

)

(57)

since 119902 le 1199021 This gives the inclusion

Proposition 21 (1) If 119906 belongs to B119904120573

119901119902(R119889+1

+) cap

B119905120573

119901119902(R119889+1

+)

then 119906 belongs to B120579119904+(1minus120579)119905120573

119901119902 (R119889+1

+) for all 120579 isin [0 1] and

119906B120579119904+(1minus120579)119905120573

119901119902(R119889+1+

)le 119906

120579

B119904120573

119901119902(R119889+1+

)119906

1minus120579

B119905120573

119901119902(R119889+1+

) (58)

(2) If 119906 belongs to B119904120573

119901infin(R119889+1

+) cap

B119905120573

119901infin(R119889+1

+) and 119904 lt 119905

then 119906 belongs to B120579119904+(1minus120579)119905120573

1199011(R119889+1

+) for all 120579 isin (0 1) and there

exists a positive constant 119862(119905 119904) such that

119906B120579119904+(1minus120579)119905120573

1199011(R119889+1+

)le 119862 (119905 119904) 119906

120579

B119904120573

119901infin(R119889+1+

)119906

1minus120579

B119905120573

119901infin(R119889+1+

) (59)

(3) If 119906 belongs to B119904120573

119901infin(R119889+1

+) cap

B119904+120576120573

119901infin (R119889+1

+) and 120576 gt

0 then 119906 belongs to B119904120573

1199011(R119889+1

+) and there exists a positive

constant 119862 such that

119906B119904120573

1199011(R119889+1+

)le

119862

120576

119906B119904120573

119901infin(R119889+1+

)log

2(119890 +

119906B119904+120576120573

119901infin(R119889+1+

)

119906B119904120573

119901infin(R119889+1+

)

)

(60)

Proof (1) is obvious from the Holderrsquos inequality As for (2)we write 119906

B120579119904+(1minus120579)119905120573

1199011(R119889+1+

)as

sum

119895le119873

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ sum

119895gt119873

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

(61)

where 119873 is chosen here after By the definition of thehomogeneous Weinstein-Besov norms we see that

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 2

119895(1minus120579)(119905minus119904)119906

B119904120573

119901infin(R119889+1+

)

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 2

minus119895120579(119905minus119904)119906

B119905120573

119901infin(R119889+1+

)

(62)

and thus 119906B120579119904+(1minus120579)119905120573

1199011(R119889+1+

)is dominated by

119906B119904120573

119901infin(R119889+1+

)sum

119895le119873

2119895(1minus120579)(119905minus119904)

+ 119906B119905120573

119901infin(R119889+1+

)sum

119895gt119873

2minus119895120579(119905minus119904)

le 119862119906B119904120573

119901infin(R119889+1+

)

2(119873+1)(1minus120579)(119905minus119904)

2(1minus120579)(119905minus119904)

minus 1

+ 119906B119905120573

119901infin(R119889+1+

)

2minus119873120579(119905minus119904)

1 minus 2minus120579(119905minus119904)

(63)

Hence in order to complete the proof of (2) it suffices tochoose119873 such that

119906B119905120573

119901infin(R119889)

119906B119904120573

119901infin(R119889)

le 2119873(119905minus119904)

lt 2

119906B119905120573

119901infin(R119889)

119906B119904120573

119901infin(R119889)

(64)

As for (3) it is easy to see that 119906B119904120573

1199011(R119889+1+

)is dominated as

sum

119895le119873minus1

211989511990410038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ sum

119895ge119873

211989511990410038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le (119873 + 1) 119906B119904120573

119901infin(R119889+1+

)+

2minus(119873minus1)120576

2120576minus 1

119906B119904+120576120573

119901infin(R119889+1+

)

(65)

Hence letting

119873 = 1 + [

[

1

120576

log2

119906B119904+120576120573

119901infin(R119889+1+

)

119906B119904120573

119901infin(R119889+1+

)

]

]

(66)

we can obtain the desired estimate

Proposition 22 Let 119902 isin (1infin) and let 119904 isin R such that0 lt 119904 lt (119889 + 2120573 + 2)119902 then one has

B119904120573

119902119902(R

119889+1

+) =

F119904120573

119902119902(R

119889+1

+) 997893rarr

F119904120573

119902infin(R

119889+1

+)

997893rarrF119904minus((119889+2120573+2)119902)120573

infininfin(R

119889+1

+)

(67)

H119904

119902120573(R

119889+1

+) =

F119904120573

1199022(R

119889+1

+) 997893rarr

F119904120573

119902infin(R

119889+1

+)

997893rarrF119904minus((119889+2120573+2)119902)120573

infininfin(R

119889+1

+)

(68)

Proof We obtain these results by the similar ideas used in thenonhomogeneous case (cf [6])

Theorem 23 Let 119886 119887 gt 0 and let 1199021 1199022 isin [1infin] Let120579 = 119886(119886 + 119887) isin (0 1) and let 1119901 = (1 minus 120579)1199021 +

1205791199022 Then there exists a constant 119862 such that for every119891 isin

F119886120573

1199021infin(R119889+1

+) cap

Fminus119887120573

1199022infin(R119889+1

+) then one has

1003816100381610038161003816119891 (119909)

1003816100381610038161003816le 119862(sup

119895isinZ

2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

1minus120579

(sup119895isinZ

2minus119887119895 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

120579

(69)

In particular one gets

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

F119886120573

1199021infin(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

120579

Fminus119887120573

1199022infin(R119889+1+

) (70)

Journal of Function Spaces and Applications 7

Proof Let 119891 be a Schwartz class we have

1003816100381610038161003816119891 (119909)

1003816100381610038161003816le sum

119895isinZ

10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816

le sum

119895isinZ

min(2minus119886119895sup119895isinZ

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816))

(71)

We define119873(119909) as the largest index such that

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816) le 2

minus119886119895sup119895isinZ

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816) (72)

and we write1003816100381610038161003816119891 (119909)

1003816100381610038161003816le sum

119895le119873(119909)

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

+ sum

119895gt119873(119909)

2minus119886119895sup

119895

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

le 119862(sup119895isinZ

2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

119887(119886+119887)

times (sup119895isinZ

2minus119887119895 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

119886(119886+119887)

(73)

Thus (69) is proved In order to obtain (70) it is enough toapply the Holder inequality in the expression above since wehave 120579 = 119886(119886+119887) isin (0 1) and let 1119901 = (1minus120579)1199021+1205791199022

Corollary 24 Let 119902 isin (1infin) and let 119904 isin R such that 0 lt 119904 lt

(119889 + 2120573 + 2)119902 then one has

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus(119902119901)

Bminus((2120573+2+119889)119902minus119904)120573

infininfin (R119889+1

+ )

10038171003817100381710038171198911003817100381710038171003817

119902119901

B119904120573

119902119902(R119889+1

+ )

(74)

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus(119902119901)

Bminus((2120573+2+119889)119902minus119904)120573

infininfin (R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

119902119901

H119904119902120573(R119889+1+

) (75)

where 119901 = 119902(2120573 + 2 + 119889)(2120573 + 2 + 119889 minus 119902119904)

Proof By choosing 119886 = 119904 gt 0 minus119887 = 119904 minus (119889 + 2120573 + 2)119902 lt 01199021 = 119902 and 1199022 = infin we deduce (74) from the relations (70)and (67) In the same way we deduce (75) from the relations(70) and (68)

4 Generalized Heat Equation

41 Characterization for the Weinstein-Besov Spaces TheWeinstein heat equation reads

120597119905119906 (119905 119909) minus Δ 120573119906 (119905 119909) = 119891 (119905 119909) (119905 119909) isin [0infin) timesR119889+1

+

119906|119905=0 = 119892

(76)

We introduce the Weinstein heat semigroup 119867120573(119905) for theWeinstein-Laplace operator

119867120573 (119905) 119892 (119909) =

int

R119889+1+

Γ120573 (119905 119909 119910) 119892 (119910) 119889120583120573 (119910) if 119905 gt 0

119892 (119909) if 119905 = 0(77)

where Γ120573 is the Weinstein heat kernel defined by

Γ120573 (119905 119909 119910) = 120591119909 (119864(120573)

119905) (119910) (78)

where

119864(120573)

119905(119910) =

2

1205871198892Γ (120573 + 1) (4119905)

120573+1+1198892119890minus11991024119905 (79)

Thus

119867120573 (119905) 119892 (119909) = 119892lowast119882119864(120573)

119905 (119909) (80)

In practice we use the integral formulation of (76)

119906 (119905 119909) = 119867120573 (119905) 119892 (119909) + 119866120573 (119891) (119905 119909)

= 119867120573 (119905) 119892 (119909) + int

119905

0

119867120573 (119905 minus 119904) 119891 (119904 119909) 119889119904

(81)

Remark 25 The function 119864(120573)119905

is the Gauss kernel associatedwith Weinstein operators This function satisfies

forall120585 isin R119889+1

+ F119882 (119864

(120573)

119905) (120585) = 119890

minus1199051205852

(82)

Proposition 26 Let 1 le 119901 le 119903 le infin and let 119891 isin 119871119901

120573(R119889+1

+)

Then the operator 119867120573(119905) maps 119871119901

120573(R119889+1

+) continuously to

119871119903

120573(R119889+1

+) and

10038171003817100381710038171003817119867120573 (119905) 119891

10038171003817100381710038171003817119871119903120573(R119889)

le 119862119905minus((119889+2120573+2)2)(1119901minus1119903)1003817

1003817100381710038171198911003817100381710038171003817119871119901

120573(R119889)

(83)

Moreover1003817100381710038171003817100381710038171003817

(minusΔ 120573)

1205752

119867120573 (119905) 119891

1003817100381710038171003817100381710038171003817119871119903120573(R119889)

le 119862119905minus1205752minus((119889+2120573+2)2)(1119901minus1119903)1003817

1003817100381710038171198911003817100381710038171003817119871119901

120573(R119889)

(84)

for all 120575 gt 0

Proof It follows from the relations (80) and (29) combinedwith scaling property of the kernel 119864(120573)

119905

In this section we prove estimates for the Weinstein heatsemigroupThese estimates are based on the following result

Lemma 27 Let C be an annulus Positive constants 119888 and 119862exist such that for any 119901 in [1infin] and any couple (119905 120582) ofpositive real numbers one has

suppF119882 (119906) sub 120582C 997904rArr

10038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le 119862119890minus1198881199051205822

119906119871119901

120573(R119889+1+

)

(85)

8 Journal of Function Spaces and Applications

Proof We again consider a function Θ in 119863(R119889+1

+ 0) the

value of which is identically 1 in neighborhood of annulusCWe can also assume without loss of generality that 120582 = 1 Wethen have

119867120573 (119905) 119906 = 119892 (119905 sdot) lowast119882119906 (86)where

119892 (119905 sdot) = Fminus1

119882(Θ (120585) 119890

minus1199051205852

) (87)

The lemma is proved provided that we can find positive realnumbers 119888 and 119862 such that

forall119905 gt 01003817100381710038171003817119892 (119905 sdot)

10038171003817100381710038171198711120573(R119889+1+

)le 119862119890

minus119888119905 (88)

To begin we perform integrations by parts in (87) We get1003816100381610038161003816119892 (119905 119909)

1003816100381610038161003816=

1

119888120573

(1 + 1199092)

minus(119889+[2120573]+3)

times int

R119889+1+

Λ (119909 120585) (119868119889 minus Δ 120573)

119889+[2120573]+3

times (Θ (120585) 119890minus1199051205852

) 119889120583120573 (120585)

(89)

Using Leibnizrsquos formula we obtain1003816100381610038161003816119892 (119905 119909)

1003816100381610038161003816le 119862(1 + 119909

2)

minus(119889+[2120573]+3)

119890minus119888119905 (90)

and (88) follows

For any interval 119868 of R (bounded or unbounded) wedefine themixed space-time 119871119901(119868 119871119902

120573(R119889+1

+)) Banach space of

(classes of) measurable functions 119906 119868 rarr 119871119902

120573(R119889+1

+) such

that 119906119871119901(119868119871

119902

120573(R119889+1+

))lt infin with

119906119871119901(119868119871119902

120573(R119889+1+

))= (int

119868

119906 (119905 sdot)119901

119871119902

120573(R119889+1+

)119889119905)

1119901

if 1 le 119901 119902 lt infin

119906119871infin(119868119871119902

120573(R119889+1+

))= ess sup

119905isin119868

119906 (119905 sdot)119871119902

120573(R119889+1+

) if 1 le 119902 lt infin

(91)Corollary 28 Let C be an annulus and 120582 a positive realnumber Let 1199060 (resp 119891 = 119891(119905 119909)) satisfy suppF119882(1199060) sub 120582C(resp suppF119882(119891(119905 sdot)) sub 120582C for all 119905 in [0 119879]) Consider 119906 asolution of

120597119905119906 minus Δ 120573119906 = 0 119906|119905=0 = 1199060 (92)and V a solution of

120597119905V minus Δ 120573V = 119891 (119905 sdot) V|119905=0 = 0 (93)There exist positive constants 119888 and 119862 depending only on Csuch that for any 1 le 119886 le 119887 le infin and 1 le 119901 le 119902 le infin we have

119906119871119902([0119879]119871119887120573(R119889+1+

))le 119862120582

(119889+2120573+2)(1119886minus1119887)120582minus21199021003817

1003817100381710038171199060

1003817100381710038171003817119871119886120573(R119889+1+

)

V119871119902([0119879]119871119887120573(R119889+1+

))le 119862120582

minus2(1+1119902minus1119901)120582(119889+2120573+2)(1119886minus1119887)

times10038171003817100381710038171198911003817100381710038171003817119871119901([0119879]119871119886

120573(R119889+1+

))

(94)

Proof It suffices to use the fact that

119906 (119905 sdot) = 119867120573 (119905) 1199060 V (119905 sdot) = int119905

0

119867120573 (119905 minus 119904) 119891 (119904 sdot) 119889119904

(95)

Combining Lemma 27 and Youngrsquos inequality (29) withscaling property of the kernel 119864(120573)

119905now yields the result

Theorem 29 Let 119904 be a positive real number and (119901 119903) isin

[1infin]2 A constant 119862 exists which satisfies the following

property For 119906 isin Bminus2119904120573

119901119903 (R119889+1

+) one has

119862minus1119906

Bminus2119904120573

119901119903(R119889+1+

)le

10038171003817100381710038171003817100381710038171003817

10038171003817100381710038171003817119905119904119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862119906Bminus2119904120573

119901119903(R119889+1+

)

(96)

To prove this result we need the following lemma

Lemma 30 There exist two positive constants 120581 and 119862

depending only on 120593 such that for all 1 le 119901 le infin 120591 ge 0 and119895 isin Z one has

10038171003817100381710038171003817Δ 119895 (119867120573 (120591) 119906)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862119890

minus1205812211989512059110038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

) (97)

Proof The result follows immediately by applying Lemma 27and because Δ 119895(119867120573(120591))119906 = (119867120573(120591)Δ 119895)119906

Proof of Theorem 29 Using Lemma 30 and considering thefact that the operator Δ 119895 commutes with the operator 119867120573(119905)

and the definition of the homogeneous Weinstein-Besov(semi) norm we get

10038171003817100381710038171003817119905119904119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862119906

Bminus2119904120573

119901119903(R119889+1+

)sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903119895

(98)

where (119888119903119895)119895isinZ denotes as in all this proof a generic elementof the unit sphere of 119897119903(Z) In the case when 119903 = infin therequired inequality comes immediately from the followingeasy result For any positive 119904 we have

sup119905gt0

sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

lt infin (99)

In the case 119903 lt infin using the Holder inequality with theweight 22119895119904119890minus1205811199052

2119895

(99) and the Fubini theorem we obtain

int

infin

0

11990511990311990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

119889119905

119905

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)int

infin

0

(sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

)

119903minus1

times (sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903

119903119895)

119889119905

119905

Journal of Function Spaces and Applications 9

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)int

infin

0

(sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903

119903119895)

119889119905

119905

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)sum

119895isinZ

119888119903

119903119895int

infin

0

(11990511990422119895119904119890minus12058111990522119895

)

119889119905

119905

le 119862Γ (119904) 119906119903

Bminus2119904120573

119901119903(R119889+1+

)

(100)

In order to prove the other inequality let us observe thatfor any 119904 greater than minus1 we have

Δ 119895119906 =

1

Γ (119904 + 1)

int

infin

0

119905119904(minusΔ 120573)

119904+1

119867120573 (119905) Δ 119895119906 119889119905 (101)

Then Lemma 30 Proposition 9 and the fact that the operatorΔ 119895 commutes with the operator119867120573(119905) lead to the following

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862int

infin

0

11990511990422119895(119904+1)

119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889119905

(102)

In the case 119903 = infin we simply write

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862(sup

119905gt0

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

times int

infin

0

22119895(119904+1)

119890minus12058111990522119895

119889119905

le 11986222119895119904(sup119905gt0

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

(103)

In the case 119903 lt infin Holderrsquos inequality with the weight 119890minus12058111990522119895

gives

(int

infin

0

119905119904119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889119905)

119903

le 1198622minus2119895(119903minus1)

int

infin

0

119905119903119904119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)119889119905

(104)

Thanks to (99) and Fubinirsquos theorem we infer from (102) that

sum

119895isinZ

2minus211989511990311990410038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)le 119862int

infin

0

11990511990311990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

119889119905

119905

(105)

The theorem is proved

Second Proof ofTheorem 29We only consider the case 1 le 119903 ltinfinThe case 119903 = infin can be shown similarlyWe first prove that

119862minus1119906

Bminus2119904120573

119901119903(R119889+1+

)le

10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

(106)

It is easy to see that

Δ 119895119906 = 120594119895lowast119882119864(120573)

2minus2119895lowast119882119906

(107)

where 120594119895 = Fminus1

119882(120593(2

minus119895120585)119890

2minus21198951205852

) and119864(120573)2minus2119895

is the Gauss kernelassociated with Weinstein operators By relation (29) we get

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le

10038171003817100381710038171003817120594119895

100381710038171003817100381710038171198711120573(R119889+1+

)

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

) (108)

As10038171003817100381710038171003817120594119895

100381710038171003817100381710038171198711120573(R119889+1+

)= int

R119889+1+

100381610038161003816100381610038161003816F

minus1

119882(120593 (120585) 119890

1205852

)

100381610038161003816100381610038161003816119889120583120573 (120585) lt infin (109)

we obtain10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

) (110)

Moreover simple calculations give that

119864(120573)

2minus2119895lowast119882119906 = 119867120573 (2

minus4119895minus 119905

2) (119864

(120573)

1199052lowast119882119906)

(111)

Thus from Proposition 26 it follows that100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1

+ )

(112)

for any 119905 isin [2minus119895minus1 2minus119895] which implies that

sum

119895isinZ

2minus211990411989511990310038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

le 119862 sum

119895isinZ

int

2minus119895

2minus119895minus1(119905

2119904100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862int

infin

0

(1199052119904100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862int

infin

0

(11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

(113)

where we have used the fact that 119864(120573)1199052lowast119882119906 = 119867120573(119905

2)119906

We now prove that10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862119906Bminus2119904120573

119901119903(R119889+1+

) (114)

Indeed one has

119864(120573)

2minus2119895lowast119882119906 = sum

119899isinZ

119864(120573)

2minus2119895lowast119882Δ 119899+119895119906 (115)

Arguing as above we have100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1

+ )

(116)

for any 119905 isin [2minus119895 21minus119895] Thus10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

= 2

100381710038171003817100381710038171003817100381710038171003817

1003817100381710038171003817100381710038171199052119904119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)

100381710038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862 sum

119895isinZ

int

21minus119895

2minus119895(2

minus2119895119904100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862 sum

119895isinZ

(2minus2119895119904

sum

119899isinZ

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

(117)

10 Journal of Function Spaces and Applications

On the other hand it is easy to see that100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 1198622

minus211989911990410038171003817100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

) (118)

for any 119904 gt 0 For 0 lt 1199041 lt 119904 lt 1199042 and by using theMinkowskiinequality we have

sum

119895isinZ

(2minus2119895119904

sum

119899isinZ

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

le sum

119895isinZ

(2minus2119895119904

0

sum

minusinfin

2minus21198991199041

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

+ sum

119895isinZ

(2minus2119895119904

sum

119899isinN

2minus21198991199042

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

le 119862(

0

sum

minusinfin

2minus2119899(119904

1minus119904)(sum

119895isinZ

(2minus2(119899+119895)11990311990410038171003817

100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

)

1119903

)

119903

+ 119862(sum

N

2minus2119899(119904

2minus119904)(sum

119895isinZ

(2minus2(119899+119895)11990311990410038171003817

100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

)

1119903

)

119903

le 119862 sum

119895isinZ

2minus211989511990311990410038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

(119)

The result is immediately from (117) and (119)

42 Embedding Sobolev Theorems

Theorem 31 Let 1 lt 119901 lt infin and let 0 lt 119904 lt (119889 + 2120573 +

2)119901There exists a positive constant119862 such that for all function119891 isin

H119904

119901120573(R119889+1

+) one has

10038171003817100381710038171198911003817100381710038171003817119871119902

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

H119904119901120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

120579

B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

) (120)

where 120579 = 119904119901(119889+2120573+2) and 119902 = 119901(119889+2120573+2)(119889+2120573+2minus119901119904)

Proof Bydensity we can suppose that119891belongs toSlowast(R119889+1

)It is easy to see that

119891 = int

infin

0

119867120573 (119905) Δ 120573119891119889119905(121)

and decompose the integral in two parts as follows

119891 = int

119860

0

119867120573 (119905) Δ 120573119891119889119905 + int

infin

119860

119867120573 (119905) Δ 120573119891119889119905(122)

where 119860 is a constant to be fixed laterOn the other hand byTheorem 29 we obtain10038171003817100381710038171003817119867120573 (119905) Δ 120573119891

10038171003817100381710038171003817119871infin120573(R119889+1+

)

le

119862

1199051minus(12)(119904minus(119889+2120573+2)119901)

10038171003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(123)

Therefore after integrating we get

int

infin

119860

10038171003817100381710038171003817119867120573 (119905) Δ 120573119891

10038171003817100381710038171003817119871infin120573(R119889+1+

)119889119905

le 119860(12)(119904minus(119889+2120573+2)119901)1003817

1003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(124)

On the other hand denoting 119892 = (minusΔ 120573)1199042119891 we have

119867120573 (119905) Δ 120573119891 =

1

(minus119905)1minus1199042

119867120573 (119905) (minus119905Δ 120573)

1minus1199042

119892 (125)

We proceed as in [8] we prove that1003816100381610038161003816100381610038161003816

119867120573 (119905) (minus119905Δ 120573)

1minus1199042

119892 (119909)

1003816100381610038161003816100381610038161003816

le 119862 (119904)119872120573 (119892) (119909) (126)

where119872120573(119892) is a maximal function of 119892 associated with theWeinstein operators (cf [12])

This leads to100381610038161003816100381610038161003816100381610038161003816

int

119860

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

100381610038161003816100381610038161003816100381610038161003816

le 1198621198601199042119872120573 (119892) (119909) (127)

In conclusion we get10038161003816100381610038161003816100381610038161003816

int

infin

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

10038161003816100381610038161003816100381610038161003816

le 119862 (1198601199042119872120573 (119892) (119909)

+119860(12)(119904minus(119889+2120573+2)119901)1003817

1003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

))

(128)

and the choice of 119860 such that

119860(119889+2120573+2)2119901

119872120573 (119892) (119909) =10038171003817100381710038171198911003817100381710038171003817B119904minus(119889+2120573+2)119901120573

infininfin (R119889+1+

)(129)

ensures that10038161003816100381610038161003816100381610038161003816

int

infin

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

10038161003816100381610038161003816100381610038161003816

le 119862(119872120573(119892)(119909))

1minus(119901119904(119889+2120573+2))10038171003817100381710038171198911003817100381710038171003817

119901119904(119889+2120573+2)

B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(130)

Finally taking the 119871119902120573norm with 119902 = 119901(119889 + 2120573 + 2)(119889 + 2120573 +

2minus119901119904) ends the proof thanks to the fact themaximal function119872120573 is bounded of 119871119902

120573(R119889+1

+) into itself for 119902 gt 1

Theorem 32 Let 1 lt 119901 lt 119902 lt infin For all function 119891 such that119891 isin

H1199041

119901120573(R119889+1

+)⋂

Bminus120573120573

infininfin(R119889+1

+) one has

10038171003817100381710038171198911003817100381710038171003817H119904119901120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

120579

H1199041

119901120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573120573

infininfin(R119889+1

+) (131)

where 120579 = 119901119902 119904 = 1205791199041 minus (1 minus 120579)120573 with 120573 gt 0 minus120573 lt 119904 lt 1199041

Proof It suffices to prove that1003817100381710038171003817100381710038171003817

(minusΔ 120573)

(119904minus1199041)2

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le 11986210038171003817100381710038171198911003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(132)

Journal of Function Spaces and Applications 11

Indeed we use the following identity (which may be easilyproven by taking the Weinstein transform in 119909 of both sides)

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

infin

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905 (133)

with 120575 = 1199041 minus 119904 gt 0We decompose the integral in two parts as follows

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

119879

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

+

1

Γ (1205752)

int

infin

119879

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

(134)

where 119879 is a constant to be fixed laterWe proceed as in [8] we obtain

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119872120573 (119891) (119909) (135)

On the other hand we use Theorem 29 and the fact that 119891belongs to Bminus120573minus119904

1120573

infininfin(R119889+1

+) to deduce that

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119905

(minus120573minus1199041)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

) (136)

Thus by applying the preceding estimates on the right part of(134) we obtain

1003816100381610038161003816100381610038161003816

(minusΔ 120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621

Γ (1205752)

1198791205752119872120573 (119891) (119909)

+

1198622

Γ (1205752)

119879(120575minus120573minus119904

1)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

(137)

We fix now

119879 = (

10038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

119872120573 (119891) (119909)

)

2(120573+1199041)

(138)

We obtain

1003816100381610038161003816100381610038161003816

(minusΔ120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621 + 1198622

Γ (1205752)

(119872120573(119891)(119909))

12057910038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(139)

Thus we deduce that1003817100381710038171003817100381710038171003817

(minusΔ120573)

minus1205752

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le

1198621 + 1198622

Γ (1205752)

10038171003817100381710038171003817119872120573 (119891)

10038171003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(140)

To conclude we used the fact that the maximal function119872120573

is bounded of 119871119902120573(R119889+1

+) into itself for 119902 gt 1

43 Estimates in Generalized Besov Spaces For any interval 119868ofR (bounded or unbounded) and a normed space 119865(R119889+1

+)

we define the mixed space-time 119871119901(119868 119865(R119889+1

+)) space of

(classes of) measurable functions 119906 119868 rarr 119865(R119889+1

+) such that

||119906||119871119901(119868119865(R119889+1+

)) lt infin with

119906119871119901(119868119865(R119889+1+

)) = (int

119868

119906 (119905 sdot)119901

119865(R119889+1+

)119889119905)

1119901

if 1 le 119901 lt infin

119906119871infin(119868119865(R119889+1+

)) = ess sup119905isin119868

119906 (119905 sdot)119865(R119889+1+

)

(141)

For any interval 119868 of R (bounded or unbounded) anda Banach space 119883 we define the mixed space-time 119862(119868119883)space of continuous functions 119868 rarr 119883 When 119868 is bounded119862(119868119883) is a Banach space with the norm of 119871infin(119868 119883)

Theorem 33 Let 119904 isin R and 1 le 119901 119902 119903 le infin Let 119879 gt 0 119892 isin

B119904120573

119901119903(R119889+1

+) and119891 in 119871119902((0 119879) B

119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76)

has a unique solution

119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903(R

119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R

119889+1

+))

(142)

and there exists a constant 119862 such that for all 1199021 isin [119902infin] onehas

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862(10038171003817100381710038171198921003817100381710038171003817B119904120573

119901119903(R119889+1+

)+10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)B

119904minus2+(2119902)120573

119901119903(R119889+1+

)))

(143)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] B119904120573

119901119903(R119889+1

+))

Proof Since 119892 and 119891 are temperate distributions (76) has aunique solution 119906 in S1015840

((0 119879) timesR119889+1

+) which satisfies

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(144)

Next we notice that applying Δ 119895 to (76) and using formula(81) yield

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(145)

Therefore

10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(146)

12 Journal of Function Spaces and Applications

By virtue of Lemma 30 we thus have for some 120581 gt 0

10038171003817100381710038171003817Δ 119895119906(119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

119890minus12058122119895(119905minus120591) 10038171003817

100381710038171003817Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(147)

Applying convolution inequalities we get

10038171003817100381710038171003817Δ 119895119906

100381710038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862[

[

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

)

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

]

]

(148)

with 11199022 = 1+ 11199021 minus1119902 Finally taking the 119897119903(Z) norm we

conclude that (with the usual convention if 119903 = infin)

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

1199031199021

)(211989511990410038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

]

]

1119903

+ 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

1199031199022

)

times (2119895(119904minus2+2119902)10038171003817

100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902(0119879119871

119901

120573(R119889+1+

))

119903

]

]

1119903

(149)

which insures that 119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903 (R119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R119889+1

+)) and yields the desired inequality

Since 119906 belongs to 119862([0 119879]B119904120573

119901119903(R119889+1

+)) in the case where

119903 is finite may be easily deduced from the density ofSlowast(R

119889+1)⋂

B119904120573

119901119903(R119889+1

+) in B

119904120573

119901119903(R)

Theorem 34 Let 119904 isin R 119879 gt 0 and 1 le 119901 119902 119903 le infinOne supposes that 119892 isin 119861

119904120573

119901119903(R119889+1

+) and 119891 isin 119871

119902((0 119879)

119861119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76) has a unique solution 119906 belong-

ing to

119871119902((0 119879) 119861

119904+(2119902)120573

119901119903(R

119889+1

+))⋂119871

infin((0 119879) 119861

119904120573

119901119903(R

119889+1

+))

(150)

and there exists a constant 119862 such that for all 119902 le 1199021 le infin

1199061198711199021 ((0119879)119861

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862 [ (1 + 11987911199021)10038171003817100381710038171198921003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (1 + 1198791+1119902

1minus1119902

)10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)119861

119904minus2+(2119902)120573

119901119903(R119889+1+

))]

(151)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] 119861119904120573119901119903(R119889+1

+))

Proof Since 119892 119891 are tempered (76) has a unique solution 119906in S1015840

((0 119879) timesR119889+1

+) satisfying

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(152)

Hence applying Δ 119895 119895 ge 0 to (81) we see that

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(153)

and thus by Lemma 30 we can deduce that10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+int

119905

0

119890minus12058122119895(119905minus120591)10038171003817

100381710038171003817Δ 119895119891(120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(154)

Then it follows from convolution inequalities thatΔ 1198951199061198711199021 ((0119879)119871

119901

120573(R119889+1+

))is dominated by

(

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

(155)

with 11199022 = 1 + 11199021 minus 1119902 Moreover similarly as above wecan obtain that

1003817100381710038171003817Δminus1119906(119905 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)le1003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

1003817100381710038171003817Δminus1119891 (120591 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(156)

and thus if 1 le 119902 le 1199021 le infin1003817100381710038171003817Δminus1119906

10038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862(119879111990211003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)+ 119879

111990221003817100381710038171003817Δminus1119891

1003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

)))

(157)

Journal of Function Spaces and Applications 13

Finally taking the 119897119903-normwith respect to 119895 in (155) and (157)with the usual convention if 119903 = infin we can deduce the desiredestimate

Acknowledgment

Theauthor gratefully acknowledges theDeanship of ScientificResearch at the University of Taibah The author is deeplyindebted to the referee for providing constructive commentsand help in improving the contents of this paper

References

[1] Z Ben Nahia and N Ben Salem ldquoSpherical harmonics andapplications associated with the Weinstein operatorrdquo in Pro-ceedings of the International Conference on PotentialTheory heldin Kouty Czech Republic (ICPT rsquo94) pp 235ndash241 1996

[2] Z Ben Nahia and N Ben Salem ldquoOn a mean value propertyassociated with the Weinstein operatorrdquo in Proceedings of theInternational Conference on Potential Theory held in KoutyCzech Republic (ICPT rsquo94) pp 243ndash253 1996

[3] M Brelot ldquoEquation de Weinstein et potentiels de MarcelRieszrdquo in Seminaire de Theorie de Potentiel Paris No 3 vol 681of Lecture Notes in Mathematics pp 18ndash38 Springer BerlinGermany 1978

[4] H Mejjaoli and M Salhi ldquoUncertainty principles for theweinstein transformrdquo Czechoslovak Mathematical Journal vol61 no 4 pp 941ndash974 2011

[5] H Mejjaoli and A Ould Ahmed Salem ldquoWeinstein Gabortransform and applicationsrdquo Advanced Studies in Pure Mathe-matics vol 2 no 3 pp 203ndash210 2012

[6] H Mejjaoli ldquoBesov spaces associated withthe Weinstein opera-tor and applicationsrdquo In press

[7] T Kawazoe and H Mejjaoli ldquoGeneralized Besov spaces andtheir applicationsrdquo Tokyo Journal of Mathematics vol 35 no 2pp 297ndash320 2012

[8] H Mejjaoli ldquoLittlewood-Paley decomposition associated withthe Dunkl operators and paraproduct operatorsrdquo Journal ofInequalities in Pure and Applied Mathematics vol 9 no 4 pp1ndash25 2008

[9] H Mejjaoli and N Sraeib ldquoGeneralized sobolev spaces inquantum calculus and applicationsrdquo Journal of Inequalities andSpecial Functions vol 1 no 4 pp 43ndash64 2012

[10] H Mejjaoli ldquoGeneralized homogeneous Besov spaces and theirapplicationsrdquo Serdica Mathematical Journal vol 38 no 4 pp575ndash614 2012

[11] H Triebel Interpolation Theory Functions Spaces DifferentialOperators North-Holland AmsterdamThe Netherlands 1978

[12] V S Guliev ldquoOn maximal function and fractional integralassociated with the Bessel differential operatorrdquo MathematicalInequalities and Applications vol 6 no 2 pp 317ndash330 2003

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 5

Definition 10 Let 119904 isin R and 119901 119902 isin [1infin]The homogeneousWeinstein-Besov spaces B

119904120573

119901119902(R119889+1

+) are the spaces of distri-

bution in S1015840

ℎ120573lowast(R119889+1

) such that

10038171003817100381710038171198911003817100381710038171003817B119904120573

119901119902(R119889+1+

)= (sum

119895isinZ

(211990411989510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119902

)

1119902

lt infin (44)

Proposition 11 (see [6]) Let 119904 isin R and 119901 and 119902 two elementsof [1infin] the space B

119904120573

119901119902(R119889+1

+) is the set of 119891 isin S1015840

ℎ120573lowast(R119889+1

)

verifying

10038171003817100381710038171003817

119891

10038171003817100381710038171003817B119904120573

119901119902(R119889+1+

)= (int

infin

0

(119905minus1199041003817100381710038171003817119891lowast119882120601119905

1003817100381710038171003817119871119901

120573(R119889+1+

))

119902119889119905

119905

)

1119902

lt infin

(45)

where 120601119905(119909) = (11199052120573+2+119889

)120601(119909119905) for all 119905 isin (0infin) and 119909 isin

R119889+1

+

Definition 12 For 119904 isin R and 119901 119902 isin [1infin] one writes10038171003817100381710038171198911003817100381710038171003817119861119904120573

119901119902(R119889+1+

)=10038171003817100381710038171198780119891

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ (sum

119895ge1

(211990411989510038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119902

)

1119902

(46)

The nonhomogeneous Besov space 119861119904120573119901119902(R119889+1

+) associated

with the Weinstein operators is defined by

119861119904120573

119901119902(R

119889+1

+) = 119891 isin S

1015840(R

119889)

10038171003817100381710038171198911003817100381710038171003817119861119904120573

119901119902(R119889+1+

)lt infin (47)

We give now another definition equivalent to the nonho-mogeneous Besov space 119861119904120573119901119902(R119889+1

+)

Proposition 13 Let 119904 isin R and119901 and 119902 two elements of [1infin]the space 119861119904120573119901119902(R119889+1

+) is the set of 119891 isin S1015840

(R119889) verifying

10038171003817100381710038171003817

119891

10038171003817100381710038171003817119861119904120573

119901119902(R119889+1+

)=1003817100381710038171003817119891lowast119882120595

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ (int

1

0

(119905minus1199041003817100381710038171003817119891lowast119882120601119905

1003817100381710038171003817119871119901

120573(R119889+1+

))

119902119889119905

119905

)

1119902

lt infin

(48)

Definition 14 Let 119904 isin R and 1 le 119901 119902 le infin the homogeneousWeinstein-Triebel-Lizorkin space F

119904120573

119901119902(R119889+1

+) is the space of

distribution in S1015840

ℎ120573lowast(R119889+1

) such that

10038171003817100381710038171198911003817100381710038171003817F119904120573

119901119902(R119889+1+

)=

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(sum

119895isinZ

211990411989511990210038161003816100381610038161003816Δ 119895119891

10038161003816100381610038161003816

119902

)

1119902100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)

lt infin (49)

Definition 15 For 119904 isin R the operatorR119904

120573from S1015840

ℎ120573lowast(R119889+1

)

to S1015840

ℎ120573lowast(R119889+1

) is defined by

R119904

120573(119891) = F

minus1

119882(sdot

119904F119882119891) (50)

The operatorRminus119904

120573is called Weinstein-Riesz potential space

Definition 16 For 119904 isin R and 1 le 119901 le infin the homogeneousWeinstein-Riesz potential space H119904

119901120573(R119889+1

+) is defined

as the space Rminus119904

120573(119871

119901

120573(R119889+1

+)) equipped with the norm

119891H119904119901120573(R119889+1+

)= R119904

120573(119891)

119871119901

120573(R119889+1+

)

Proposition 17 Let 119904 isin R and 1 le 119901 119902 le infinThe operator Δ 120573 is a linear continuous operator from

B119904120573

119901119902(R119889+1

+) into B

119904minus2120573

119901119902 (R119889+1

+) and from H119904

119901120573(R119889+1

+) into

H119904minus2

119901120573(R119889+1

+)

Proof We obtain these results by the similar ideas used in thenonhomogeneous case (cf [6])

Proposition 18 Let 119904 119905 isin R and 1 le 119901 119902 le infin The operatorR119905

120573is a linear continuous injective operator from B

119904120573

119901119902(R119889+1

+)

onto B119904minus119905120573

119901119902 (R119889+1

+) and from H119904

119901120573(R119889+1

+) onto H119904minus119905

119901120573(R119889+1

+)

Proof We obtain these results by the similar ideas used in thenonhomogeneous case (cf [6])

33 Embeddings As in the Euclidean case (cf [11]) themono-tone character of 119897119902-spaces and the Minkowskis inequalityyield the following

Proposition 19 If 1 le 1199021 lt 1199022 le infin one has

B119904120573

1199011199021

(R119889+1

+) 997893rarr

B119904120573

1199011199022

(R119889+1

+) (1 le 119901 le infin 119904 isin R)

(51)

MoreoverB119904120573

1199011(R

119889+1

+) 997893rarr

H119904

119901120573(R

119889+1

+) 997893rarr

B119904120573

119901infin(R

119889+1

+)

(1 le 119901 le infin 119904 isin R)

(52)

If 1199040 = 1199041 one also has

(H1199040

119901120573(R

119889+1

+)

H1199041

119901120573(R

119889+1

+))

120579119902=

B119904120573

119901119902(R

119889+1

+)

(1 le 119901 119902 le infin 120579 isin (0 1))

(53)

where 119904 = (1 minus 120579)1199040 + 1205791199041

Proposition 20 One assumes that 119904 minus (119889 + 2120573 + 2)119901 = 1199041 minus

(119889 + 2120573 + 2)1199011 Then the following inclusion holds

B119904120573

119901119902(R

119889+1

+) 997893rarr

B1199041120573

11990111199021

(R119889+1

+)

(1 le 119901 le 1199011 le infin 1 le 119902 le 1199021 le infin 119904 1199041 isin R)

(54)

Proof In order to prove the inclusion we use the estimate

Δ 119895119891 = 2119895(119889+2120573+2)

120601 (2119895sdot) lowast119882Δ 119895119891 (55)

Proposition 9(i) gives that10038171003817100381710038171003817Δ 119895119891

100381710038171003817100381710038171198711199011

120573(R119889+1+

)=

100381710038171003817100381710038172119895(119889+2120573+2)

120601 (2119895sdot) lowast119882Δ 119895119891

100381710038171003817100381710038171198711199011

120573(R119889+1+

)

le 1198622119895(119889+2120573+2)(1119901minus1119901

1)10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

(56)

6 Journal of Function Spaces and Applications

By definition of the homogeneous Weinstein-Besov spaceswe therefore infer10038171003817100381710038171198911003817100381710038171003817B1199041120573

11990111199021(R119889+1+

)

= (

infin

sum

119895=minusinfin

(2119895119904110038171003817100381710038171003817Δ 119895119891

100381710038171003817100381710038171198711199011

120573(R119889+1+

))

1199021

)

11199021

le 119862(sum

119895isinZ

(211989511990412119895(119889+2120573+2)(1119901minus1119901

1)10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

1199021

)

11199021

le 119862(sum

119895isinZ

(211989511990410038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

1199021

)

11199021

le 11986210038171003817100381710038171198911003817100381710038171003817B119904120573

119901119902(R119889+1+

)

(57)

since 119902 le 1199021 This gives the inclusion

Proposition 21 (1) If 119906 belongs to B119904120573

119901119902(R119889+1

+) cap

B119905120573

119901119902(R119889+1

+)

then 119906 belongs to B120579119904+(1minus120579)119905120573

119901119902 (R119889+1

+) for all 120579 isin [0 1] and

119906B120579119904+(1minus120579)119905120573

119901119902(R119889+1+

)le 119906

120579

B119904120573

119901119902(R119889+1+

)119906

1minus120579

B119905120573

119901119902(R119889+1+

) (58)

(2) If 119906 belongs to B119904120573

119901infin(R119889+1

+) cap

B119905120573

119901infin(R119889+1

+) and 119904 lt 119905

then 119906 belongs to B120579119904+(1minus120579)119905120573

1199011(R119889+1

+) for all 120579 isin (0 1) and there

exists a positive constant 119862(119905 119904) such that

119906B120579119904+(1minus120579)119905120573

1199011(R119889+1+

)le 119862 (119905 119904) 119906

120579

B119904120573

119901infin(R119889+1+

)119906

1minus120579

B119905120573

119901infin(R119889+1+

) (59)

(3) If 119906 belongs to B119904120573

119901infin(R119889+1

+) cap

B119904+120576120573

119901infin (R119889+1

+) and 120576 gt

0 then 119906 belongs to B119904120573

1199011(R119889+1

+) and there exists a positive

constant 119862 such that

119906B119904120573

1199011(R119889+1+

)le

119862

120576

119906B119904120573

119901infin(R119889+1+

)log

2(119890 +

119906B119904+120576120573

119901infin(R119889+1+

)

119906B119904120573

119901infin(R119889+1+

)

)

(60)

Proof (1) is obvious from the Holderrsquos inequality As for (2)we write 119906

B120579119904+(1minus120579)119905120573

1199011(R119889+1+

)as

sum

119895le119873

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ sum

119895gt119873

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

(61)

where 119873 is chosen here after By the definition of thehomogeneous Weinstein-Besov norms we see that

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 2

119895(1minus120579)(119905minus119904)119906

B119904120573

119901infin(R119889+1+

)

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 2

minus119895120579(119905minus119904)119906

B119905120573

119901infin(R119889+1+

)

(62)

and thus 119906B120579119904+(1minus120579)119905120573

1199011(R119889+1+

)is dominated by

119906B119904120573

119901infin(R119889+1+

)sum

119895le119873

2119895(1minus120579)(119905minus119904)

+ 119906B119905120573

119901infin(R119889+1+

)sum

119895gt119873

2minus119895120579(119905minus119904)

le 119862119906B119904120573

119901infin(R119889+1+

)

2(119873+1)(1minus120579)(119905minus119904)

2(1minus120579)(119905minus119904)

minus 1

+ 119906B119905120573

119901infin(R119889+1+

)

2minus119873120579(119905minus119904)

1 minus 2minus120579(119905minus119904)

(63)

Hence in order to complete the proof of (2) it suffices tochoose119873 such that

119906B119905120573

119901infin(R119889)

119906B119904120573

119901infin(R119889)

le 2119873(119905minus119904)

lt 2

119906B119905120573

119901infin(R119889)

119906B119904120573

119901infin(R119889)

(64)

As for (3) it is easy to see that 119906B119904120573

1199011(R119889+1+

)is dominated as

sum

119895le119873minus1

211989511990410038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ sum

119895ge119873

211989511990410038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le (119873 + 1) 119906B119904120573

119901infin(R119889+1+

)+

2minus(119873minus1)120576

2120576minus 1

119906B119904+120576120573

119901infin(R119889+1+

)

(65)

Hence letting

119873 = 1 + [

[

1

120576

log2

119906B119904+120576120573

119901infin(R119889+1+

)

119906B119904120573

119901infin(R119889+1+

)

]

]

(66)

we can obtain the desired estimate

Proposition 22 Let 119902 isin (1infin) and let 119904 isin R such that0 lt 119904 lt (119889 + 2120573 + 2)119902 then one has

B119904120573

119902119902(R

119889+1

+) =

F119904120573

119902119902(R

119889+1

+) 997893rarr

F119904120573

119902infin(R

119889+1

+)

997893rarrF119904minus((119889+2120573+2)119902)120573

infininfin(R

119889+1

+)

(67)

H119904

119902120573(R

119889+1

+) =

F119904120573

1199022(R

119889+1

+) 997893rarr

F119904120573

119902infin(R

119889+1

+)

997893rarrF119904minus((119889+2120573+2)119902)120573

infininfin(R

119889+1

+)

(68)

Proof We obtain these results by the similar ideas used in thenonhomogeneous case (cf [6])

Theorem 23 Let 119886 119887 gt 0 and let 1199021 1199022 isin [1infin] Let120579 = 119886(119886 + 119887) isin (0 1) and let 1119901 = (1 minus 120579)1199021 +

1205791199022 Then there exists a constant 119862 such that for every119891 isin

F119886120573

1199021infin(R119889+1

+) cap

Fminus119887120573

1199022infin(R119889+1

+) then one has

1003816100381610038161003816119891 (119909)

1003816100381610038161003816le 119862(sup

119895isinZ

2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

1minus120579

(sup119895isinZ

2minus119887119895 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

120579

(69)

In particular one gets

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

F119886120573

1199021infin(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

120579

Fminus119887120573

1199022infin(R119889+1+

) (70)

Journal of Function Spaces and Applications 7

Proof Let 119891 be a Schwartz class we have

1003816100381610038161003816119891 (119909)

1003816100381610038161003816le sum

119895isinZ

10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816

le sum

119895isinZ

min(2minus119886119895sup119895isinZ

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816))

(71)

We define119873(119909) as the largest index such that

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816) le 2

minus119886119895sup119895isinZ

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816) (72)

and we write1003816100381610038161003816119891 (119909)

1003816100381610038161003816le sum

119895le119873(119909)

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

+ sum

119895gt119873(119909)

2minus119886119895sup

119895

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

le 119862(sup119895isinZ

2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

119887(119886+119887)

times (sup119895isinZ

2minus119887119895 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

119886(119886+119887)

(73)

Thus (69) is proved In order to obtain (70) it is enough toapply the Holder inequality in the expression above since wehave 120579 = 119886(119886+119887) isin (0 1) and let 1119901 = (1minus120579)1199021+1205791199022

Corollary 24 Let 119902 isin (1infin) and let 119904 isin R such that 0 lt 119904 lt

(119889 + 2120573 + 2)119902 then one has

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus(119902119901)

Bminus((2120573+2+119889)119902minus119904)120573

infininfin (R119889+1

+ )

10038171003817100381710038171198911003817100381710038171003817

119902119901

B119904120573

119902119902(R119889+1

+ )

(74)

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus(119902119901)

Bminus((2120573+2+119889)119902minus119904)120573

infininfin (R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

119902119901

H119904119902120573(R119889+1+

) (75)

where 119901 = 119902(2120573 + 2 + 119889)(2120573 + 2 + 119889 minus 119902119904)

Proof By choosing 119886 = 119904 gt 0 minus119887 = 119904 minus (119889 + 2120573 + 2)119902 lt 01199021 = 119902 and 1199022 = infin we deduce (74) from the relations (70)and (67) In the same way we deduce (75) from the relations(70) and (68)

4 Generalized Heat Equation

41 Characterization for the Weinstein-Besov Spaces TheWeinstein heat equation reads

120597119905119906 (119905 119909) minus Δ 120573119906 (119905 119909) = 119891 (119905 119909) (119905 119909) isin [0infin) timesR119889+1

+

119906|119905=0 = 119892

(76)

We introduce the Weinstein heat semigroup 119867120573(119905) for theWeinstein-Laplace operator

119867120573 (119905) 119892 (119909) =

int

R119889+1+

Γ120573 (119905 119909 119910) 119892 (119910) 119889120583120573 (119910) if 119905 gt 0

119892 (119909) if 119905 = 0(77)

where Γ120573 is the Weinstein heat kernel defined by

Γ120573 (119905 119909 119910) = 120591119909 (119864(120573)

119905) (119910) (78)

where

119864(120573)

119905(119910) =

2

1205871198892Γ (120573 + 1) (4119905)

120573+1+1198892119890minus11991024119905 (79)

Thus

119867120573 (119905) 119892 (119909) = 119892lowast119882119864(120573)

119905 (119909) (80)

In practice we use the integral formulation of (76)

119906 (119905 119909) = 119867120573 (119905) 119892 (119909) + 119866120573 (119891) (119905 119909)

= 119867120573 (119905) 119892 (119909) + int

119905

0

119867120573 (119905 minus 119904) 119891 (119904 119909) 119889119904

(81)

Remark 25 The function 119864(120573)119905

is the Gauss kernel associatedwith Weinstein operators This function satisfies

forall120585 isin R119889+1

+ F119882 (119864

(120573)

119905) (120585) = 119890

minus1199051205852

(82)

Proposition 26 Let 1 le 119901 le 119903 le infin and let 119891 isin 119871119901

120573(R119889+1

+)

Then the operator 119867120573(119905) maps 119871119901

120573(R119889+1

+) continuously to

119871119903

120573(R119889+1

+) and

10038171003817100381710038171003817119867120573 (119905) 119891

10038171003817100381710038171003817119871119903120573(R119889)

le 119862119905minus((119889+2120573+2)2)(1119901minus1119903)1003817

1003817100381710038171198911003817100381710038171003817119871119901

120573(R119889)

(83)

Moreover1003817100381710038171003817100381710038171003817

(minusΔ 120573)

1205752

119867120573 (119905) 119891

1003817100381710038171003817100381710038171003817119871119903120573(R119889)

le 119862119905minus1205752minus((119889+2120573+2)2)(1119901minus1119903)1003817

1003817100381710038171198911003817100381710038171003817119871119901

120573(R119889)

(84)

for all 120575 gt 0

Proof It follows from the relations (80) and (29) combinedwith scaling property of the kernel 119864(120573)

119905

In this section we prove estimates for the Weinstein heatsemigroupThese estimates are based on the following result

Lemma 27 Let C be an annulus Positive constants 119888 and 119862exist such that for any 119901 in [1infin] and any couple (119905 120582) ofpositive real numbers one has

suppF119882 (119906) sub 120582C 997904rArr

10038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le 119862119890minus1198881199051205822

119906119871119901

120573(R119889+1+

)

(85)

8 Journal of Function Spaces and Applications

Proof We again consider a function Θ in 119863(R119889+1

+ 0) the

value of which is identically 1 in neighborhood of annulusCWe can also assume without loss of generality that 120582 = 1 Wethen have

119867120573 (119905) 119906 = 119892 (119905 sdot) lowast119882119906 (86)where

119892 (119905 sdot) = Fminus1

119882(Θ (120585) 119890

minus1199051205852

) (87)

The lemma is proved provided that we can find positive realnumbers 119888 and 119862 such that

forall119905 gt 01003817100381710038171003817119892 (119905 sdot)

10038171003817100381710038171198711120573(R119889+1+

)le 119862119890

minus119888119905 (88)

To begin we perform integrations by parts in (87) We get1003816100381610038161003816119892 (119905 119909)

1003816100381610038161003816=

1

119888120573

(1 + 1199092)

minus(119889+[2120573]+3)

times int

R119889+1+

Λ (119909 120585) (119868119889 minus Δ 120573)

119889+[2120573]+3

times (Θ (120585) 119890minus1199051205852

) 119889120583120573 (120585)

(89)

Using Leibnizrsquos formula we obtain1003816100381610038161003816119892 (119905 119909)

1003816100381610038161003816le 119862(1 + 119909

2)

minus(119889+[2120573]+3)

119890minus119888119905 (90)

and (88) follows

For any interval 119868 of R (bounded or unbounded) wedefine themixed space-time 119871119901(119868 119871119902

120573(R119889+1

+)) Banach space of

(classes of) measurable functions 119906 119868 rarr 119871119902

120573(R119889+1

+) such

that 119906119871119901(119868119871

119902

120573(R119889+1+

))lt infin with

119906119871119901(119868119871119902

120573(R119889+1+

))= (int

119868

119906 (119905 sdot)119901

119871119902

120573(R119889+1+

)119889119905)

1119901

if 1 le 119901 119902 lt infin

119906119871infin(119868119871119902

120573(R119889+1+

))= ess sup

119905isin119868

119906 (119905 sdot)119871119902

120573(R119889+1+

) if 1 le 119902 lt infin

(91)Corollary 28 Let C be an annulus and 120582 a positive realnumber Let 1199060 (resp 119891 = 119891(119905 119909)) satisfy suppF119882(1199060) sub 120582C(resp suppF119882(119891(119905 sdot)) sub 120582C for all 119905 in [0 119879]) Consider 119906 asolution of

120597119905119906 minus Δ 120573119906 = 0 119906|119905=0 = 1199060 (92)and V a solution of

120597119905V minus Δ 120573V = 119891 (119905 sdot) V|119905=0 = 0 (93)There exist positive constants 119888 and 119862 depending only on Csuch that for any 1 le 119886 le 119887 le infin and 1 le 119901 le 119902 le infin we have

119906119871119902([0119879]119871119887120573(R119889+1+

))le 119862120582

(119889+2120573+2)(1119886minus1119887)120582minus21199021003817

1003817100381710038171199060

1003817100381710038171003817119871119886120573(R119889+1+

)

V119871119902([0119879]119871119887120573(R119889+1+

))le 119862120582

minus2(1+1119902minus1119901)120582(119889+2120573+2)(1119886minus1119887)

times10038171003817100381710038171198911003817100381710038171003817119871119901([0119879]119871119886

120573(R119889+1+

))

(94)

Proof It suffices to use the fact that

119906 (119905 sdot) = 119867120573 (119905) 1199060 V (119905 sdot) = int119905

0

119867120573 (119905 minus 119904) 119891 (119904 sdot) 119889119904

(95)

Combining Lemma 27 and Youngrsquos inequality (29) withscaling property of the kernel 119864(120573)

119905now yields the result

Theorem 29 Let 119904 be a positive real number and (119901 119903) isin

[1infin]2 A constant 119862 exists which satisfies the following

property For 119906 isin Bminus2119904120573

119901119903 (R119889+1

+) one has

119862minus1119906

Bminus2119904120573

119901119903(R119889+1+

)le

10038171003817100381710038171003817100381710038171003817

10038171003817100381710038171003817119905119904119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862119906Bminus2119904120573

119901119903(R119889+1+

)

(96)

To prove this result we need the following lemma

Lemma 30 There exist two positive constants 120581 and 119862

depending only on 120593 such that for all 1 le 119901 le infin 120591 ge 0 and119895 isin Z one has

10038171003817100381710038171003817Δ 119895 (119867120573 (120591) 119906)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862119890

minus1205812211989512059110038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

) (97)

Proof The result follows immediately by applying Lemma 27and because Δ 119895(119867120573(120591))119906 = (119867120573(120591)Δ 119895)119906

Proof of Theorem 29 Using Lemma 30 and considering thefact that the operator Δ 119895 commutes with the operator 119867120573(119905)

and the definition of the homogeneous Weinstein-Besov(semi) norm we get

10038171003817100381710038171003817119905119904119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862119906

Bminus2119904120573

119901119903(R119889+1+

)sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903119895

(98)

where (119888119903119895)119895isinZ denotes as in all this proof a generic elementof the unit sphere of 119897119903(Z) In the case when 119903 = infin therequired inequality comes immediately from the followingeasy result For any positive 119904 we have

sup119905gt0

sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

lt infin (99)

In the case 119903 lt infin using the Holder inequality with theweight 22119895119904119890minus1205811199052

2119895

(99) and the Fubini theorem we obtain

int

infin

0

11990511990311990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

119889119905

119905

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)int

infin

0

(sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

)

119903minus1

times (sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903

119903119895)

119889119905

119905

Journal of Function Spaces and Applications 9

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)int

infin

0

(sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903

119903119895)

119889119905

119905

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)sum

119895isinZ

119888119903

119903119895int

infin

0

(11990511990422119895119904119890minus12058111990522119895

)

119889119905

119905

le 119862Γ (119904) 119906119903

Bminus2119904120573

119901119903(R119889+1+

)

(100)

In order to prove the other inequality let us observe thatfor any 119904 greater than minus1 we have

Δ 119895119906 =

1

Γ (119904 + 1)

int

infin

0

119905119904(minusΔ 120573)

119904+1

119867120573 (119905) Δ 119895119906 119889119905 (101)

Then Lemma 30 Proposition 9 and the fact that the operatorΔ 119895 commutes with the operator119867120573(119905) lead to the following

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862int

infin

0

11990511990422119895(119904+1)

119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889119905

(102)

In the case 119903 = infin we simply write

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862(sup

119905gt0

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

times int

infin

0

22119895(119904+1)

119890minus12058111990522119895

119889119905

le 11986222119895119904(sup119905gt0

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

(103)

In the case 119903 lt infin Holderrsquos inequality with the weight 119890minus12058111990522119895

gives

(int

infin

0

119905119904119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889119905)

119903

le 1198622minus2119895(119903minus1)

int

infin

0

119905119903119904119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)119889119905

(104)

Thanks to (99) and Fubinirsquos theorem we infer from (102) that

sum

119895isinZ

2minus211989511990311990410038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)le 119862int

infin

0

11990511990311990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

119889119905

119905

(105)

The theorem is proved

Second Proof ofTheorem 29We only consider the case 1 le 119903 ltinfinThe case 119903 = infin can be shown similarlyWe first prove that

119862minus1119906

Bminus2119904120573

119901119903(R119889+1+

)le

10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

(106)

It is easy to see that

Δ 119895119906 = 120594119895lowast119882119864(120573)

2minus2119895lowast119882119906

(107)

where 120594119895 = Fminus1

119882(120593(2

minus119895120585)119890

2minus21198951205852

) and119864(120573)2minus2119895

is the Gauss kernelassociated with Weinstein operators By relation (29) we get

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le

10038171003817100381710038171003817120594119895

100381710038171003817100381710038171198711120573(R119889+1+

)

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

) (108)

As10038171003817100381710038171003817120594119895

100381710038171003817100381710038171198711120573(R119889+1+

)= int

R119889+1+

100381610038161003816100381610038161003816F

minus1

119882(120593 (120585) 119890

1205852

)

100381610038161003816100381610038161003816119889120583120573 (120585) lt infin (109)

we obtain10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

) (110)

Moreover simple calculations give that

119864(120573)

2minus2119895lowast119882119906 = 119867120573 (2

minus4119895minus 119905

2) (119864

(120573)

1199052lowast119882119906)

(111)

Thus from Proposition 26 it follows that100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1

+ )

(112)

for any 119905 isin [2minus119895minus1 2minus119895] which implies that

sum

119895isinZ

2minus211990411989511990310038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

le 119862 sum

119895isinZ

int

2minus119895

2minus119895minus1(119905

2119904100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862int

infin

0

(1199052119904100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862int

infin

0

(11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

(113)

where we have used the fact that 119864(120573)1199052lowast119882119906 = 119867120573(119905

2)119906

We now prove that10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862119906Bminus2119904120573

119901119903(R119889+1+

) (114)

Indeed one has

119864(120573)

2minus2119895lowast119882119906 = sum

119899isinZ

119864(120573)

2minus2119895lowast119882Δ 119899+119895119906 (115)

Arguing as above we have100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1

+ )

(116)

for any 119905 isin [2minus119895 21minus119895] Thus10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

= 2

100381710038171003817100381710038171003817100381710038171003817

1003817100381710038171003817100381710038171199052119904119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)

100381710038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862 sum

119895isinZ

int

21minus119895

2minus119895(2

minus2119895119904100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862 sum

119895isinZ

(2minus2119895119904

sum

119899isinZ

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

(117)

10 Journal of Function Spaces and Applications

On the other hand it is easy to see that100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 1198622

minus211989911990410038171003817100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

) (118)

for any 119904 gt 0 For 0 lt 1199041 lt 119904 lt 1199042 and by using theMinkowskiinequality we have

sum

119895isinZ

(2minus2119895119904

sum

119899isinZ

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

le sum

119895isinZ

(2minus2119895119904

0

sum

minusinfin

2minus21198991199041

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

+ sum

119895isinZ

(2minus2119895119904

sum

119899isinN

2minus21198991199042

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

le 119862(

0

sum

minusinfin

2minus2119899(119904

1minus119904)(sum

119895isinZ

(2minus2(119899+119895)11990311990410038171003817

100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

)

1119903

)

119903

+ 119862(sum

N

2minus2119899(119904

2minus119904)(sum

119895isinZ

(2minus2(119899+119895)11990311990410038171003817

100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

)

1119903

)

119903

le 119862 sum

119895isinZ

2minus211989511990311990410038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

(119)

The result is immediately from (117) and (119)

42 Embedding Sobolev Theorems

Theorem 31 Let 1 lt 119901 lt infin and let 0 lt 119904 lt (119889 + 2120573 +

2)119901There exists a positive constant119862 such that for all function119891 isin

H119904

119901120573(R119889+1

+) one has

10038171003817100381710038171198911003817100381710038171003817119871119902

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

H119904119901120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

120579

B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

) (120)

where 120579 = 119904119901(119889+2120573+2) and 119902 = 119901(119889+2120573+2)(119889+2120573+2minus119901119904)

Proof Bydensity we can suppose that119891belongs toSlowast(R119889+1

)It is easy to see that

119891 = int

infin

0

119867120573 (119905) Δ 120573119891119889119905(121)

and decompose the integral in two parts as follows

119891 = int

119860

0

119867120573 (119905) Δ 120573119891119889119905 + int

infin

119860

119867120573 (119905) Δ 120573119891119889119905(122)

where 119860 is a constant to be fixed laterOn the other hand byTheorem 29 we obtain10038171003817100381710038171003817119867120573 (119905) Δ 120573119891

10038171003817100381710038171003817119871infin120573(R119889+1+

)

le

119862

1199051minus(12)(119904minus(119889+2120573+2)119901)

10038171003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(123)

Therefore after integrating we get

int

infin

119860

10038171003817100381710038171003817119867120573 (119905) Δ 120573119891

10038171003817100381710038171003817119871infin120573(R119889+1+

)119889119905

le 119860(12)(119904minus(119889+2120573+2)119901)1003817

1003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(124)

On the other hand denoting 119892 = (minusΔ 120573)1199042119891 we have

119867120573 (119905) Δ 120573119891 =

1

(minus119905)1minus1199042

119867120573 (119905) (minus119905Δ 120573)

1minus1199042

119892 (125)

We proceed as in [8] we prove that1003816100381610038161003816100381610038161003816

119867120573 (119905) (minus119905Δ 120573)

1minus1199042

119892 (119909)

1003816100381610038161003816100381610038161003816

le 119862 (119904)119872120573 (119892) (119909) (126)

where119872120573(119892) is a maximal function of 119892 associated with theWeinstein operators (cf [12])

This leads to100381610038161003816100381610038161003816100381610038161003816

int

119860

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

100381610038161003816100381610038161003816100381610038161003816

le 1198621198601199042119872120573 (119892) (119909) (127)

In conclusion we get10038161003816100381610038161003816100381610038161003816

int

infin

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

10038161003816100381610038161003816100381610038161003816

le 119862 (1198601199042119872120573 (119892) (119909)

+119860(12)(119904minus(119889+2120573+2)119901)1003817

1003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

))

(128)

and the choice of 119860 such that

119860(119889+2120573+2)2119901

119872120573 (119892) (119909) =10038171003817100381710038171198911003817100381710038171003817B119904minus(119889+2120573+2)119901120573

infininfin (R119889+1+

)(129)

ensures that10038161003816100381610038161003816100381610038161003816

int

infin

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

10038161003816100381610038161003816100381610038161003816

le 119862(119872120573(119892)(119909))

1minus(119901119904(119889+2120573+2))10038171003817100381710038171198911003817100381710038171003817

119901119904(119889+2120573+2)

B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(130)

Finally taking the 119871119902120573norm with 119902 = 119901(119889 + 2120573 + 2)(119889 + 2120573 +

2minus119901119904) ends the proof thanks to the fact themaximal function119872120573 is bounded of 119871119902

120573(R119889+1

+) into itself for 119902 gt 1

Theorem 32 Let 1 lt 119901 lt 119902 lt infin For all function 119891 such that119891 isin

H1199041

119901120573(R119889+1

+)⋂

Bminus120573120573

infininfin(R119889+1

+) one has

10038171003817100381710038171198911003817100381710038171003817H119904119901120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

120579

H1199041

119901120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573120573

infininfin(R119889+1

+) (131)

where 120579 = 119901119902 119904 = 1205791199041 minus (1 minus 120579)120573 with 120573 gt 0 minus120573 lt 119904 lt 1199041

Proof It suffices to prove that1003817100381710038171003817100381710038171003817

(minusΔ 120573)

(119904minus1199041)2

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le 11986210038171003817100381710038171198911003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(132)

Journal of Function Spaces and Applications 11

Indeed we use the following identity (which may be easilyproven by taking the Weinstein transform in 119909 of both sides)

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

infin

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905 (133)

with 120575 = 1199041 minus 119904 gt 0We decompose the integral in two parts as follows

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

119879

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

+

1

Γ (1205752)

int

infin

119879

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

(134)

where 119879 is a constant to be fixed laterWe proceed as in [8] we obtain

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119872120573 (119891) (119909) (135)

On the other hand we use Theorem 29 and the fact that 119891belongs to Bminus120573minus119904

1120573

infininfin(R119889+1

+) to deduce that

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119905

(minus120573minus1199041)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

) (136)

Thus by applying the preceding estimates on the right part of(134) we obtain

1003816100381610038161003816100381610038161003816

(minusΔ 120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621

Γ (1205752)

1198791205752119872120573 (119891) (119909)

+

1198622

Γ (1205752)

119879(120575minus120573minus119904

1)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

(137)

We fix now

119879 = (

10038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

119872120573 (119891) (119909)

)

2(120573+1199041)

(138)

We obtain

1003816100381610038161003816100381610038161003816

(minusΔ120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621 + 1198622

Γ (1205752)

(119872120573(119891)(119909))

12057910038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(139)

Thus we deduce that1003817100381710038171003817100381710038171003817

(minusΔ120573)

minus1205752

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le

1198621 + 1198622

Γ (1205752)

10038171003817100381710038171003817119872120573 (119891)

10038171003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(140)

To conclude we used the fact that the maximal function119872120573

is bounded of 119871119902120573(R119889+1

+) into itself for 119902 gt 1

43 Estimates in Generalized Besov Spaces For any interval 119868ofR (bounded or unbounded) and a normed space 119865(R119889+1

+)

we define the mixed space-time 119871119901(119868 119865(R119889+1

+)) space of

(classes of) measurable functions 119906 119868 rarr 119865(R119889+1

+) such that

||119906||119871119901(119868119865(R119889+1+

)) lt infin with

119906119871119901(119868119865(R119889+1+

)) = (int

119868

119906 (119905 sdot)119901

119865(R119889+1+

)119889119905)

1119901

if 1 le 119901 lt infin

119906119871infin(119868119865(R119889+1+

)) = ess sup119905isin119868

119906 (119905 sdot)119865(R119889+1+

)

(141)

For any interval 119868 of R (bounded or unbounded) anda Banach space 119883 we define the mixed space-time 119862(119868119883)space of continuous functions 119868 rarr 119883 When 119868 is bounded119862(119868119883) is a Banach space with the norm of 119871infin(119868 119883)

Theorem 33 Let 119904 isin R and 1 le 119901 119902 119903 le infin Let 119879 gt 0 119892 isin

B119904120573

119901119903(R119889+1

+) and119891 in 119871119902((0 119879) B

119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76)

has a unique solution

119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903(R

119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R

119889+1

+))

(142)

and there exists a constant 119862 such that for all 1199021 isin [119902infin] onehas

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862(10038171003817100381710038171198921003817100381710038171003817B119904120573

119901119903(R119889+1+

)+10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)B

119904minus2+(2119902)120573

119901119903(R119889+1+

)))

(143)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] B119904120573

119901119903(R119889+1

+))

Proof Since 119892 and 119891 are temperate distributions (76) has aunique solution 119906 in S1015840

((0 119879) timesR119889+1

+) which satisfies

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(144)

Next we notice that applying Δ 119895 to (76) and using formula(81) yield

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(145)

Therefore

10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(146)

12 Journal of Function Spaces and Applications

By virtue of Lemma 30 we thus have for some 120581 gt 0

10038171003817100381710038171003817Δ 119895119906(119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

119890minus12058122119895(119905minus120591) 10038171003817

100381710038171003817Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(147)

Applying convolution inequalities we get

10038171003817100381710038171003817Δ 119895119906

100381710038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862[

[

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

)

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

]

]

(148)

with 11199022 = 1+ 11199021 minus1119902 Finally taking the 119897119903(Z) norm we

conclude that (with the usual convention if 119903 = infin)

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

1199031199021

)(211989511990410038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

]

]

1119903

+ 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

1199031199022

)

times (2119895(119904minus2+2119902)10038171003817

100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902(0119879119871

119901

120573(R119889+1+

))

119903

]

]

1119903

(149)

which insures that 119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903 (R119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R119889+1

+)) and yields the desired inequality

Since 119906 belongs to 119862([0 119879]B119904120573

119901119903(R119889+1

+)) in the case where

119903 is finite may be easily deduced from the density ofSlowast(R

119889+1)⋂

B119904120573

119901119903(R119889+1

+) in B

119904120573

119901119903(R)

Theorem 34 Let 119904 isin R 119879 gt 0 and 1 le 119901 119902 119903 le infinOne supposes that 119892 isin 119861

119904120573

119901119903(R119889+1

+) and 119891 isin 119871

119902((0 119879)

119861119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76) has a unique solution 119906 belong-

ing to

119871119902((0 119879) 119861

119904+(2119902)120573

119901119903(R

119889+1

+))⋂119871

infin((0 119879) 119861

119904120573

119901119903(R

119889+1

+))

(150)

and there exists a constant 119862 such that for all 119902 le 1199021 le infin

1199061198711199021 ((0119879)119861

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862 [ (1 + 11987911199021)10038171003817100381710038171198921003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (1 + 1198791+1119902

1minus1119902

)10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)119861

119904minus2+(2119902)120573

119901119903(R119889+1+

))]

(151)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] 119861119904120573119901119903(R119889+1

+))

Proof Since 119892 119891 are tempered (76) has a unique solution 119906in S1015840

((0 119879) timesR119889+1

+) satisfying

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(152)

Hence applying Δ 119895 119895 ge 0 to (81) we see that

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(153)

and thus by Lemma 30 we can deduce that10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+int

119905

0

119890minus12058122119895(119905minus120591)10038171003817

100381710038171003817Δ 119895119891(120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(154)

Then it follows from convolution inequalities thatΔ 1198951199061198711199021 ((0119879)119871

119901

120573(R119889+1+

))is dominated by

(

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

(155)

with 11199022 = 1 + 11199021 minus 1119902 Moreover similarly as above wecan obtain that

1003817100381710038171003817Δminus1119906(119905 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)le1003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

1003817100381710038171003817Δminus1119891 (120591 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(156)

and thus if 1 le 119902 le 1199021 le infin1003817100381710038171003817Δminus1119906

10038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862(119879111990211003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)+ 119879

111990221003817100381710038171003817Δminus1119891

1003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

)))

(157)

Journal of Function Spaces and Applications 13

Finally taking the 119897119903-normwith respect to 119895 in (155) and (157)with the usual convention if 119903 = infin we can deduce the desiredestimate

Acknowledgment

Theauthor gratefully acknowledges theDeanship of ScientificResearch at the University of Taibah The author is deeplyindebted to the referee for providing constructive commentsand help in improving the contents of this paper

References

[1] Z Ben Nahia and N Ben Salem ldquoSpherical harmonics andapplications associated with the Weinstein operatorrdquo in Pro-ceedings of the International Conference on PotentialTheory heldin Kouty Czech Republic (ICPT rsquo94) pp 235ndash241 1996

[2] Z Ben Nahia and N Ben Salem ldquoOn a mean value propertyassociated with the Weinstein operatorrdquo in Proceedings of theInternational Conference on Potential Theory held in KoutyCzech Republic (ICPT rsquo94) pp 243ndash253 1996

[3] M Brelot ldquoEquation de Weinstein et potentiels de MarcelRieszrdquo in Seminaire de Theorie de Potentiel Paris No 3 vol 681of Lecture Notes in Mathematics pp 18ndash38 Springer BerlinGermany 1978

[4] H Mejjaoli and M Salhi ldquoUncertainty principles for theweinstein transformrdquo Czechoslovak Mathematical Journal vol61 no 4 pp 941ndash974 2011

[5] H Mejjaoli and A Ould Ahmed Salem ldquoWeinstein Gabortransform and applicationsrdquo Advanced Studies in Pure Mathe-matics vol 2 no 3 pp 203ndash210 2012

[6] H Mejjaoli ldquoBesov spaces associated withthe Weinstein opera-tor and applicationsrdquo In press

[7] T Kawazoe and H Mejjaoli ldquoGeneralized Besov spaces andtheir applicationsrdquo Tokyo Journal of Mathematics vol 35 no 2pp 297ndash320 2012

[8] H Mejjaoli ldquoLittlewood-Paley decomposition associated withthe Dunkl operators and paraproduct operatorsrdquo Journal ofInequalities in Pure and Applied Mathematics vol 9 no 4 pp1ndash25 2008

[9] H Mejjaoli and N Sraeib ldquoGeneralized sobolev spaces inquantum calculus and applicationsrdquo Journal of Inequalities andSpecial Functions vol 1 no 4 pp 43ndash64 2012

[10] H Mejjaoli ldquoGeneralized homogeneous Besov spaces and theirapplicationsrdquo Serdica Mathematical Journal vol 38 no 4 pp575ndash614 2012

[11] H Triebel Interpolation Theory Functions Spaces DifferentialOperators North-Holland AmsterdamThe Netherlands 1978

[12] V S Guliev ldquoOn maximal function and fractional integralassociated with the Bessel differential operatorrdquo MathematicalInequalities and Applications vol 6 no 2 pp 317ndash330 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Function Spaces and Applications

By definition of the homogeneous Weinstein-Besov spaceswe therefore infer10038171003817100381710038171198911003817100381710038171003817B1199041120573

11990111199021(R119889+1+

)

= (

infin

sum

119895=minusinfin

(2119895119904110038171003817100381710038171003817Δ 119895119891

100381710038171003817100381710038171198711199011

120573(R119889+1+

))

1199021

)

11199021

le 119862(sum

119895isinZ

(211989511990412119895(119889+2120573+2)(1119901minus1119901

1)10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

1199021

)

11199021

le 119862(sum

119895isinZ

(211989511990410038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

1199021

)

11199021

le 11986210038171003817100381710038171198911003817100381710038171003817B119904120573

119901119902(R119889+1+

)

(57)

since 119902 le 1199021 This gives the inclusion

Proposition 21 (1) If 119906 belongs to B119904120573

119901119902(R119889+1

+) cap

B119905120573

119901119902(R119889+1

+)

then 119906 belongs to B120579119904+(1minus120579)119905120573

119901119902 (R119889+1

+) for all 120579 isin [0 1] and

119906B120579119904+(1minus120579)119905120573

119901119902(R119889+1+

)le 119906

120579

B119904120573

119901119902(R119889+1+

)119906

1minus120579

B119905120573

119901119902(R119889+1+

) (58)

(2) If 119906 belongs to B119904120573

119901infin(R119889+1

+) cap

B119905120573

119901infin(R119889+1

+) and 119904 lt 119905

then 119906 belongs to B120579119904+(1minus120579)119905120573

1199011(R119889+1

+) for all 120579 isin (0 1) and there

exists a positive constant 119862(119905 119904) such that

119906B120579119904+(1minus120579)119905120573

1199011(R119889+1+

)le 119862 (119905 119904) 119906

120579

B119904120573

119901infin(R119889+1+

)119906

1minus120579

B119905120573

119901infin(R119889+1+

) (59)

(3) If 119906 belongs to B119904120573

119901infin(R119889+1

+) cap

B119904+120576120573

119901infin (R119889+1

+) and 120576 gt

0 then 119906 belongs to B119904120573

1199011(R119889+1

+) and there exists a positive

constant 119862 such that

119906B119904120573

1199011(R119889+1+

)le

119862

120576

119906B119904120573

119901infin(R119889+1+

)log

2(119890 +

119906B119904+120576120573

119901infin(R119889+1+

)

119906B119904120573

119901infin(R119889+1+

)

)

(60)

Proof (1) is obvious from the Holderrsquos inequality As for (2)we write 119906

B120579119904+(1minus120579)119905120573

1199011(R119889+1+

)as

sum

119895le119873

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ sum

119895gt119873

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

(61)

where 119873 is chosen here after By the definition of thehomogeneous Weinstein-Besov norms we see that

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 2

119895(1minus120579)(119905minus119904)119906

B119904120573

119901infin(R119889+1+

)

2119895(120579119904+(1minus120579)119905)10038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 2

minus119895120579(119905minus119904)119906

B119905120573

119901infin(R119889+1+

)

(62)

and thus 119906B120579119904+(1minus120579)119905120573

1199011(R119889+1+

)is dominated by

119906B119904120573

119901infin(R119889+1+

)sum

119895le119873

2119895(1minus120579)(119905minus119904)

+ 119906B119905120573

119901infin(R119889+1+

)sum

119895gt119873

2minus119895120579(119905minus119904)

le 119862119906B119904120573

119901infin(R119889+1+

)

2(119873+1)(1minus120579)(119905minus119904)

2(1minus120579)(119905minus119904)

minus 1

+ 119906B119905120573

119901infin(R119889+1+

)

2minus119873120579(119905minus119904)

1 minus 2minus120579(119905minus119904)

(63)

Hence in order to complete the proof of (2) it suffices tochoose119873 such that

119906B119905120573

119901infin(R119889)

119906B119904120573

119901infin(R119889)

le 2119873(119905minus119904)

lt 2

119906B119905120573

119901infin(R119889)

119906B119904120573

119901infin(R119889)

(64)

As for (3) it is easy to see that 119906B119904120573

1199011(R119889+1+

)is dominated as

sum

119895le119873minus1

211989511990410038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ sum

119895ge119873

211989511990410038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le (119873 + 1) 119906B119904120573

119901infin(R119889+1+

)+

2minus(119873minus1)120576

2120576minus 1

119906B119904+120576120573

119901infin(R119889+1+

)

(65)

Hence letting

119873 = 1 + [

[

1

120576

log2

119906B119904+120576120573

119901infin(R119889+1+

)

119906B119904120573

119901infin(R119889+1+

)

]

]

(66)

we can obtain the desired estimate

Proposition 22 Let 119902 isin (1infin) and let 119904 isin R such that0 lt 119904 lt (119889 + 2120573 + 2)119902 then one has

B119904120573

119902119902(R

119889+1

+) =

F119904120573

119902119902(R

119889+1

+) 997893rarr

F119904120573

119902infin(R

119889+1

+)

997893rarrF119904minus((119889+2120573+2)119902)120573

infininfin(R

119889+1

+)

(67)

H119904

119902120573(R

119889+1

+) =

F119904120573

1199022(R

119889+1

+) 997893rarr

F119904120573

119902infin(R

119889+1

+)

997893rarrF119904minus((119889+2120573+2)119902)120573

infininfin(R

119889+1

+)

(68)

Proof We obtain these results by the similar ideas used in thenonhomogeneous case (cf [6])

Theorem 23 Let 119886 119887 gt 0 and let 1199021 1199022 isin [1infin] Let120579 = 119886(119886 + 119887) isin (0 1) and let 1119901 = (1 minus 120579)1199021 +

1205791199022 Then there exists a constant 119862 such that for every119891 isin

F119886120573

1199021infin(R119889+1

+) cap

Fminus119887120573

1199022infin(R119889+1

+) then one has

1003816100381610038161003816119891 (119909)

1003816100381610038161003816le 119862(sup

119895isinZ

2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

1minus120579

(sup119895isinZ

2minus119887119895 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

120579

(69)

In particular one gets

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

F119886120573

1199021infin(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

120579

Fminus119887120573

1199022infin(R119889+1+

) (70)

Journal of Function Spaces and Applications 7

Proof Let 119891 be a Schwartz class we have

1003816100381610038161003816119891 (119909)

1003816100381610038161003816le sum

119895isinZ

10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816

le sum

119895isinZ

min(2minus119886119895sup119895isinZ

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816))

(71)

We define119873(119909) as the largest index such that

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816) le 2

minus119886119895sup119895isinZ

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816) (72)

and we write1003816100381610038161003816119891 (119909)

1003816100381610038161003816le sum

119895le119873(119909)

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

+ sum

119895gt119873(119909)

2minus119886119895sup

119895

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

le 119862(sup119895isinZ

2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

119887(119886+119887)

times (sup119895isinZ

2minus119887119895 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

119886(119886+119887)

(73)

Thus (69) is proved In order to obtain (70) it is enough toapply the Holder inequality in the expression above since wehave 120579 = 119886(119886+119887) isin (0 1) and let 1119901 = (1minus120579)1199021+1205791199022

Corollary 24 Let 119902 isin (1infin) and let 119904 isin R such that 0 lt 119904 lt

(119889 + 2120573 + 2)119902 then one has

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus(119902119901)

Bminus((2120573+2+119889)119902minus119904)120573

infininfin (R119889+1

+ )

10038171003817100381710038171198911003817100381710038171003817

119902119901

B119904120573

119902119902(R119889+1

+ )

(74)

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus(119902119901)

Bminus((2120573+2+119889)119902minus119904)120573

infininfin (R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

119902119901

H119904119902120573(R119889+1+

) (75)

where 119901 = 119902(2120573 + 2 + 119889)(2120573 + 2 + 119889 minus 119902119904)

Proof By choosing 119886 = 119904 gt 0 minus119887 = 119904 minus (119889 + 2120573 + 2)119902 lt 01199021 = 119902 and 1199022 = infin we deduce (74) from the relations (70)and (67) In the same way we deduce (75) from the relations(70) and (68)

4 Generalized Heat Equation

41 Characterization for the Weinstein-Besov Spaces TheWeinstein heat equation reads

120597119905119906 (119905 119909) minus Δ 120573119906 (119905 119909) = 119891 (119905 119909) (119905 119909) isin [0infin) timesR119889+1

+

119906|119905=0 = 119892

(76)

We introduce the Weinstein heat semigroup 119867120573(119905) for theWeinstein-Laplace operator

119867120573 (119905) 119892 (119909) =

int

R119889+1+

Γ120573 (119905 119909 119910) 119892 (119910) 119889120583120573 (119910) if 119905 gt 0

119892 (119909) if 119905 = 0(77)

where Γ120573 is the Weinstein heat kernel defined by

Γ120573 (119905 119909 119910) = 120591119909 (119864(120573)

119905) (119910) (78)

where

119864(120573)

119905(119910) =

2

1205871198892Γ (120573 + 1) (4119905)

120573+1+1198892119890minus11991024119905 (79)

Thus

119867120573 (119905) 119892 (119909) = 119892lowast119882119864(120573)

119905 (119909) (80)

In practice we use the integral formulation of (76)

119906 (119905 119909) = 119867120573 (119905) 119892 (119909) + 119866120573 (119891) (119905 119909)

= 119867120573 (119905) 119892 (119909) + int

119905

0

119867120573 (119905 minus 119904) 119891 (119904 119909) 119889119904

(81)

Remark 25 The function 119864(120573)119905

is the Gauss kernel associatedwith Weinstein operators This function satisfies

forall120585 isin R119889+1

+ F119882 (119864

(120573)

119905) (120585) = 119890

minus1199051205852

(82)

Proposition 26 Let 1 le 119901 le 119903 le infin and let 119891 isin 119871119901

120573(R119889+1

+)

Then the operator 119867120573(119905) maps 119871119901

120573(R119889+1

+) continuously to

119871119903

120573(R119889+1

+) and

10038171003817100381710038171003817119867120573 (119905) 119891

10038171003817100381710038171003817119871119903120573(R119889)

le 119862119905minus((119889+2120573+2)2)(1119901minus1119903)1003817

1003817100381710038171198911003817100381710038171003817119871119901

120573(R119889)

(83)

Moreover1003817100381710038171003817100381710038171003817

(minusΔ 120573)

1205752

119867120573 (119905) 119891

1003817100381710038171003817100381710038171003817119871119903120573(R119889)

le 119862119905minus1205752minus((119889+2120573+2)2)(1119901minus1119903)1003817

1003817100381710038171198911003817100381710038171003817119871119901

120573(R119889)

(84)

for all 120575 gt 0

Proof It follows from the relations (80) and (29) combinedwith scaling property of the kernel 119864(120573)

119905

In this section we prove estimates for the Weinstein heatsemigroupThese estimates are based on the following result

Lemma 27 Let C be an annulus Positive constants 119888 and 119862exist such that for any 119901 in [1infin] and any couple (119905 120582) ofpositive real numbers one has

suppF119882 (119906) sub 120582C 997904rArr

10038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le 119862119890minus1198881199051205822

119906119871119901

120573(R119889+1+

)

(85)

8 Journal of Function Spaces and Applications

Proof We again consider a function Θ in 119863(R119889+1

+ 0) the

value of which is identically 1 in neighborhood of annulusCWe can also assume without loss of generality that 120582 = 1 Wethen have

119867120573 (119905) 119906 = 119892 (119905 sdot) lowast119882119906 (86)where

119892 (119905 sdot) = Fminus1

119882(Θ (120585) 119890

minus1199051205852

) (87)

The lemma is proved provided that we can find positive realnumbers 119888 and 119862 such that

forall119905 gt 01003817100381710038171003817119892 (119905 sdot)

10038171003817100381710038171198711120573(R119889+1+

)le 119862119890

minus119888119905 (88)

To begin we perform integrations by parts in (87) We get1003816100381610038161003816119892 (119905 119909)

1003816100381610038161003816=

1

119888120573

(1 + 1199092)

minus(119889+[2120573]+3)

times int

R119889+1+

Λ (119909 120585) (119868119889 minus Δ 120573)

119889+[2120573]+3

times (Θ (120585) 119890minus1199051205852

) 119889120583120573 (120585)

(89)

Using Leibnizrsquos formula we obtain1003816100381610038161003816119892 (119905 119909)

1003816100381610038161003816le 119862(1 + 119909

2)

minus(119889+[2120573]+3)

119890minus119888119905 (90)

and (88) follows

For any interval 119868 of R (bounded or unbounded) wedefine themixed space-time 119871119901(119868 119871119902

120573(R119889+1

+)) Banach space of

(classes of) measurable functions 119906 119868 rarr 119871119902

120573(R119889+1

+) such

that 119906119871119901(119868119871

119902

120573(R119889+1+

))lt infin with

119906119871119901(119868119871119902

120573(R119889+1+

))= (int

119868

119906 (119905 sdot)119901

119871119902

120573(R119889+1+

)119889119905)

1119901

if 1 le 119901 119902 lt infin

119906119871infin(119868119871119902

120573(R119889+1+

))= ess sup

119905isin119868

119906 (119905 sdot)119871119902

120573(R119889+1+

) if 1 le 119902 lt infin

(91)Corollary 28 Let C be an annulus and 120582 a positive realnumber Let 1199060 (resp 119891 = 119891(119905 119909)) satisfy suppF119882(1199060) sub 120582C(resp suppF119882(119891(119905 sdot)) sub 120582C for all 119905 in [0 119879]) Consider 119906 asolution of

120597119905119906 minus Δ 120573119906 = 0 119906|119905=0 = 1199060 (92)and V a solution of

120597119905V minus Δ 120573V = 119891 (119905 sdot) V|119905=0 = 0 (93)There exist positive constants 119888 and 119862 depending only on Csuch that for any 1 le 119886 le 119887 le infin and 1 le 119901 le 119902 le infin we have

119906119871119902([0119879]119871119887120573(R119889+1+

))le 119862120582

(119889+2120573+2)(1119886minus1119887)120582minus21199021003817

1003817100381710038171199060

1003817100381710038171003817119871119886120573(R119889+1+

)

V119871119902([0119879]119871119887120573(R119889+1+

))le 119862120582

minus2(1+1119902minus1119901)120582(119889+2120573+2)(1119886minus1119887)

times10038171003817100381710038171198911003817100381710038171003817119871119901([0119879]119871119886

120573(R119889+1+

))

(94)

Proof It suffices to use the fact that

119906 (119905 sdot) = 119867120573 (119905) 1199060 V (119905 sdot) = int119905

0

119867120573 (119905 minus 119904) 119891 (119904 sdot) 119889119904

(95)

Combining Lemma 27 and Youngrsquos inequality (29) withscaling property of the kernel 119864(120573)

119905now yields the result

Theorem 29 Let 119904 be a positive real number and (119901 119903) isin

[1infin]2 A constant 119862 exists which satisfies the following

property For 119906 isin Bminus2119904120573

119901119903 (R119889+1

+) one has

119862minus1119906

Bminus2119904120573

119901119903(R119889+1+

)le

10038171003817100381710038171003817100381710038171003817

10038171003817100381710038171003817119905119904119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862119906Bminus2119904120573

119901119903(R119889+1+

)

(96)

To prove this result we need the following lemma

Lemma 30 There exist two positive constants 120581 and 119862

depending only on 120593 such that for all 1 le 119901 le infin 120591 ge 0 and119895 isin Z one has

10038171003817100381710038171003817Δ 119895 (119867120573 (120591) 119906)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862119890

minus1205812211989512059110038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

) (97)

Proof The result follows immediately by applying Lemma 27and because Δ 119895(119867120573(120591))119906 = (119867120573(120591)Δ 119895)119906

Proof of Theorem 29 Using Lemma 30 and considering thefact that the operator Δ 119895 commutes with the operator 119867120573(119905)

and the definition of the homogeneous Weinstein-Besov(semi) norm we get

10038171003817100381710038171003817119905119904119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862119906

Bminus2119904120573

119901119903(R119889+1+

)sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903119895

(98)

where (119888119903119895)119895isinZ denotes as in all this proof a generic elementof the unit sphere of 119897119903(Z) In the case when 119903 = infin therequired inequality comes immediately from the followingeasy result For any positive 119904 we have

sup119905gt0

sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

lt infin (99)

In the case 119903 lt infin using the Holder inequality with theweight 22119895119904119890minus1205811199052

2119895

(99) and the Fubini theorem we obtain

int

infin

0

11990511990311990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

119889119905

119905

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)int

infin

0

(sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

)

119903minus1

times (sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903

119903119895)

119889119905

119905

Journal of Function Spaces and Applications 9

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)int

infin

0

(sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903

119903119895)

119889119905

119905

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)sum

119895isinZ

119888119903

119903119895int

infin

0

(11990511990422119895119904119890minus12058111990522119895

)

119889119905

119905

le 119862Γ (119904) 119906119903

Bminus2119904120573

119901119903(R119889+1+

)

(100)

In order to prove the other inequality let us observe thatfor any 119904 greater than minus1 we have

Δ 119895119906 =

1

Γ (119904 + 1)

int

infin

0

119905119904(minusΔ 120573)

119904+1

119867120573 (119905) Δ 119895119906 119889119905 (101)

Then Lemma 30 Proposition 9 and the fact that the operatorΔ 119895 commutes with the operator119867120573(119905) lead to the following

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862int

infin

0

11990511990422119895(119904+1)

119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889119905

(102)

In the case 119903 = infin we simply write

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862(sup

119905gt0

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

times int

infin

0

22119895(119904+1)

119890minus12058111990522119895

119889119905

le 11986222119895119904(sup119905gt0

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

(103)

In the case 119903 lt infin Holderrsquos inequality with the weight 119890minus12058111990522119895

gives

(int

infin

0

119905119904119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889119905)

119903

le 1198622minus2119895(119903minus1)

int

infin

0

119905119903119904119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)119889119905

(104)

Thanks to (99) and Fubinirsquos theorem we infer from (102) that

sum

119895isinZ

2minus211989511990311990410038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)le 119862int

infin

0

11990511990311990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

119889119905

119905

(105)

The theorem is proved

Second Proof ofTheorem 29We only consider the case 1 le 119903 ltinfinThe case 119903 = infin can be shown similarlyWe first prove that

119862minus1119906

Bminus2119904120573

119901119903(R119889+1+

)le

10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

(106)

It is easy to see that

Δ 119895119906 = 120594119895lowast119882119864(120573)

2minus2119895lowast119882119906

(107)

where 120594119895 = Fminus1

119882(120593(2

minus119895120585)119890

2minus21198951205852

) and119864(120573)2minus2119895

is the Gauss kernelassociated with Weinstein operators By relation (29) we get

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le

10038171003817100381710038171003817120594119895

100381710038171003817100381710038171198711120573(R119889+1+

)

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

) (108)

As10038171003817100381710038171003817120594119895

100381710038171003817100381710038171198711120573(R119889+1+

)= int

R119889+1+

100381610038161003816100381610038161003816F

minus1

119882(120593 (120585) 119890

1205852

)

100381610038161003816100381610038161003816119889120583120573 (120585) lt infin (109)

we obtain10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

) (110)

Moreover simple calculations give that

119864(120573)

2minus2119895lowast119882119906 = 119867120573 (2

minus4119895minus 119905

2) (119864

(120573)

1199052lowast119882119906)

(111)

Thus from Proposition 26 it follows that100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1

+ )

(112)

for any 119905 isin [2minus119895minus1 2minus119895] which implies that

sum

119895isinZ

2minus211990411989511990310038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

le 119862 sum

119895isinZ

int

2minus119895

2minus119895minus1(119905

2119904100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862int

infin

0

(1199052119904100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862int

infin

0

(11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

(113)

where we have used the fact that 119864(120573)1199052lowast119882119906 = 119867120573(119905

2)119906

We now prove that10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862119906Bminus2119904120573

119901119903(R119889+1+

) (114)

Indeed one has

119864(120573)

2minus2119895lowast119882119906 = sum

119899isinZ

119864(120573)

2minus2119895lowast119882Δ 119899+119895119906 (115)

Arguing as above we have100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1

+ )

(116)

for any 119905 isin [2minus119895 21minus119895] Thus10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

= 2

100381710038171003817100381710038171003817100381710038171003817

1003817100381710038171003817100381710038171199052119904119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)

100381710038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862 sum

119895isinZ

int

21minus119895

2minus119895(2

minus2119895119904100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862 sum

119895isinZ

(2minus2119895119904

sum

119899isinZ

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

(117)

10 Journal of Function Spaces and Applications

On the other hand it is easy to see that100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 1198622

minus211989911990410038171003817100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

) (118)

for any 119904 gt 0 For 0 lt 1199041 lt 119904 lt 1199042 and by using theMinkowskiinequality we have

sum

119895isinZ

(2minus2119895119904

sum

119899isinZ

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

le sum

119895isinZ

(2minus2119895119904

0

sum

minusinfin

2minus21198991199041

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

+ sum

119895isinZ

(2minus2119895119904

sum

119899isinN

2minus21198991199042

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

le 119862(

0

sum

minusinfin

2minus2119899(119904

1minus119904)(sum

119895isinZ

(2minus2(119899+119895)11990311990410038171003817

100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

)

1119903

)

119903

+ 119862(sum

N

2minus2119899(119904

2minus119904)(sum

119895isinZ

(2minus2(119899+119895)11990311990410038171003817

100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

)

1119903

)

119903

le 119862 sum

119895isinZ

2minus211989511990311990410038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

(119)

The result is immediately from (117) and (119)

42 Embedding Sobolev Theorems

Theorem 31 Let 1 lt 119901 lt infin and let 0 lt 119904 lt (119889 + 2120573 +

2)119901There exists a positive constant119862 such that for all function119891 isin

H119904

119901120573(R119889+1

+) one has

10038171003817100381710038171198911003817100381710038171003817119871119902

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

H119904119901120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

120579

B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

) (120)

where 120579 = 119904119901(119889+2120573+2) and 119902 = 119901(119889+2120573+2)(119889+2120573+2minus119901119904)

Proof Bydensity we can suppose that119891belongs toSlowast(R119889+1

)It is easy to see that

119891 = int

infin

0

119867120573 (119905) Δ 120573119891119889119905(121)

and decompose the integral in two parts as follows

119891 = int

119860

0

119867120573 (119905) Δ 120573119891119889119905 + int

infin

119860

119867120573 (119905) Δ 120573119891119889119905(122)

where 119860 is a constant to be fixed laterOn the other hand byTheorem 29 we obtain10038171003817100381710038171003817119867120573 (119905) Δ 120573119891

10038171003817100381710038171003817119871infin120573(R119889+1+

)

le

119862

1199051minus(12)(119904minus(119889+2120573+2)119901)

10038171003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(123)

Therefore after integrating we get

int

infin

119860

10038171003817100381710038171003817119867120573 (119905) Δ 120573119891

10038171003817100381710038171003817119871infin120573(R119889+1+

)119889119905

le 119860(12)(119904minus(119889+2120573+2)119901)1003817

1003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(124)

On the other hand denoting 119892 = (minusΔ 120573)1199042119891 we have

119867120573 (119905) Δ 120573119891 =

1

(minus119905)1minus1199042

119867120573 (119905) (minus119905Δ 120573)

1minus1199042

119892 (125)

We proceed as in [8] we prove that1003816100381610038161003816100381610038161003816

119867120573 (119905) (minus119905Δ 120573)

1minus1199042

119892 (119909)

1003816100381610038161003816100381610038161003816

le 119862 (119904)119872120573 (119892) (119909) (126)

where119872120573(119892) is a maximal function of 119892 associated with theWeinstein operators (cf [12])

This leads to100381610038161003816100381610038161003816100381610038161003816

int

119860

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

100381610038161003816100381610038161003816100381610038161003816

le 1198621198601199042119872120573 (119892) (119909) (127)

In conclusion we get10038161003816100381610038161003816100381610038161003816

int

infin

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

10038161003816100381610038161003816100381610038161003816

le 119862 (1198601199042119872120573 (119892) (119909)

+119860(12)(119904minus(119889+2120573+2)119901)1003817

1003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

))

(128)

and the choice of 119860 such that

119860(119889+2120573+2)2119901

119872120573 (119892) (119909) =10038171003817100381710038171198911003817100381710038171003817B119904minus(119889+2120573+2)119901120573

infininfin (R119889+1+

)(129)

ensures that10038161003816100381610038161003816100381610038161003816

int

infin

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

10038161003816100381610038161003816100381610038161003816

le 119862(119872120573(119892)(119909))

1minus(119901119904(119889+2120573+2))10038171003817100381710038171198911003817100381710038171003817

119901119904(119889+2120573+2)

B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(130)

Finally taking the 119871119902120573norm with 119902 = 119901(119889 + 2120573 + 2)(119889 + 2120573 +

2minus119901119904) ends the proof thanks to the fact themaximal function119872120573 is bounded of 119871119902

120573(R119889+1

+) into itself for 119902 gt 1

Theorem 32 Let 1 lt 119901 lt 119902 lt infin For all function 119891 such that119891 isin

H1199041

119901120573(R119889+1

+)⋂

Bminus120573120573

infininfin(R119889+1

+) one has

10038171003817100381710038171198911003817100381710038171003817H119904119901120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

120579

H1199041

119901120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573120573

infininfin(R119889+1

+) (131)

where 120579 = 119901119902 119904 = 1205791199041 minus (1 minus 120579)120573 with 120573 gt 0 minus120573 lt 119904 lt 1199041

Proof It suffices to prove that1003817100381710038171003817100381710038171003817

(minusΔ 120573)

(119904minus1199041)2

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le 11986210038171003817100381710038171198911003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(132)

Journal of Function Spaces and Applications 11

Indeed we use the following identity (which may be easilyproven by taking the Weinstein transform in 119909 of both sides)

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

infin

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905 (133)

with 120575 = 1199041 minus 119904 gt 0We decompose the integral in two parts as follows

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

119879

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

+

1

Γ (1205752)

int

infin

119879

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

(134)

where 119879 is a constant to be fixed laterWe proceed as in [8] we obtain

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119872120573 (119891) (119909) (135)

On the other hand we use Theorem 29 and the fact that 119891belongs to Bminus120573minus119904

1120573

infininfin(R119889+1

+) to deduce that

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119905

(minus120573minus1199041)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

) (136)

Thus by applying the preceding estimates on the right part of(134) we obtain

1003816100381610038161003816100381610038161003816

(minusΔ 120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621

Γ (1205752)

1198791205752119872120573 (119891) (119909)

+

1198622

Γ (1205752)

119879(120575minus120573minus119904

1)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

(137)

We fix now

119879 = (

10038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

119872120573 (119891) (119909)

)

2(120573+1199041)

(138)

We obtain

1003816100381610038161003816100381610038161003816

(minusΔ120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621 + 1198622

Γ (1205752)

(119872120573(119891)(119909))

12057910038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(139)

Thus we deduce that1003817100381710038171003817100381710038171003817

(minusΔ120573)

minus1205752

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le

1198621 + 1198622

Γ (1205752)

10038171003817100381710038171003817119872120573 (119891)

10038171003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(140)

To conclude we used the fact that the maximal function119872120573

is bounded of 119871119902120573(R119889+1

+) into itself for 119902 gt 1

43 Estimates in Generalized Besov Spaces For any interval 119868ofR (bounded or unbounded) and a normed space 119865(R119889+1

+)

we define the mixed space-time 119871119901(119868 119865(R119889+1

+)) space of

(classes of) measurable functions 119906 119868 rarr 119865(R119889+1

+) such that

||119906||119871119901(119868119865(R119889+1+

)) lt infin with

119906119871119901(119868119865(R119889+1+

)) = (int

119868

119906 (119905 sdot)119901

119865(R119889+1+

)119889119905)

1119901

if 1 le 119901 lt infin

119906119871infin(119868119865(R119889+1+

)) = ess sup119905isin119868

119906 (119905 sdot)119865(R119889+1+

)

(141)

For any interval 119868 of R (bounded or unbounded) anda Banach space 119883 we define the mixed space-time 119862(119868119883)space of continuous functions 119868 rarr 119883 When 119868 is bounded119862(119868119883) is a Banach space with the norm of 119871infin(119868 119883)

Theorem 33 Let 119904 isin R and 1 le 119901 119902 119903 le infin Let 119879 gt 0 119892 isin

B119904120573

119901119903(R119889+1

+) and119891 in 119871119902((0 119879) B

119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76)

has a unique solution

119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903(R

119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R

119889+1

+))

(142)

and there exists a constant 119862 such that for all 1199021 isin [119902infin] onehas

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862(10038171003817100381710038171198921003817100381710038171003817B119904120573

119901119903(R119889+1+

)+10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)B

119904minus2+(2119902)120573

119901119903(R119889+1+

)))

(143)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] B119904120573

119901119903(R119889+1

+))

Proof Since 119892 and 119891 are temperate distributions (76) has aunique solution 119906 in S1015840

((0 119879) timesR119889+1

+) which satisfies

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(144)

Next we notice that applying Δ 119895 to (76) and using formula(81) yield

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(145)

Therefore

10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(146)

12 Journal of Function Spaces and Applications

By virtue of Lemma 30 we thus have for some 120581 gt 0

10038171003817100381710038171003817Δ 119895119906(119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

119890minus12058122119895(119905minus120591) 10038171003817

100381710038171003817Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(147)

Applying convolution inequalities we get

10038171003817100381710038171003817Δ 119895119906

100381710038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862[

[

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

)

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

]

]

(148)

with 11199022 = 1+ 11199021 minus1119902 Finally taking the 119897119903(Z) norm we

conclude that (with the usual convention if 119903 = infin)

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

1199031199021

)(211989511990410038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

]

]

1119903

+ 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

1199031199022

)

times (2119895(119904minus2+2119902)10038171003817

100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902(0119879119871

119901

120573(R119889+1+

))

119903

]

]

1119903

(149)

which insures that 119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903 (R119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R119889+1

+)) and yields the desired inequality

Since 119906 belongs to 119862([0 119879]B119904120573

119901119903(R119889+1

+)) in the case where

119903 is finite may be easily deduced from the density ofSlowast(R

119889+1)⋂

B119904120573

119901119903(R119889+1

+) in B

119904120573

119901119903(R)

Theorem 34 Let 119904 isin R 119879 gt 0 and 1 le 119901 119902 119903 le infinOne supposes that 119892 isin 119861

119904120573

119901119903(R119889+1

+) and 119891 isin 119871

119902((0 119879)

119861119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76) has a unique solution 119906 belong-

ing to

119871119902((0 119879) 119861

119904+(2119902)120573

119901119903(R

119889+1

+))⋂119871

infin((0 119879) 119861

119904120573

119901119903(R

119889+1

+))

(150)

and there exists a constant 119862 such that for all 119902 le 1199021 le infin

1199061198711199021 ((0119879)119861

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862 [ (1 + 11987911199021)10038171003817100381710038171198921003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (1 + 1198791+1119902

1minus1119902

)10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)119861

119904minus2+(2119902)120573

119901119903(R119889+1+

))]

(151)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] 119861119904120573119901119903(R119889+1

+))

Proof Since 119892 119891 are tempered (76) has a unique solution 119906in S1015840

((0 119879) timesR119889+1

+) satisfying

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(152)

Hence applying Δ 119895 119895 ge 0 to (81) we see that

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(153)

and thus by Lemma 30 we can deduce that10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+int

119905

0

119890minus12058122119895(119905minus120591)10038171003817

100381710038171003817Δ 119895119891(120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(154)

Then it follows from convolution inequalities thatΔ 1198951199061198711199021 ((0119879)119871

119901

120573(R119889+1+

))is dominated by

(

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

(155)

with 11199022 = 1 + 11199021 minus 1119902 Moreover similarly as above wecan obtain that

1003817100381710038171003817Δminus1119906(119905 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)le1003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

1003817100381710038171003817Δminus1119891 (120591 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(156)

and thus if 1 le 119902 le 1199021 le infin1003817100381710038171003817Δminus1119906

10038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862(119879111990211003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)+ 119879

111990221003817100381710038171003817Δminus1119891

1003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

)))

(157)

Journal of Function Spaces and Applications 13

Finally taking the 119897119903-normwith respect to 119895 in (155) and (157)with the usual convention if 119903 = infin we can deduce the desiredestimate

Acknowledgment

Theauthor gratefully acknowledges theDeanship of ScientificResearch at the University of Taibah The author is deeplyindebted to the referee for providing constructive commentsand help in improving the contents of this paper

References

[1] Z Ben Nahia and N Ben Salem ldquoSpherical harmonics andapplications associated with the Weinstein operatorrdquo in Pro-ceedings of the International Conference on PotentialTheory heldin Kouty Czech Republic (ICPT rsquo94) pp 235ndash241 1996

[2] Z Ben Nahia and N Ben Salem ldquoOn a mean value propertyassociated with the Weinstein operatorrdquo in Proceedings of theInternational Conference on Potential Theory held in KoutyCzech Republic (ICPT rsquo94) pp 243ndash253 1996

[3] M Brelot ldquoEquation de Weinstein et potentiels de MarcelRieszrdquo in Seminaire de Theorie de Potentiel Paris No 3 vol 681of Lecture Notes in Mathematics pp 18ndash38 Springer BerlinGermany 1978

[4] H Mejjaoli and M Salhi ldquoUncertainty principles for theweinstein transformrdquo Czechoslovak Mathematical Journal vol61 no 4 pp 941ndash974 2011

[5] H Mejjaoli and A Ould Ahmed Salem ldquoWeinstein Gabortransform and applicationsrdquo Advanced Studies in Pure Mathe-matics vol 2 no 3 pp 203ndash210 2012

[6] H Mejjaoli ldquoBesov spaces associated withthe Weinstein opera-tor and applicationsrdquo In press

[7] T Kawazoe and H Mejjaoli ldquoGeneralized Besov spaces andtheir applicationsrdquo Tokyo Journal of Mathematics vol 35 no 2pp 297ndash320 2012

[8] H Mejjaoli ldquoLittlewood-Paley decomposition associated withthe Dunkl operators and paraproduct operatorsrdquo Journal ofInequalities in Pure and Applied Mathematics vol 9 no 4 pp1ndash25 2008

[9] H Mejjaoli and N Sraeib ldquoGeneralized sobolev spaces inquantum calculus and applicationsrdquo Journal of Inequalities andSpecial Functions vol 1 no 4 pp 43ndash64 2012

[10] H Mejjaoli ldquoGeneralized homogeneous Besov spaces and theirapplicationsrdquo Serdica Mathematical Journal vol 38 no 4 pp575ndash614 2012

[11] H Triebel Interpolation Theory Functions Spaces DifferentialOperators North-Holland AmsterdamThe Netherlands 1978

[12] V S Guliev ldquoOn maximal function and fractional integralassociated with the Bessel differential operatorrdquo MathematicalInequalities and Applications vol 6 no 2 pp 317ndash330 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 7

Proof Let 119891 be a Schwartz class we have

1003816100381610038161003816119891 (119909)

1003816100381610038161003816le sum

119895isinZ

10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816

le sum

119895isinZ

min(2minus119886119895sup119895isinZ

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816))

(71)

We define119873(119909) as the largest index such that

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816) le 2

minus119886119895sup119895isinZ

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816) (72)

and we write1003816100381610038161003816119891 (119909)

1003816100381610038161003816le sum

119895le119873(119909)

2119895119887sup119895isinZ

(2minus119895119887 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

+ sum

119895gt119873(119909)

2minus119886119895sup

119895

(2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

le 119862(sup119895isinZ

2119886119895 10038161003816100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

119887(119886+119887)

times (sup119895isinZ

2minus119887119895 10038161003816

100381610038161003816Δ 119895119891 (119909)

10038161003816100381610038161003816)

119886(119886+119887)

(73)

Thus (69) is proved In order to obtain (70) it is enough toapply the Holder inequality in the expression above since wehave 120579 = 119886(119886+119887) isin (0 1) and let 1119901 = (1minus120579)1199021+1205791199022

Corollary 24 Let 119902 isin (1infin) and let 119904 isin R such that 0 lt 119904 lt

(119889 + 2120573 + 2)119902 then one has

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus(119902119901)

Bminus((2120573+2+119889)119902minus119904)120573

infininfin (R119889+1

+ )

10038171003817100381710038171198911003817100381710038171003817

119902119901

B119904120573

119902119902(R119889+1

+ )

(74)

10038171003817100381710038171198911003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus(119902119901)

Bminus((2120573+2+119889)119902minus119904)120573

infininfin (R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

119902119901

H119904119902120573(R119889+1+

) (75)

where 119901 = 119902(2120573 + 2 + 119889)(2120573 + 2 + 119889 minus 119902119904)

Proof By choosing 119886 = 119904 gt 0 minus119887 = 119904 minus (119889 + 2120573 + 2)119902 lt 01199021 = 119902 and 1199022 = infin we deduce (74) from the relations (70)and (67) In the same way we deduce (75) from the relations(70) and (68)

4 Generalized Heat Equation

41 Characterization for the Weinstein-Besov Spaces TheWeinstein heat equation reads

120597119905119906 (119905 119909) minus Δ 120573119906 (119905 119909) = 119891 (119905 119909) (119905 119909) isin [0infin) timesR119889+1

+

119906|119905=0 = 119892

(76)

We introduce the Weinstein heat semigroup 119867120573(119905) for theWeinstein-Laplace operator

119867120573 (119905) 119892 (119909) =

int

R119889+1+

Γ120573 (119905 119909 119910) 119892 (119910) 119889120583120573 (119910) if 119905 gt 0

119892 (119909) if 119905 = 0(77)

where Γ120573 is the Weinstein heat kernel defined by

Γ120573 (119905 119909 119910) = 120591119909 (119864(120573)

119905) (119910) (78)

where

119864(120573)

119905(119910) =

2

1205871198892Γ (120573 + 1) (4119905)

120573+1+1198892119890minus11991024119905 (79)

Thus

119867120573 (119905) 119892 (119909) = 119892lowast119882119864(120573)

119905 (119909) (80)

In practice we use the integral formulation of (76)

119906 (119905 119909) = 119867120573 (119905) 119892 (119909) + 119866120573 (119891) (119905 119909)

= 119867120573 (119905) 119892 (119909) + int

119905

0

119867120573 (119905 minus 119904) 119891 (119904 119909) 119889119904

(81)

Remark 25 The function 119864(120573)119905

is the Gauss kernel associatedwith Weinstein operators This function satisfies

forall120585 isin R119889+1

+ F119882 (119864

(120573)

119905) (120585) = 119890

minus1199051205852

(82)

Proposition 26 Let 1 le 119901 le 119903 le infin and let 119891 isin 119871119901

120573(R119889+1

+)

Then the operator 119867120573(119905) maps 119871119901

120573(R119889+1

+) continuously to

119871119903

120573(R119889+1

+) and

10038171003817100381710038171003817119867120573 (119905) 119891

10038171003817100381710038171003817119871119903120573(R119889)

le 119862119905minus((119889+2120573+2)2)(1119901minus1119903)1003817

1003817100381710038171198911003817100381710038171003817119871119901

120573(R119889)

(83)

Moreover1003817100381710038171003817100381710038171003817

(minusΔ 120573)

1205752

119867120573 (119905) 119891

1003817100381710038171003817100381710038171003817119871119903120573(R119889)

le 119862119905minus1205752minus((119889+2120573+2)2)(1119901minus1119903)1003817

1003817100381710038171198911003817100381710038171003817119871119901

120573(R119889)

(84)

for all 120575 gt 0

Proof It follows from the relations (80) and (29) combinedwith scaling property of the kernel 119864(120573)

119905

In this section we prove estimates for the Weinstein heatsemigroupThese estimates are based on the following result

Lemma 27 Let C be an annulus Positive constants 119888 and 119862exist such that for any 119901 in [1infin] and any couple (119905 120582) ofpositive real numbers one has

suppF119882 (119906) sub 120582C 997904rArr

10038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le 119862119890minus1198881199051205822

119906119871119901

120573(R119889+1+

)

(85)

8 Journal of Function Spaces and Applications

Proof We again consider a function Θ in 119863(R119889+1

+ 0) the

value of which is identically 1 in neighborhood of annulusCWe can also assume without loss of generality that 120582 = 1 Wethen have

119867120573 (119905) 119906 = 119892 (119905 sdot) lowast119882119906 (86)where

119892 (119905 sdot) = Fminus1

119882(Θ (120585) 119890

minus1199051205852

) (87)

The lemma is proved provided that we can find positive realnumbers 119888 and 119862 such that

forall119905 gt 01003817100381710038171003817119892 (119905 sdot)

10038171003817100381710038171198711120573(R119889+1+

)le 119862119890

minus119888119905 (88)

To begin we perform integrations by parts in (87) We get1003816100381610038161003816119892 (119905 119909)

1003816100381610038161003816=

1

119888120573

(1 + 1199092)

minus(119889+[2120573]+3)

times int

R119889+1+

Λ (119909 120585) (119868119889 minus Δ 120573)

119889+[2120573]+3

times (Θ (120585) 119890minus1199051205852

) 119889120583120573 (120585)

(89)

Using Leibnizrsquos formula we obtain1003816100381610038161003816119892 (119905 119909)

1003816100381610038161003816le 119862(1 + 119909

2)

minus(119889+[2120573]+3)

119890minus119888119905 (90)

and (88) follows

For any interval 119868 of R (bounded or unbounded) wedefine themixed space-time 119871119901(119868 119871119902

120573(R119889+1

+)) Banach space of

(classes of) measurable functions 119906 119868 rarr 119871119902

120573(R119889+1

+) such

that 119906119871119901(119868119871

119902

120573(R119889+1+

))lt infin with

119906119871119901(119868119871119902

120573(R119889+1+

))= (int

119868

119906 (119905 sdot)119901

119871119902

120573(R119889+1+

)119889119905)

1119901

if 1 le 119901 119902 lt infin

119906119871infin(119868119871119902

120573(R119889+1+

))= ess sup

119905isin119868

119906 (119905 sdot)119871119902

120573(R119889+1+

) if 1 le 119902 lt infin

(91)Corollary 28 Let C be an annulus and 120582 a positive realnumber Let 1199060 (resp 119891 = 119891(119905 119909)) satisfy suppF119882(1199060) sub 120582C(resp suppF119882(119891(119905 sdot)) sub 120582C for all 119905 in [0 119879]) Consider 119906 asolution of

120597119905119906 minus Δ 120573119906 = 0 119906|119905=0 = 1199060 (92)and V a solution of

120597119905V minus Δ 120573V = 119891 (119905 sdot) V|119905=0 = 0 (93)There exist positive constants 119888 and 119862 depending only on Csuch that for any 1 le 119886 le 119887 le infin and 1 le 119901 le 119902 le infin we have

119906119871119902([0119879]119871119887120573(R119889+1+

))le 119862120582

(119889+2120573+2)(1119886minus1119887)120582minus21199021003817

1003817100381710038171199060

1003817100381710038171003817119871119886120573(R119889+1+

)

V119871119902([0119879]119871119887120573(R119889+1+

))le 119862120582

minus2(1+1119902minus1119901)120582(119889+2120573+2)(1119886minus1119887)

times10038171003817100381710038171198911003817100381710038171003817119871119901([0119879]119871119886

120573(R119889+1+

))

(94)

Proof It suffices to use the fact that

119906 (119905 sdot) = 119867120573 (119905) 1199060 V (119905 sdot) = int119905

0

119867120573 (119905 minus 119904) 119891 (119904 sdot) 119889119904

(95)

Combining Lemma 27 and Youngrsquos inequality (29) withscaling property of the kernel 119864(120573)

119905now yields the result

Theorem 29 Let 119904 be a positive real number and (119901 119903) isin

[1infin]2 A constant 119862 exists which satisfies the following

property For 119906 isin Bminus2119904120573

119901119903 (R119889+1

+) one has

119862minus1119906

Bminus2119904120573

119901119903(R119889+1+

)le

10038171003817100381710038171003817100381710038171003817

10038171003817100381710038171003817119905119904119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862119906Bminus2119904120573

119901119903(R119889+1+

)

(96)

To prove this result we need the following lemma

Lemma 30 There exist two positive constants 120581 and 119862

depending only on 120593 such that for all 1 le 119901 le infin 120591 ge 0 and119895 isin Z one has

10038171003817100381710038171003817Δ 119895 (119867120573 (120591) 119906)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862119890

minus1205812211989512059110038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

) (97)

Proof The result follows immediately by applying Lemma 27and because Δ 119895(119867120573(120591))119906 = (119867120573(120591)Δ 119895)119906

Proof of Theorem 29 Using Lemma 30 and considering thefact that the operator Δ 119895 commutes with the operator 119867120573(119905)

and the definition of the homogeneous Weinstein-Besov(semi) norm we get

10038171003817100381710038171003817119905119904119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862119906

Bminus2119904120573

119901119903(R119889+1+

)sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903119895

(98)

where (119888119903119895)119895isinZ denotes as in all this proof a generic elementof the unit sphere of 119897119903(Z) In the case when 119903 = infin therequired inequality comes immediately from the followingeasy result For any positive 119904 we have

sup119905gt0

sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

lt infin (99)

In the case 119903 lt infin using the Holder inequality with theweight 22119895119904119890minus1205811199052

2119895

(99) and the Fubini theorem we obtain

int

infin

0

11990511990311990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

119889119905

119905

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)int

infin

0

(sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

)

119903minus1

times (sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903

119903119895)

119889119905

119905

Journal of Function Spaces and Applications 9

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)int

infin

0

(sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903

119903119895)

119889119905

119905

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)sum

119895isinZ

119888119903

119903119895int

infin

0

(11990511990422119895119904119890minus12058111990522119895

)

119889119905

119905

le 119862Γ (119904) 119906119903

Bminus2119904120573

119901119903(R119889+1+

)

(100)

In order to prove the other inequality let us observe thatfor any 119904 greater than minus1 we have

Δ 119895119906 =

1

Γ (119904 + 1)

int

infin

0

119905119904(minusΔ 120573)

119904+1

119867120573 (119905) Δ 119895119906 119889119905 (101)

Then Lemma 30 Proposition 9 and the fact that the operatorΔ 119895 commutes with the operator119867120573(119905) lead to the following

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862int

infin

0

11990511990422119895(119904+1)

119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889119905

(102)

In the case 119903 = infin we simply write

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862(sup

119905gt0

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

times int

infin

0

22119895(119904+1)

119890minus12058111990522119895

119889119905

le 11986222119895119904(sup119905gt0

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

(103)

In the case 119903 lt infin Holderrsquos inequality with the weight 119890minus12058111990522119895

gives

(int

infin

0

119905119904119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889119905)

119903

le 1198622minus2119895(119903minus1)

int

infin

0

119905119903119904119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)119889119905

(104)

Thanks to (99) and Fubinirsquos theorem we infer from (102) that

sum

119895isinZ

2minus211989511990311990410038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)le 119862int

infin

0

11990511990311990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

119889119905

119905

(105)

The theorem is proved

Second Proof ofTheorem 29We only consider the case 1 le 119903 ltinfinThe case 119903 = infin can be shown similarlyWe first prove that

119862minus1119906

Bminus2119904120573

119901119903(R119889+1+

)le

10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

(106)

It is easy to see that

Δ 119895119906 = 120594119895lowast119882119864(120573)

2minus2119895lowast119882119906

(107)

where 120594119895 = Fminus1

119882(120593(2

minus119895120585)119890

2minus21198951205852

) and119864(120573)2minus2119895

is the Gauss kernelassociated with Weinstein operators By relation (29) we get

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le

10038171003817100381710038171003817120594119895

100381710038171003817100381710038171198711120573(R119889+1+

)

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

) (108)

As10038171003817100381710038171003817120594119895

100381710038171003817100381710038171198711120573(R119889+1+

)= int

R119889+1+

100381610038161003816100381610038161003816F

minus1

119882(120593 (120585) 119890

1205852

)

100381610038161003816100381610038161003816119889120583120573 (120585) lt infin (109)

we obtain10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

) (110)

Moreover simple calculations give that

119864(120573)

2minus2119895lowast119882119906 = 119867120573 (2

minus4119895minus 119905

2) (119864

(120573)

1199052lowast119882119906)

(111)

Thus from Proposition 26 it follows that100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1

+ )

(112)

for any 119905 isin [2minus119895minus1 2minus119895] which implies that

sum

119895isinZ

2minus211990411989511990310038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

le 119862 sum

119895isinZ

int

2minus119895

2minus119895minus1(119905

2119904100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862int

infin

0

(1199052119904100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862int

infin

0

(11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

(113)

where we have used the fact that 119864(120573)1199052lowast119882119906 = 119867120573(119905

2)119906

We now prove that10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862119906Bminus2119904120573

119901119903(R119889+1+

) (114)

Indeed one has

119864(120573)

2minus2119895lowast119882119906 = sum

119899isinZ

119864(120573)

2minus2119895lowast119882Δ 119899+119895119906 (115)

Arguing as above we have100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1

+ )

(116)

for any 119905 isin [2minus119895 21minus119895] Thus10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

= 2

100381710038171003817100381710038171003817100381710038171003817

1003817100381710038171003817100381710038171199052119904119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)

100381710038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862 sum

119895isinZ

int

21minus119895

2minus119895(2

minus2119895119904100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862 sum

119895isinZ

(2minus2119895119904

sum

119899isinZ

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

(117)

10 Journal of Function Spaces and Applications

On the other hand it is easy to see that100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 1198622

minus211989911990410038171003817100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

) (118)

for any 119904 gt 0 For 0 lt 1199041 lt 119904 lt 1199042 and by using theMinkowskiinequality we have

sum

119895isinZ

(2minus2119895119904

sum

119899isinZ

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

le sum

119895isinZ

(2minus2119895119904

0

sum

minusinfin

2minus21198991199041

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

+ sum

119895isinZ

(2minus2119895119904

sum

119899isinN

2minus21198991199042

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

le 119862(

0

sum

minusinfin

2minus2119899(119904

1minus119904)(sum

119895isinZ

(2minus2(119899+119895)11990311990410038171003817

100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

)

1119903

)

119903

+ 119862(sum

N

2minus2119899(119904

2minus119904)(sum

119895isinZ

(2minus2(119899+119895)11990311990410038171003817

100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

)

1119903

)

119903

le 119862 sum

119895isinZ

2minus211989511990311990410038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

(119)

The result is immediately from (117) and (119)

42 Embedding Sobolev Theorems

Theorem 31 Let 1 lt 119901 lt infin and let 0 lt 119904 lt (119889 + 2120573 +

2)119901There exists a positive constant119862 such that for all function119891 isin

H119904

119901120573(R119889+1

+) one has

10038171003817100381710038171198911003817100381710038171003817119871119902

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

H119904119901120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

120579

B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

) (120)

where 120579 = 119904119901(119889+2120573+2) and 119902 = 119901(119889+2120573+2)(119889+2120573+2minus119901119904)

Proof Bydensity we can suppose that119891belongs toSlowast(R119889+1

)It is easy to see that

119891 = int

infin

0

119867120573 (119905) Δ 120573119891119889119905(121)

and decompose the integral in two parts as follows

119891 = int

119860

0

119867120573 (119905) Δ 120573119891119889119905 + int

infin

119860

119867120573 (119905) Δ 120573119891119889119905(122)

where 119860 is a constant to be fixed laterOn the other hand byTheorem 29 we obtain10038171003817100381710038171003817119867120573 (119905) Δ 120573119891

10038171003817100381710038171003817119871infin120573(R119889+1+

)

le

119862

1199051minus(12)(119904minus(119889+2120573+2)119901)

10038171003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(123)

Therefore after integrating we get

int

infin

119860

10038171003817100381710038171003817119867120573 (119905) Δ 120573119891

10038171003817100381710038171003817119871infin120573(R119889+1+

)119889119905

le 119860(12)(119904minus(119889+2120573+2)119901)1003817

1003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(124)

On the other hand denoting 119892 = (minusΔ 120573)1199042119891 we have

119867120573 (119905) Δ 120573119891 =

1

(minus119905)1minus1199042

119867120573 (119905) (minus119905Δ 120573)

1minus1199042

119892 (125)

We proceed as in [8] we prove that1003816100381610038161003816100381610038161003816

119867120573 (119905) (minus119905Δ 120573)

1minus1199042

119892 (119909)

1003816100381610038161003816100381610038161003816

le 119862 (119904)119872120573 (119892) (119909) (126)

where119872120573(119892) is a maximal function of 119892 associated with theWeinstein operators (cf [12])

This leads to100381610038161003816100381610038161003816100381610038161003816

int

119860

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

100381610038161003816100381610038161003816100381610038161003816

le 1198621198601199042119872120573 (119892) (119909) (127)

In conclusion we get10038161003816100381610038161003816100381610038161003816

int

infin

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

10038161003816100381610038161003816100381610038161003816

le 119862 (1198601199042119872120573 (119892) (119909)

+119860(12)(119904minus(119889+2120573+2)119901)1003817

1003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

))

(128)

and the choice of 119860 such that

119860(119889+2120573+2)2119901

119872120573 (119892) (119909) =10038171003817100381710038171198911003817100381710038171003817B119904minus(119889+2120573+2)119901120573

infininfin (R119889+1+

)(129)

ensures that10038161003816100381610038161003816100381610038161003816

int

infin

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

10038161003816100381610038161003816100381610038161003816

le 119862(119872120573(119892)(119909))

1minus(119901119904(119889+2120573+2))10038171003817100381710038171198911003817100381710038171003817

119901119904(119889+2120573+2)

B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(130)

Finally taking the 119871119902120573norm with 119902 = 119901(119889 + 2120573 + 2)(119889 + 2120573 +

2minus119901119904) ends the proof thanks to the fact themaximal function119872120573 is bounded of 119871119902

120573(R119889+1

+) into itself for 119902 gt 1

Theorem 32 Let 1 lt 119901 lt 119902 lt infin For all function 119891 such that119891 isin

H1199041

119901120573(R119889+1

+)⋂

Bminus120573120573

infininfin(R119889+1

+) one has

10038171003817100381710038171198911003817100381710038171003817H119904119901120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

120579

H1199041

119901120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573120573

infininfin(R119889+1

+) (131)

where 120579 = 119901119902 119904 = 1205791199041 minus (1 minus 120579)120573 with 120573 gt 0 minus120573 lt 119904 lt 1199041

Proof It suffices to prove that1003817100381710038171003817100381710038171003817

(minusΔ 120573)

(119904minus1199041)2

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le 11986210038171003817100381710038171198911003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(132)

Journal of Function Spaces and Applications 11

Indeed we use the following identity (which may be easilyproven by taking the Weinstein transform in 119909 of both sides)

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

infin

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905 (133)

with 120575 = 1199041 minus 119904 gt 0We decompose the integral in two parts as follows

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

119879

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

+

1

Γ (1205752)

int

infin

119879

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

(134)

where 119879 is a constant to be fixed laterWe proceed as in [8] we obtain

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119872120573 (119891) (119909) (135)

On the other hand we use Theorem 29 and the fact that 119891belongs to Bminus120573minus119904

1120573

infininfin(R119889+1

+) to deduce that

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119905

(minus120573minus1199041)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

) (136)

Thus by applying the preceding estimates on the right part of(134) we obtain

1003816100381610038161003816100381610038161003816

(minusΔ 120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621

Γ (1205752)

1198791205752119872120573 (119891) (119909)

+

1198622

Γ (1205752)

119879(120575minus120573minus119904

1)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

(137)

We fix now

119879 = (

10038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

119872120573 (119891) (119909)

)

2(120573+1199041)

(138)

We obtain

1003816100381610038161003816100381610038161003816

(minusΔ120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621 + 1198622

Γ (1205752)

(119872120573(119891)(119909))

12057910038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(139)

Thus we deduce that1003817100381710038171003817100381710038171003817

(minusΔ120573)

minus1205752

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le

1198621 + 1198622

Γ (1205752)

10038171003817100381710038171003817119872120573 (119891)

10038171003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(140)

To conclude we used the fact that the maximal function119872120573

is bounded of 119871119902120573(R119889+1

+) into itself for 119902 gt 1

43 Estimates in Generalized Besov Spaces For any interval 119868ofR (bounded or unbounded) and a normed space 119865(R119889+1

+)

we define the mixed space-time 119871119901(119868 119865(R119889+1

+)) space of

(classes of) measurable functions 119906 119868 rarr 119865(R119889+1

+) such that

||119906||119871119901(119868119865(R119889+1+

)) lt infin with

119906119871119901(119868119865(R119889+1+

)) = (int

119868

119906 (119905 sdot)119901

119865(R119889+1+

)119889119905)

1119901

if 1 le 119901 lt infin

119906119871infin(119868119865(R119889+1+

)) = ess sup119905isin119868

119906 (119905 sdot)119865(R119889+1+

)

(141)

For any interval 119868 of R (bounded or unbounded) anda Banach space 119883 we define the mixed space-time 119862(119868119883)space of continuous functions 119868 rarr 119883 When 119868 is bounded119862(119868119883) is a Banach space with the norm of 119871infin(119868 119883)

Theorem 33 Let 119904 isin R and 1 le 119901 119902 119903 le infin Let 119879 gt 0 119892 isin

B119904120573

119901119903(R119889+1

+) and119891 in 119871119902((0 119879) B

119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76)

has a unique solution

119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903(R

119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R

119889+1

+))

(142)

and there exists a constant 119862 such that for all 1199021 isin [119902infin] onehas

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862(10038171003817100381710038171198921003817100381710038171003817B119904120573

119901119903(R119889+1+

)+10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)B

119904minus2+(2119902)120573

119901119903(R119889+1+

)))

(143)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] B119904120573

119901119903(R119889+1

+))

Proof Since 119892 and 119891 are temperate distributions (76) has aunique solution 119906 in S1015840

((0 119879) timesR119889+1

+) which satisfies

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(144)

Next we notice that applying Δ 119895 to (76) and using formula(81) yield

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(145)

Therefore

10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(146)

12 Journal of Function Spaces and Applications

By virtue of Lemma 30 we thus have for some 120581 gt 0

10038171003817100381710038171003817Δ 119895119906(119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

119890minus12058122119895(119905minus120591) 10038171003817

100381710038171003817Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(147)

Applying convolution inequalities we get

10038171003817100381710038171003817Δ 119895119906

100381710038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862[

[

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

)

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

]

]

(148)

with 11199022 = 1+ 11199021 minus1119902 Finally taking the 119897119903(Z) norm we

conclude that (with the usual convention if 119903 = infin)

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

1199031199021

)(211989511990410038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

]

]

1119903

+ 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

1199031199022

)

times (2119895(119904minus2+2119902)10038171003817

100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902(0119879119871

119901

120573(R119889+1+

))

119903

]

]

1119903

(149)

which insures that 119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903 (R119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R119889+1

+)) and yields the desired inequality

Since 119906 belongs to 119862([0 119879]B119904120573

119901119903(R119889+1

+)) in the case where

119903 is finite may be easily deduced from the density ofSlowast(R

119889+1)⋂

B119904120573

119901119903(R119889+1

+) in B

119904120573

119901119903(R)

Theorem 34 Let 119904 isin R 119879 gt 0 and 1 le 119901 119902 119903 le infinOne supposes that 119892 isin 119861

119904120573

119901119903(R119889+1

+) and 119891 isin 119871

119902((0 119879)

119861119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76) has a unique solution 119906 belong-

ing to

119871119902((0 119879) 119861

119904+(2119902)120573

119901119903(R

119889+1

+))⋂119871

infin((0 119879) 119861

119904120573

119901119903(R

119889+1

+))

(150)

and there exists a constant 119862 such that for all 119902 le 1199021 le infin

1199061198711199021 ((0119879)119861

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862 [ (1 + 11987911199021)10038171003817100381710038171198921003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (1 + 1198791+1119902

1minus1119902

)10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)119861

119904minus2+(2119902)120573

119901119903(R119889+1+

))]

(151)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] 119861119904120573119901119903(R119889+1

+))

Proof Since 119892 119891 are tempered (76) has a unique solution 119906in S1015840

((0 119879) timesR119889+1

+) satisfying

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(152)

Hence applying Δ 119895 119895 ge 0 to (81) we see that

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(153)

and thus by Lemma 30 we can deduce that10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+int

119905

0

119890minus12058122119895(119905minus120591)10038171003817

100381710038171003817Δ 119895119891(120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(154)

Then it follows from convolution inequalities thatΔ 1198951199061198711199021 ((0119879)119871

119901

120573(R119889+1+

))is dominated by

(

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

(155)

with 11199022 = 1 + 11199021 minus 1119902 Moreover similarly as above wecan obtain that

1003817100381710038171003817Δminus1119906(119905 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)le1003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

1003817100381710038171003817Δminus1119891 (120591 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(156)

and thus if 1 le 119902 le 1199021 le infin1003817100381710038171003817Δminus1119906

10038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862(119879111990211003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)+ 119879

111990221003817100381710038171003817Δminus1119891

1003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

)))

(157)

Journal of Function Spaces and Applications 13

Finally taking the 119897119903-normwith respect to 119895 in (155) and (157)with the usual convention if 119903 = infin we can deduce the desiredestimate

Acknowledgment

Theauthor gratefully acknowledges theDeanship of ScientificResearch at the University of Taibah The author is deeplyindebted to the referee for providing constructive commentsand help in improving the contents of this paper

References

[1] Z Ben Nahia and N Ben Salem ldquoSpherical harmonics andapplications associated with the Weinstein operatorrdquo in Pro-ceedings of the International Conference on PotentialTheory heldin Kouty Czech Republic (ICPT rsquo94) pp 235ndash241 1996

[2] Z Ben Nahia and N Ben Salem ldquoOn a mean value propertyassociated with the Weinstein operatorrdquo in Proceedings of theInternational Conference on Potential Theory held in KoutyCzech Republic (ICPT rsquo94) pp 243ndash253 1996

[3] M Brelot ldquoEquation de Weinstein et potentiels de MarcelRieszrdquo in Seminaire de Theorie de Potentiel Paris No 3 vol 681of Lecture Notes in Mathematics pp 18ndash38 Springer BerlinGermany 1978

[4] H Mejjaoli and M Salhi ldquoUncertainty principles for theweinstein transformrdquo Czechoslovak Mathematical Journal vol61 no 4 pp 941ndash974 2011

[5] H Mejjaoli and A Ould Ahmed Salem ldquoWeinstein Gabortransform and applicationsrdquo Advanced Studies in Pure Mathe-matics vol 2 no 3 pp 203ndash210 2012

[6] H Mejjaoli ldquoBesov spaces associated withthe Weinstein opera-tor and applicationsrdquo In press

[7] T Kawazoe and H Mejjaoli ldquoGeneralized Besov spaces andtheir applicationsrdquo Tokyo Journal of Mathematics vol 35 no 2pp 297ndash320 2012

[8] H Mejjaoli ldquoLittlewood-Paley decomposition associated withthe Dunkl operators and paraproduct operatorsrdquo Journal ofInequalities in Pure and Applied Mathematics vol 9 no 4 pp1ndash25 2008

[9] H Mejjaoli and N Sraeib ldquoGeneralized sobolev spaces inquantum calculus and applicationsrdquo Journal of Inequalities andSpecial Functions vol 1 no 4 pp 43ndash64 2012

[10] H Mejjaoli ldquoGeneralized homogeneous Besov spaces and theirapplicationsrdquo Serdica Mathematical Journal vol 38 no 4 pp575ndash614 2012

[11] H Triebel Interpolation Theory Functions Spaces DifferentialOperators North-Holland AmsterdamThe Netherlands 1978

[12] V S Guliev ldquoOn maximal function and fractional integralassociated with the Bessel differential operatorrdquo MathematicalInequalities and Applications vol 6 no 2 pp 317ndash330 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Function Spaces and Applications

Proof We again consider a function Θ in 119863(R119889+1

+ 0) the

value of which is identically 1 in neighborhood of annulusCWe can also assume without loss of generality that 120582 = 1 Wethen have

119867120573 (119905) 119906 = 119892 (119905 sdot) lowast119882119906 (86)where

119892 (119905 sdot) = Fminus1

119882(Θ (120585) 119890

minus1199051205852

) (87)

The lemma is proved provided that we can find positive realnumbers 119888 and 119862 such that

forall119905 gt 01003817100381710038171003817119892 (119905 sdot)

10038171003817100381710038171198711120573(R119889+1+

)le 119862119890

minus119888119905 (88)

To begin we perform integrations by parts in (87) We get1003816100381610038161003816119892 (119905 119909)

1003816100381610038161003816=

1

119888120573

(1 + 1199092)

minus(119889+[2120573]+3)

times int

R119889+1+

Λ (119909 120585) (119868119889 minus Δ 120573)

119889+[2120573]+3

times (Θ (120585) 119890minus1199051205852

) 119889120583120573 (120585)

(89)

Using Leibnizrsquos formula we obtain1003816100381610038161003816119892 (119905 119909)

1003816100381610038161003816le 119862(1 + 119909

2)

minus(119889+[2120573]+3)

119890minus119888119905 (90)

and (88) follows

For any interval 119868 of R (bounded or unbounded) wedefine themixed space-time 119871119901(119868 119871119902

120573(R119889+1

+)) Banach space of

(classes of) measurable functions 119906 119868 rarr 119871119902

120573(R119889+1

+) such

that 119906119871119901(119868119871

119902

120573(R119889+1+

))lt infin with

119906119871119901(119868119871119902

120573(R119889+1+

))= (int

119868

119906 (119905 sdot)119901

119871119902

120573(R119889+1+

)119889119905)

1119901

if 1 le 119901 119902 lt infin

119906119871infin(119868119871119902

120573(R119889+1+

))= ess sup

119905isin119868

119906 (119905 sdot)119871119902

120573(R119889+1+

) if 1 le 119902 lt infin

(91)Corollary 28 Let C be an annulus and 120582 a positive realnumber Let 1199060 (resp 119891 = 119891(119905 119909)) satisfy suppF119882(1199060) sub 120582C(resp suppF119882(119891(119905 sdot)) sub 120582C for all 119905 in [0 119879]) Consider 119906 asolution of

120597119905119906 minus Δ 120573119906 = 0 119906|119905=0 = 1199060 (92)and V a solution of

120597119905V minus Δ 120573V = 119891 (119905 sdot) V|119905=0 = 0 (93)There exist positive constants 119888 and 119862 depending only on Csuch that for any 1 le 119886 le 119887 le infin and 1 le 119901 le 119902 le infin we have

119906119871119902([0119879]119871119887120573(R119889+1+

))le 119862120582

(119889+2120573+2)(1119886minus1119887)120582minus21199021003817

1003817100381710038171199060

1003817100381710038171003817119871119886120573(R119889+1+

)

V119871119902([0119879]119871119887120573(R119889+1+

))le 119862120582

minus2(1+1119902minus1119901)120582(119889+2120573+2)(1119886minus1119887)

times10038171003817100381710038171198911003817100381710038171003817119871119901([0119879]119871119886

120573(R119889+1+

))

(94)

Proof It suffices to use the fact that

119906 (119905 sdot) = 119867120573 (119905) 1199060 V (119905 sdot) = int119905

0

119867120573 (119905 minus 119904) 119891 (119904 sdot) 119889119904

(95)

Combining Lemma 27 and Youngrsquos inequality (29) withscaling property of the kernel 119864(120573)

119905now yields the result

Theorem 29 Let 119904 be a positive real number and (119901 119903) isin

[1infin]2 A constant 119862 exists which satisfies the following

property For 119906 isin Bminus2119904120573

119901119903 (R119889+1

+) one has

119862minus1119906

Bminus2119904120573

119901119903(R119889+1+

)le

10038171003817100381710038171003817100381710038171003817

10038171003817100381710038171003817119905119904119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862119906Bminus2119904120573

119901119903(R119889+1+

)

(96)

To prove this result we need the following lemma

Lemma 30 There exist two positive constants 120581 and 119862

depending only on 120593 such that for all 1 le 119901 le infin 120591 ge 0 and119895 isin Z one has

10038171003817100381710038171003817Δ 119895 (119867120573 (120591) 119906)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862119890

minus1205812211989512059110038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

) (97)

Proof The result follows immediately by applying Lemma 27and because Δ 119895(119867120573(120591))119906 = (119867120573(120591)Δ 119895)119906

Proof of Theorem 29 Using Lemma 30 and considering thefact that the operator Δ 119895 commutes with the operator 119867120573(119905)

and the definition of the homogeneous Weinstein-Besov(semi) norm we get

10038171003817100381710038171003817119905119904119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862119906

Bminus2119904120573

119901119903(R119889+1+

)sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903119895

(98)

where (119888119903119895)119895isinZ denotes as in all this proof a generic elementof the unit sphere of 119897119903(Z) In the case when 119903 = infin therequired inequality comes immediately from the followingeasy result For any positive 119904 we have

sup119905gt0

sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

lt infin (99)

In the case 119903 lt infin using the Holder inequality with theweight 22119895119904119890minus1205811199052

2119895

(99) and the Fubini theorem we obtain

int

infin

0

11990511990311990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

119889119905

119905

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)int

infin

0

(sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

)

119903minus1

times (sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903

119903119895)

119889119905

119905

Journal of Function Spaces and Applications 9

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)int

infin

0

(sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903

119903119895)

119889119905

119905

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)sum

119895isinZ

119888119903

119903119895int

infin

0

(11990511990422119895119904119890minus12058111990522119895

)

119889119905

119905

le 119862Γ (119904) 119906119903

Bminus2119904120573

119901119903(R119889+1+

)

(100)

In order to prove the other inequality let us observe thatfor any 119904 greater than minus1 we have

Δ 119895119906 =

1

Γ (119904 + 1)

int

infin

0

119905119904(minusΔ 120573)

119904+1

119867120573 (119905) Δ 119895119906 119889119905 (101)

Then Lemma 30 Proposition 9 and the fact that the operatorΔ 119895 commutes with the operator119867120573(119905) lead to the following

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862int

infin

0

11990511990422119895(119904+1)

119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889119905

(102)

In the case 119903 = infin we simply write

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862(sup

119905gt0

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

times int

infin

0

22119895(119904+1)

119890minus12058111990522119895

119889119905

le 11986222119895119904(sup119905gt0

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

(103)

In the case 119903 lt infin Holderrsquos inequality with the weight 119890minus12058111990522119895

gives

(int

infin

0

119905119904119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889119905)

119903

le 1198622minus2119895(119903minus1)

int

infin

0

119905119903119904119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)119889119905

(104)

Thanks to (99) and Fubinirsquos theorem we infer from (102) that

sum

119895isinZ

2minus211989511990311990410038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)le 119862int

infin

0

11990511990311990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

119889119905

119905

(105)

The theorem is proved

Second Proof ofTheorem 29We only consider the case 1 le 119903 ltinfinThe case 119903 = infin can be shown similarlyWe first prove that

119862minus1119906

Bminus2119904120573

119901119903(R119889+1+

)le

10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

(106)

It is easy to see that

Δ 119895119906 = 120594119895lowast119882119864(120573)

2minus2119895lowast119882119906

(107)

where 120594119895 = Fminus1

119882(120593(2

minus119895120585)119890

2minus21198951205852

) and119864(120573)2minus2119895

is the Gauss kernelassociated with Weinstein operators By relation (29) we get

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le

10038171003817100381710038171003817120594119895

100381710038171003817100381710038171198711120573(R119889+1+

)

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

) (108)

As10038171003817100381710038171003817120594119895

100381710038171003817100381710038171198711120573(R119889+1+

)= int

R119889+1+

100381610038161003816100381610038161003816F

minus1

119882(120593 (120585) 119890

1205852

)

100381610038161003816100381610038161003816119889120583120573 (120585) lt infin (109)

we obtain10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

) (110)

Moreover simple calculations give that

119864(120573)

2minus2119895lowast119882119906 = 119867120573 (2

minus4119895minus 119905

2) (119864

(120573)

1199052lowast119882119906)

(111)

Thus from Proposition 26 it follows that100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1

+ )

(112)

for any 119905 isin [2minus119895minus1 2minus119895] which implies that

sum

119895isinZ

2minus211990411989511990310038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

le 119862 sum

119895isinZ

int

2minus119895

2minus119895minus1(119905

2119904100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862int

infin

0

(1199052119904100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862int

infin

0

(11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

(113)

where we have used the fact that 119864(120573)1199052lowast119882119906 = 119867120573(119905

2)119906

We now prove that10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862119906Bminus2119904120573

119901119903(R119889+1+

) (114)

Indeed one has

119864(120573)

2minus2119895lowast119882119906 = sum

119899isinZ

119864(120573)

2minus2119895lowast119882Δ 119899+119895119906 (115)

Arguing as above we have100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1

+ )

(116)

for any 119905 isin [2minus119895 21minus119895] Thus10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

= 2

100381710038171003817100381710038171003817100381710038171003817

1003817100381710038171003817100381710038171199052119904119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)

100381710038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862 sum

119895isinZ

int

21minus119895

2minus119895(2

minus2119895119904100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862 sum

119895isinZ

(2minus2119895119904

sum

119899isinZ

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

(117)

10 Journal of Function Spaces and Applications

On the other hand it is easy to see that100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 1198622

minus211989911990410038171003817100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

) (118)

for any 119904 gt 0 For 0 lt 1199041 lt 119904 lt 1199042 and by using theMinkowskiinequality we have

sum

119895isinZ

(2minus2119895119904

sum

119899isinZ

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

le sum

119895isinZ

(2minus2119895119904

0

sum

minusinfin

2minus21198991199041

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

+ sum

119895isinZ

(2minus2119895119904

sum

119899isinN

2minus21198991199042

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

le 119862(

0

sum

minusinfin

2minus2119899(119904

1minus119904)(sum

119895isinZ

(2minus2(119899+119895)11990311990410038171003817

100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

)

1119903

)

119903

+ 119862(sum

N

2minus2119899(119904

2minus119904)(sum

119895isinZ

(2minus2(119899+119895)11990311990410038171003817

100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

)

1119903

)

119903

le 119862 sum

119895isinZ

2minus211989511990311990410038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

(119)

The result is immediately from (117) and (119)

42 Embedding Sobolev Theorems

Theorem 31 Let 1 lt 119901 lt infin and let 0 lt 119904 lt (119889 + 2120573 +

2)119901There exists a positive constant119862 such that for all function119891 isin

H119904

119901120573(R119889+1

+) one has

10038171003817100381710038171198911003817100381710038171003817119871119902

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

H119904119901120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

120579

B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

) (120)

where 120579 = 119904119901(119889+2120573+2) and 119902 = 119901(119889+2120573+2)(119889+2120573+2minus119901119904)

Proof Bydensity we can suppose that119891belongs toSlowast(R119889+1

)It is easy to see that

119891 = int

infin

0

119867120573 (119905) Δ 120573119891119889119905(121)

and decompose the integral in two parts as follows

119891 = int

119860

0

119867120573 (119905) Δ 120573119891119889119905 + int

infin

119860

119867120573 (119905) Δ 120573119891119889119905(122)

where 119860 is a constant to be fixed laterOn the other hand byTheorem 29 we obtain10038171003817100381710038171003817119867120573 (119905) Δ 120573119891

10038171003817100381710038171003817119871infin120573(R119889+1+

)

le

119862

1199051minus(12)(119904minus(119889+2120573+2)119901)

10038171003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(123)

Therefore after integrating we get

int

infin

119860

10038171003817100381710038171003817119867120573 (119905) Δ 120573119891

10038171003817100381710038171003817119871infin120573(R119889+1+

)119889119905

le 119860(12)(119904minus(119889+2120573+2)119901)1003817

1003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(124)

On the other hand denoting 119892 = (minusΔ 120573)1199042119891 we have

119867120573 (119905) Δ 120573119891 =

1

(minus119905)1minus1199042

119867120573 (119905) (minus119905Δ 120573)

1minus1199042

119892 (125)

We proceed as in [8] we prove that1003816100381610038161003816100381610038161003816

119867120573 (119905) (minus119905Δ 120573)

1minus1199042

119892 (119909)

1003816100381610038161003816100381610038161003816

le 119862 (119904)119872120573 (119892) (119909) (126)

where119872120573(119892) is a maximal function of 119892 associated with theWeinstein operators (cf [12])

This leads to100381610038161003816100381610038161003816100381610038161003816

int

119860

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

100381610038161003816100381610038161003816100381610038161003816

le 1198621198601199042119872120573 (119892) (119909) (127)

In conclusion we get10038161003816100381610038161003816100381610038161003816

int

infin

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

10038161003816100381610038161003816100381610038161003816

le 119862 (1198601199042119872120573 (119892) (119909)

+119860(12)(119904minus(119889+2120573+2)119901)1003817

1003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

))

(128)

and the choice of 119860 such that

119860(119889+2120573+2)2119901

119872120573 (119892) (119909) =10038171003817100381710038171198911003817100381710038171003817B119904minus(119889+2120573+2)119901120573

infininfin (R119889+1+

)(129)

ensures that10038161003816100381610038161003816100381610038161003816

int

infin

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

10038161003816100381610038161003816100381610038161003816

le 119862(119872120573(119892)(119909))

1minus(119901119904(119889+2120573+2))10038171003817100381710038171198911003817100381710038171003817

119901119904(119889+2120573+2)

B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(130)

Finally taking the 119871119902120573norm with 119902 = 119901(119889 + 2120573 + 2)(119889 + 2120573 +

2minus119901119904) ends the proof thanks to the fact themaximal function119872120573 is bounded of 119871119902

120573(R119889+1

+) into itself for 119902 gt 1

Theorem 32 Let 1 lt 119901 lt 119902 lt infin For all function 119891 such that119891 isin

H1199041

119901120573(R119889+1

+)⋂

Bminus120573120573

infininfin(R119889+1

+) one has

10038171003817100381710038171198911003817100381710038171003817H119904119901120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

120579

H1199041

119901120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573120573

infininfin(R119889+1

+) (131)

where 120579 = 119901119902 119904 = 1205791199041 minus (1 minus 120579)120573 with 120573 gt 0 minus120573 lt 119904 lt 1199041

Proof It suffices to prove that1003817100381710038171003817100381710038171003817

(minusΔ 120573)

(119904minus1199041)2

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le 11986210038171003817100381710038171198911003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(132)

Journal of Function Spaces and Applications 11

Indeed we use the following identity (which may be easilyproven by taking the Weinstein transform in 119909 of both sides)

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

infin

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905 (133)

with 120575 = 1199041 minus 119904 gt 0We decompose the integral in two parts as follows

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

119879

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

+

1

Γ (1205752)

int

infin

119879

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

(134)

where 119879 is a constant to be fixed laterWe proceed as in [8] we obtain

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119872120573 (119891) (119909) (135)

On the other hand we use Theorem 29 and the fact that 119891belongs to Bminus120573minus119904

1120573

infininfin(R119889+1

+) to deduce that

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119905

(minus120573minus1199041)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

) (136)

Thus by applying the preceding estimates on the right part of(134) we obtain

1003816100381610038161003816100381610038161003816

(minusΔ 120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621

Γ (1205752)

1198791205752119872120573 (119891) (119909)

+

1198622

Γ (1205752)

119879(120575minus120573minus119904

1)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

(137)

We fix now

119879 = (

10038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

119872120573 (119891) (119909)

)

2(120573+1199041)

(138)

We obtain

1003816100381610038161003816100381610038161003816

(minusΔ120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621 + 1198622

Γ (1205752)

(119872120573(119891)(119909))

12057910038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(139)

Thus we deduce that1003817100381710038171003817100381710038171003817

(minusΔ120573)

minus1205752

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le

1198621 + 1198622

Γ (1205752)

10038171003817100381710038171003817119872120573 (119891)

10038171003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(140)

To conclude we used the fact that the maximal function119872120573

is bounded of 119871119902120573(R119889+1

+) into itself for 119902 gt 1

43 Estimates in Generalized Besov Spaces For any interval 119868ofR (bounded or unbounded) and a normed space 119865(R119889+1

+)

we define the mixed space-time 119871119901(119868 119865(R119889+1

+)) space of

(classes of) measurable functions 119906 119868 rarr 119865(R119889+1

+) such that

||119906||119871119901(119868119865(R119889+1+

)) lt infin with

119906119871119901(119868119865(R119889+1+

)) = (int

119868

119906 (119905 sdot)119901

119865(R119889+1+

)119889119905)

1119901

if 1 le 119901 lt infin

119906119871infin(119868119865(R119889+1+

)) = ess sup119905isin119868

119906 (119905 sdot)119865(R119889+1+

)

(141)

For any interval 119868 of R (bounded or unbounded) anda Banach space 119883 we define the mixed space-time 119862(119868119883)space of continuous functions 119868 rarr 119883 When 119868 is bounded119862(119868119883) is a Banach space with the norm of 119871infin(119868 119883)

Theorem 33 Let 119904 isin R and 1 le 119901 119902 119903 le infin Let 119879 gt 0 119892 isin

B119904120573

119901119903(R119889+1

+) and119891 in 119871119902((0 119879) B

119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76)

has a unique solution

119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903(R

119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R

119889+1

+))

(142)

and there exists a constant 119862 such that for all 1199021 isin [119902infin] onehas

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862(10038171003817100381710038171198921003817100381710038171003817B119904120573

119901119903(R119889+1+

)+10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)B

119904minus2+(2119902)120573

119901119903(R119889+1+

)))

(143)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] B119904120573

119901119903(R119889+1

+))

Proof Since 119892 and 119891 are temperate distributions (76) has aunique solution 119906 in S1015840

((0 119879) timesR119889+1

+) which satisfies

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(144)

Next we notice that applying Δ 119895 to (76) and using formula(81) yield

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(145)

Therefore

10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(146)

12 Journal of Function Spaces and Applications

By virtue of Lemma 30 we thus have for some 120581 gt 0

10038171003817100381710038171003817Δ 119895119906(119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

119890minus12058122119895(119905minus120591) 10038171003817

100381710038171003817Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(147)

Applying convolution inequalities we get

10038171003817100381710038171003817Δ 119895119906

100381710038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862[

[

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

)

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

]

]

(148)

with 11199022 = 1+ 11199021 minus1119902 Finally taking the 119897119903(Z) norm we

conclude that (with the usual convention if 119903 = infin)

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

1199031199021

)(211989511990410038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

]

]

1119903

+ 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

1199031199022

)

times (2119895(119904minus2+2119902)10038171003817

100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902(0119879119871

119901

120573(R119889+1+

))

119903

]

]

1119903

(149)

which insures that 119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903 (R119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R119889+1

+)) and yields the desired inequality

Since 119906 belongs to 119862([0 119879]B119904120573

119901119903(R119889+1

+)) in the case where

119903 is finite may be easily deduced from the density ofSlowast(R

119889+1)⋂

B119904120573

119901119903(R119889+1

+) in B

119904120573

119901119903(R)

Theorem 34 Let 119904 isin R 119879 gt 0 and 1 le 119901 119902 119903 le infinOne supposes that 119892 isin 119861

119904120573

119901119903(R119889+1

+) and 119891 isin 119871

119902((0 119879)

119861119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76) has a unique solution 119906 belong-

ing to

119871119902((0 119879) 119861

119904+(2119902)120573

119901119903(R

119889+1

+))⋂119871

infin((0 119879) 119861

119904120573

119901119903(R

119889+1

+))

(150)

and there exists a constant 119862 such that for all 119902 le 1199021 le infin

1199061198711199021 ((0119879)119861

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862 [ (1 + 11987911199021)10038171003817100381710038171198921003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (1 + 1198791+1119902

1minus1119902

)10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)119861

119904minus2+(2119902)120573

119901119903(R119889+1+

))]

(151)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] 119861119904120573119901119903(R119889+1

+))

Proof Since 119892 119891 are tempered (76) has a unique solution 119906in S1015840

((0 119879) timesR119889+1

+) satisfying

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(152)

Hence applying Δ 119895 119895 ge 0 to (81) we see that

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(153)

and thus by Lemma 30 we can deduce that10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+int

119905

0

119890minus12058122119895(119905minus120591)10038171003817

100381710038171003817Δ 119895119891(120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(154)

Then it follows from convolution inequalities thatΔ 1198951199061198711199021 ((0119879)119871

119901

120573(R119889+1+

))is dominated by

(

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

(155)

with 11199022 = 1 + 11199021 minus 1119902 Moreover similarly as above wecan obtain that

1003817100381710038171003817Δminus1119906(119905 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)le1003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

1003817100381710038171003817Δminus1119891 (120591 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(156)

and thus if 1 le 119902 le 1199021 le infin1003817100381710038171003817Δminus1119906

10038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862(119879111990211003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)+ 119879

111990221003817100381710038171003817Δminus1119891

1003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

)))

(157)

Journal of Function Spaces and Applications 13

Finally taking the 119897119903-normwith respect to 119895 in (155) and (157)with the usual convention if 119903 = infin we can deduce the desiredestimate

Acknowledgment

Theauthor gratefully acknowledges theDeanship of ScientificResearch at the University of Taibah The author is deeplyindebted to the referee for providing constructive commentsand help in improving the contents of this paper

References

[1] Z Ben Nahia and N Ben Salem ldquoSpherical harmonics andapplications associated with the Weinstein operatorrdquo in Pro-ceedings of the International Conference on PotentialTheory heldin Kouty Czech Republic (ICPT rsquo94) pp 235ndash241 1996

[2] Z Ben Nahia and N Ben Salem ldquoOn a mean value propertyassociated with the Weinstein operatorrdquo in Proceedings of theInternational Conference on Potential Theory held in KoutyCzech Republic (ICPT rsquo94) pp 243ndash253 1996

[3] M Brelot ldquoEquation de Weinstein et potentiels de MarcelRieszrdquo in Seminaire de Theorie de Potentiel Paris No 3 vol 681of Lecture Notes in Mathematics pp 18ndash38 Springer BerlinGermany 1978

[4] H Mejjaoli and M Salhi ldquoUncertainty principles for theweinstein transformrdquo Czechoslovak Mathematical Journal vol61 no 4 pp 941ndash974 2011

[5] H Mejjaoli and A Ould Ahmed Salem ldquoWeinstein Gabortransform and applicationsrdquo Advanced Studies in Pure Mathe-matics vol 2 no 3 pp 203ndash210 2012

[6] H Mejjaoli ldquoBesov spaces associated withthe Weinstein opera-tor and applicationsrdquo In press

[7] T Kawazoe and H Mejjaoli ldquoGeneralized Besov spaces andtheir applicationsrdquo Tokyo Journal of Mathematics vol 35 no 2pp 297ndash320 2012

[8] H Mejjaoli ldquoLittlewood-Paley decomposition associated withthe Dunkl operators and paraproduct operatorsrdquo Journal ofInequalities in Pure and Applied Mathematics vol 9 no 4 pp1ndash25 2008

[9] H Mejjaoli and N Sraeib ldquoGeneralized sobolev spaces inquantum calculus and applicationsrdquo Journal of Inequalities andSpecial Functions vol 1 no 4 pp 43ndash64 2012

[10] H Mejjaoli ldquoGeneralized homogeneous Besov spaces and theirapplicationsrdquo Serdica Mathematical Journal vol 38 no 4 pp575ndash614 2012

[11] H Triebel Interpolation Theory Functions Spaces DifferentialOperators North-Holland AmsterdamThe Netherlands 1978

[12] V S Guliev ldquoOn maximal function and fractional integralassociated with the Bessel differential operatorrdquo MathematicalInequalities and Applications vol 6 no 2 pp 317ndash330 2003

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 9

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)int

infin

0

(sum

119895isinZ

11990511990422119895119904119890minus12058111990522119895

119888119903

119903119895)

119889119905

119905

le 119862119906119903

Bminus2119904120573

119901119903(R119889+1+

)sum

119895isinZ

119888119903

119903119895int

infin

0

(11990511990422119895119904119890minus12058111990522119895

)

119889119905

119905

le 119862Γ (119904) 119906119903

Bminus2119904120573

119901119903(R119889+1+

)

(100)

In order to prove the other inequality let us observe thatfor any 119904 greater than minus1 we have

Δ 119895119906 =

1

Γ (119904 + 1)

int

infin

0

119905119904(minusΔ 120573)

119904+1

119867120573 (119905) Δ 119895119906 119889119905 (101)

Then Lemma 30 Proposition 9 and the fact that the operatorΔ 119895 commutes with the operator119867120573(119905) lead to the following

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862int

infin

0

11990511990422119895(119904+1)

119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889119905

(102)

In the case 119903 = infin we simply write

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862(sup

119905gt0

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

times int

infin

0

22119895(119904+1)

119890minus12058111990522119895

119889119905

le 11986222119895119904(sup119905gt0

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

(103)

In the case 119903 lt infin Holderrsquos inequality with the weight 119890minus12058111990522119895

gives

(int

infin

0

119905119904119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889119905)

119903

le 1198622minus2119895(119903minus1)

int

infin

0

119905119903119904119890minus1205811199052211989510038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)119889119905

(104)

Thanks to (99) and Fubinirsquos theorem we infer from (102) that

sum

119895isinZ

2minus211989511990311990410038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)le 119862int

infin

0

11990511990311990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

119889119905

119905

(105)

The theorem is proved

Second Proof ofTheorem 29We only consider the case 1 le 119903 ltinfinThe case 119903 = infin can be shown similarlyWe first prove that

119862minus1119906

Bminus2119904120573

119901119903(R119889+1+

)le

10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

(106)

It is easy to see that

Δ 119895119906 = 120594119895lowast119882119864(120573)

2minus2119895lowast119882119906

(107)

where 120594119895 = Fminus1

119882(120593(2

minus119895120585)119890

2minus21198951205852

) and119864(120573)2minus2119895

is the Gauss kernelassociated with Weinstein operators By relation (29) we get

10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le

10038171003817100381710038171003817120594119895

100381710038171003817100381710038171198711120573(R119889+1+

)

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

) (108)

As10038171003817100381710038171003817120594119895

100381710038171003817100381710038171198711120573(R119889+1+

)= int

R119889+1+

100381610038161003816100381610038161003816F

minus1

119882(120593 (120585) 119890

1205852

)

100381610038161003816100381610038161003816119889120583120573 (120585) lt infin (109)

we obtain10038171003817100381710038171003817Δ 119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

) (110)

Moreover simple calculations give that

119864(120573)

2minus2119895lowast119882119906 = 119867120573 (2

minus4119895minus 119905

2) (119864

(120573)

1199052lowast119882119906)

(111)

Thus from Proposition 26 it follows that100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1

+ )

(112)

for any 119905 isin [2minus119895minus1 2minus119895] which implies that

sum

119895isinZ

2minus211990411989511990310038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

le 119862 sum

119895isinZ

int

2minus119895

2minus119895minus1(119905

2119904100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862int

infin

0

(1199052119904100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862int

infin

0

(11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

(113)

where we have used the fact that 119864(120573)1199052lowast119882119906 = 119867120573(119905

2)119906

We now prove that10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862119906Bminus2119904120573

119901119903(R119889+1+

) (114)

Indeed one has

119864(120573)

2minus2119895lowast119882119906 = sum

119899isinZ

119864(120573)

2minus2119895lowast119882Δ 119899+119895119906 (115)

Arguing as above we have100381710038171003817100381710038171003817119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 119862

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1

+ )

(116)

for any 119905 isin [2minus119895 21minus119895] Thus10038171003817100381710038171003817100381710038171003817

11990511990410038171003817100381710038171003817119867120573 (119905) 119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

10038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

= 2

100381710038171003817100381710038171003817100381710038171003817

1003817100381710038171003817100381710038171199052119904119864(120573)

1199052lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)

100381710038171003817100381710038171003817100381710038171003817119871119903(R+ 119889119905119905)

le 119862 sum

119895isinZ

int

21minus119895

2minus119895(2

minus2119895119904100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903119889119905

119905

le 119862 sum

119895isinZ

(2minus2119895119904

sum

119899isinZ

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

(117)

10 Journal of Function Spaces and Applications

On the other hand it is easy to see that100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 1198622

minus211989911990410038171003817100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

) (118)

for any 119904 gt 0 For 0 lt 1199041 lt 119904 lt 1199042 and by using theMinkowskiinequality we have

sum

119895isinZ

(2minus2119895119904

sum

119899isinZ

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

le sum

119895isinZ

(2minus2119895119904

0

sum

minusinfin

2minus21198991199041

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

+ sum

119895isinZ

(2minus2119895119904

sum

119899isinN

2minus21198991199042

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

le 119862(

0

sum

minusinfin

2minus2119899(119904

1minus119904)(sum

119895isinZ

(2minus2(119899+119895)11990311990410038171003817

100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

)

1119903

)

119903

+ 119862(sum

N

2minus2119899(119904

2minus119904)(sum

119895isinZ

(2minus2(119899+119895)11990311990410038171003817

100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

)

1119903

)

119903

le 119862 sum

119895isinZ

2minus211989511990311990410038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

(119)

The result is immediately from (117) and (119)

42 Embedding Sobolev Theorems

Theorem 31 Let 1 lt 119901 lt infin and let 0 lt 119904 lt (119889 + 2120573 +

2)119901There exists a positive constant119862 such that for all function119891 isin

H119904

119901120573(R119889+1

+) one has

10038171003817100381710038171198911003817100381710038171003817119871119902

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

H119904119901120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

120579

B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

) (120)

where 120579 = 119904119901(119889+2120573+2) and 119902 = 119901(119889+2120573+2)(119889+2120573+2minus119901119904)

Proof Bydensity we can suppose that119891belongs toSlowast(R119889+1

)It is easy to see that

119891 = int

infin

0

119867120573 (119905) Δ 120573119891119889119905(121)

and decompose the integral in two parts as follows

119891 = int

119860

0

119867120573 (119905) Δ 120573119891119889119905 + int

infin

119860

119867120573 (119905) Δ 120573119891119889119905(122)

where 119860 is a constant to be fixed laterOn the other hand byTheorem 29 we obtain10038171003817100381710038171003817119867120573 (119905) Δ 120573119891

10038171003817100381710038171003817119871infin120573(R119889+1+

)

le

119862

1199051minus(12)(119904minus(119889+2120573+2)119901)

10038171003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(123)

Therefore after integrating we get

int

infin

119860

10038171003817100381710038171003817119867120573 (119905) Δ 120573119891

10038171003817100381710038171003817119871infin120573(R119889+1+

)119889119905

le 119860(12)(119904minus(119889+2120573+2)119901)1003817

1003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(124)

On the other hand denoting 119892 = (minusΔ 120573)1199042119891 we have

119867120573 (119905) Δ 120573119891 =

1

(minus119905)1minus1199042

119867120573 (119905) (minus119905Δ 120573)

1minus1199042

119892 (125)

We proceed as in [8] we prove that1003816100381610038161003816100381610038161003816

119867120573 (119905) (minus119905Δ 120573)

1minus1199042

119892 (119909)

1003816100381610038161003816100381610038161003816

le 119862 (119904)119872120573 (119892) (119909) (126)

where119872120573(119892) is a maximal function of 119892 associated with theWeinstein operators (cf [12])

This leads to100381610038161003816100381610038161003816100381610038161003816

int

119860

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

100381610038161003816100381610038161003816100381610038161003816

le 1198621198601199042119872120573 (119892) (119909) (127)

In conclusion we get10038161003816100381610038161003816100381610038161003816

int

infin

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

10038161003816100381610038161003816100381610038161003816

le 119862 (1198601199042119872120573 (119892) (119909)

+119860(12)(119904minus(119889+2120573+2)119901)1003817

1003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

))

(128)

and the choice of 119860 such that

119860(119889+2120573+2)2119901

119872120573 (119892) (119909) =10038171003817100381710038171198911003817100381710038171003817B119904minus(119889+2120573+2)119901120573

infininfin (R119889+1+

)(129)

ensures that10038161003816100381610038161003816100381610038161003816

int

infin

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

10038161003816100381610038161003816100381610038161003816

le 119862(119872120573(119892)(119909))

1minus(119901119904(119889+2120573+2))10038171003817100381710038171198911003817100381710038171003817

119901119904(119889+2120573+2)

B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(130)

Finally taking the 119871119902120573norm with 119902 = 119901(119889 + 2120573 + 2)(119889 + 2120573 +

2minus119901119904) ends the proof thanks to the fact themaximal function119872120573 is bounded of 119871119902

120573(R119889+1

+) into itself for 119902 gt 1

Theorem 32 Let 1 lt 119901 lt 119902 lt infin For all function 119891 such that119891 isin

H1199041

119901120573(R119889+1

+)⋂

Bminus120573120573

infininfin(R119889+1

+) one has

10038171003817100381710038171198911003817100381710038171003817H119904119901120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

120579

H1199041

119901120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573120573

infininfin(R119889+1

+) (131)

where 120579 = 119901119902 119904 = 1205791199041 minus (1 minus 120579)120573 with 120573 gt 0 minus120573 lt 119904 lt 1199041

Proof It suffices to prove that1003817100381710038171003817100381710038171003817

(minusΔ 120573)

(119904minus1199041)2

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le 11986210038171003817100381710038171198911003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(132)

Journal of Function Spaces and Applications 11

Indeed we use the following identity (which may be easilyproven by taking the Weinstein transform in 119909 of both sides)

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

infin

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905 (133)

with 120575 = 1199041 minus 119904 gt 0We decompose the integral in two parts as follows

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

119879

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

+

1

Γ (1205752)

int

infin

119879

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

(134)

where 119879 is a constant to be fixed laterWe proceed as in [8] we obtain

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119872120573 (119891) (119909) (135)

On the other hand we use Theorem 29 and the fact that 119891belongs to Bminus120573minus119904

1120573

infininfin(R119889+1

+) to deduce that

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119905

(minus120573minus1199041)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

) (136)

Thus by applying the preceding estimates on the right part of(134) we obtain

1003816100381610038161003816100381610038161003816

(minusΔ 120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621

Γ (1205752)

1198791205752119872120573 (119891) (119909)

+

1198622

Γ (1205752)

119879(120575minus120573minus119904

1)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

(137)

We fix now

119879 = (

10038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

119872120573 (119891) (119909)

)

2(120573+1199041)

(138)

We obtain

1003816100381610038161003816100381610038161003816

(minusΔ120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621 + 1198622

Γ (1205752)

(119872120573(119891)(119909))

12057910038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(139)

Thus we deduce that1003817100381710038171003817100381710038171003817

(minusΔ120573)

minus1205752

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le

1198621 + 1198622

Γ (1205752)

10038171003817100381710038171003817119872120573 (119891)

10038171003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(140)

To conclude we used the fact that the maximal function119872120573

is bounded of 119871119902120573(R119889+1

+) into itself for 119902 gt 1

43 Estimates in Generalized Besov Spaces For any interval 119868ofR (bounded or unbounded) and a normed space 119865(R119889+1

+)

we define the mixed space-time 119871119901(119868 119865(R119889+1

+)) space of

(classes of) measurable functions 119906 119868 rarr 119865(R119889+1

+) such that

||119906||119871119901(119868119865(R119889+1+

)) lt infin with

119906119871119901(119868119865(R119889+1+

)) = (int

119868

119906 (119905 sdot)119901

119865(R119889+1+

)119889119905)

1119901

if 1 le 119901 lt infin

119906119871infin(119868119865(R119889+1+

)) = ess sup119905isin119868

119906 (119905 sdot)119865(R119889+1+

)

(141)

For any interval 119868 of R (bounded or unbounded) anda Banach space 119883 we define the mixed space-time 119862(119868119883)space of continuous functions 119868 rarr 119883 When 119868 is bounded119862(119868119883) is a Banach space with the norm of 119871infin(119868 119883)

Theorem 33 Let 119904 isin R and 1 le 119901 119902 119903 le infin Let 119879 gt 0 119892 isin

B119904120573

119901119903(R119889+1

+) and119891 in 119871119902((0 119879) B

119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76)

has a unique solution

119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903(R

119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R

119889+1

+))

(142)

and there exists a constant 119862 such that for all 1199021 isin [119902infin] onehas

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862(10038171003817100381710038171198921003817100381710038171003817B119904120573

119901119903(R119889+1+

)+10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)B

119904minus2+(2119902)120573

119901119903(R119889+1+

)))

(143)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] B119904120573

119901119903(R119889+1

+))

Proof Since 119892 and 119891 are temperate distributions (76) has aunique solution 119906 in S1015840

((0 119879) timesR119889+1

+) which satisfies

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(144)

Next we notice that applying Δ 119895 to (76) and using formula(81) yield

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(145)

Therefore

10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(146)

12 Journal of Function Spaces and Applications

By virtue of Lemma 30 we thus have for some 120581 gt 0

10038171003817100381710038171003817Δ 119895119906(119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

119890minus12058122119895(119905minus120591) 10038171003817

100381710038171003817Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(147)

Applying convolution inequalities we get

10038171003817100381710038171003817Δ 119895119906

100381710038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862[

[

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

)

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

]

]

(148)

with 11199022 = 1+ 11199021 minus1119902 Finally taking the 119897119903(Z) norm we

conclude that (with the usual convention if 119903 = infin)

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

1199031199021

)(211989511990410038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

]

]

1119903

+ 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

1199031199022

)

times (2119895(119904minus2+2119902)10038171003817

100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902(0119879119871

119901

120573(R119889+1+

))

119903

]

]

1119903

(149)

which insures that 119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903 (R119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R119889+1

+)) and yields the desired inequality

Since 119906 belongs to 119862([0 119879]B119904120573

119901119903(R119889+1

+)) in the case where

119903 is finite may be easily deduced from the density ofSlowast(R

119889+1)⋂

B119904120573

119901119903(R119889+1

+) in B

119904120573

119901119903(R)

Theorem 34 Let 119904 isin R 119879 gt 0 and 1 le 119901 119902 119903 le infinOne supposes that 119892 isin 119861

119904120573

119901119903(R119889+1

+) and 119891 isin 119871

119902((0 119879)

119861119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76) has a unique solution 119906 belong-

ing to

119871119902((0 119879) 119861

119904+(2119902)120573

119901119903(R

119889+1

+))⋂119871

infin((0 119879) 119861

119904120573

119901119903(R

119889+1

+))

(150)

and there exists a constant 119862 such that for all 119902 le 1199021 le infin

1199061198711199021 ((0119879)119861

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862 [ (1 + 11987911199021)10038171003817100381710038171198921003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (1 + 1198791+1119902

1minus1119902

)10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)119861

119904minus2+(2119902)120573

119901119903(R119889+1+

))]

(151)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] 119861119904120573119901119903(R119889+1

+))

Proof Since 119892 119891 are tempered (76) has a unique solution 119906in S1015840

((0 119879) timesR119889+1

+) satisfying

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(152)

Hence applying Δ 119895 119895 ge 0 to (81) we see that

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(153)

and thus by Lemma 30 we can deduce that10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+int

119905

0

119890minus12058122119895(119905minus120591)10038171003817

100381710038171003817Δ 119895119891(120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(154)

Then it follows from convolution inequalities thatΔ 1198951199061198711199021 ((0119879)119871

119901

120573(R119889+1+

))is dominated by

(

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

(155)

with 11199022 = 1 + 11199021 minus 1119902 Moreover similarly as above wecan obtain that

1003817100381710038171003817Δminus1119906(119905 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)le1003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

1003817100381710038171003817Δminus1119891 (120591 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(156)

and thus if 1 le 119902 le 1199021 le infin1003817100381710038171003817Δminus1119906

10038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862(119879111990211003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)+ 119879

111990221003817100381710038171003817Δminus1119891

1003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

)))

(157)

Journal of Function Spaces and Applications 13

Finally taking the 119897119903-normwith respect to 119895 in (155) and (157)with the usual convention if 119903 = infin we can deduce the desiredestimate

Acknowledgment

Theauthor gratefully acknowledges theDeanship of ScientificResearch at the University of Taibah The author is deeplyindebted to the referee for providing constructive commentsand help in improving the contents of this paper

References

[1] Z Ben Nahia and N Ben Salem ldquoSpherical harmonics andapplications associated with the Weinstein operatorrdquo in Pro-ceedings of the International Conference on PotentialTheory heldin Kouty Czech Republic (ICPT rsquo94) pp 235ndash241 1996

[2] Z Ben Nahia and N Ben Salem ldquoOn a mean value propertyassociated with the Weinstein operatorrdquo in Proceedings of theInternational Conference on Potential Theory held in KoutyCzech Republic (ICPT rsquo94) pp 243ndash253 1996

[3] M Brelot ldquoEquation de Weinstein et potentiels de MarcelRieszrdquo in Seminaire de Theorie de Potentiel Paris No 3 vol 681of Lecture Notes in Mathematics pp 18ndash38 Springer BerlinGermany 1978

[4] H Mejjaoli and M Salhi ldquoUncertainty principles for theweinstein transformrdquo Czechoslovak Mathematical Journal vol61 no 4 pp 941ndash974 2011

[5] H Mejjaoli and A Ould Ahmed Salem ldquoWeinstein Gabortransform and applicationsrdquo Advanced Studies in Pure Mathe-matics vol 2 no 3 pp 203ndash210 2012

[6] H Mejjaoli ldquoBesov spaces associated withthe Weinstein opera-tor and applicationsrdquo In press

[7] T Kawazoe and H Mejjaoli ldquoGeneralized Besov spaces andtheir applicationsrdquo Tokyo Journal of Mathematics vol 35 no 2pp 297ndash320 2012

[8] H Mejjaoli ldquoLittlewood-Paley decomposition associated withthe Dunkl operators and paraproduct operatorsrdquo Journal ofInequalities in Pure and Applied Mathematics vol 9 no 4 pp1ndash25 2008

[9] H Mejjaoli and N Sraeib ldquoGeneralized sobolev spaces inquantum calculus and applicationsrdquo Journal of Inequalities andSpecial Functions vol 1 no 4 pp 43ndash64 2012

[10] H Mejjaoli ldquoGeneralized homogeneous Besov spaces and theirapplicationsrdquo Serdica Mathematical Journal vol 38 no 4 pp575ndash614 2012

[11] H Triebel Interpolation Theory Functions Spaces DifferentialOperators North-Holland AmsterdamThe Netherlands 1978

[12] V S Guliev ldquoOn maximal function and fractional integralassociated with the Bessel differential operatorrdquo MathematicalInequalities and Applications vol 6 no 2 pp 317ndash330 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Function Spaces and Applications

On the other hand it is easy to see that100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

)le 1198622

minus211989911990410038171003817100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

) (118)

for any 119904 gt 0 For 0 lt 1199041 lt 119904 lt 1199042 and by using theMinkowskiinequality we have

sum

119895isinZ

(2minus2119895119904

sum

119899isinZ

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

le sum

119895isinZ

(2minus2119895119904

0

sum

minusinfin

2minus21198991199041

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

+ sum

119895isinZ

(2minus2119895119904

sum

119899isinN

2minus21198991199042

100381710038171003817100381710038171003817119864(120573)

2minus2119895lowast119882Δ 119899+119895119906

100381710038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

le 119862(

0

sum

minusinfin

2minus2119899(119904

1minus119904)(sum

119895isinZ

(2minus2(119899+119895)11990311990410038171003817

100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

)

1119903

)

119903

+ 119862(sum

N

2minus2119899(119904

2minus119904)(sum

119895isinZ

(2minus2(119899+119895)11990311990410038171003817

100381710038171003817Δ 119899+119895119906

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

)

1119903

)

119903

le 119862 sum

119895isinZ

2minus211989511990311990410038171003817

100381710038171003817Δ 119895119906

10038171003817100381710038171003817

119903

119871119901

120573(R119889+1+

)

(119)

The result is immediately from (117) and (119)

42 Embedding Sobolev Theorems

Theorem 31 Let 1 lt 119901 lt infin and let 0 lt 119904 lt (119889 + 2120573 +

2)119901There exists a positive constant119862 such that for all function119891 isin

H119904

119901120573(R119889+1

+) one has

10038171003817100381710038171198911003817100381710038171003817119871119902

120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

1minus120579

H119904119901120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

120579

B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

) (120)

where 120579 = 119904119901(119889+2120573+2) and 119902 = 119901(119889+2120573+2)(119889+2120573+2minus119901119904)

Proof Bydensity we can suppose that119891belongs toSlowast(R119889+1

)It is easy to see that

119891 = int

infin

0

119867120573 (119905) Δ 120573119891119889119905(121)

and decompose the integral in two parts as follows

119891 = int

119860

0

119867120573 (119905) Δ 120573119891119889119905 + int

infin

119860

119867120573 (119905) Δ 120573119891119889119905(122)

where 119860 is a constant to be fixed laterOn the other hand byTheorem 29 we obtain10038171003817100381710038171003817119867120573 (119905) Δ 120573119891

10038171003817100381710038171003817119871infin120573(R119889+1+

)

le

119862

1199051minus(12)(119904minus(119889+2120573+2)119901)

10038171003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(123)

Therefore after integrating we get

int

infin

119860

10038171003817100381710038171003817119867120573 (119905) Δ 120573119891

10038171003817100381710038171003817119871infin120573(R119889+1+

)119889119905

le 119860(12)(119904minus(119889+2120573+2)119901)1003817

1003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(124)

On the other hand denoting 119892 = (minusΔ 120573)1199042119891 we have

119867120573 (119905) Δ 120573119891 =

1

(minus119905)1minus1199042

119867120573 (119905) (minus119905Δ 120573)

1minus1199042

119892 (125)

We proceed as in [8] we prove that1003816100381610038161003816100381610038161003816

119867120573 (119905) (minus119905Δ 120573)

1minus1199042

119892 (119909)

1003816100381610038161003816100381610038161003816

le 119862 (119904)119872120573 (119892) (119909) (126)

where119872120573(119892) is a maximal function of 119892 associated with theWeinstein operators (cf [12])

This leads to100381610038161003816100381610038161003816100381610038161003816

int

119860

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

100381610038161003816100381610038161003816100381610038161003816

le 1198621198601199042119872120573 (119892) (119909) (127)

In conclusion we get10038161003816100381610038161003816100381610038161003816

int

infin

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

10038161003816100381610038161003816100381610038161003816

le 119862 (1198601199042119872120573 (119892) (119909)

+119860(12)(119904minus(119889+2120573+2)119901)1003817

1003817100381710038171198911003817100381710038171003817B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

))

(128)

and the choice of 119860 such that

119860(119889+2120573+2)2119901

119872120573 (119892) (119909) =10038171003817100381710038171198911003817100381710038171003817B119904minus(119889+2120573+2)119901120573

infininfin (R119889+1+

)(129)

ensures that10038161003816100381610038161003816100381610038161003816

int

infin

0

119867120573 (119905) Δ 120573119891 (119909) 119889119905

10038161003816100381610038161003816100381610038161003816

le 119862(119872120573(119892)(119909))

1minus(119901119904(119889+2120573+2))10038171003817100381710038171198911003817100381710038171003817

119901119904(119889+2120573+2)

B119904minus((119889+2120573+2)119901)120573

infininfin (R119889+1+

)

(130)

Finally taking the 119871119902120573norm with 119902 = 119901(119889 + 2120573 + 2)(119889 + 2120573 +

2minus119901119904) ends the proof thanks to the fact themaximal function119872120573 is bounded of 119871119902

120573(R119889+1

+) into itself for 119902 gt 1

Theorem 32 Let 1 lt 119901 lt 119902 lt infin For all function 119891 such that119891 isin

H1199041

119901120573(R119889+1

+)⋂

Bminus120573120573

infininfin(R119889+1

+) one has

10038171003817100381710038171198911003817100381710038171003817H119904119901120573(R119889+1+

)le 119862

10038171003817100381710038171198911003817100381710038171003817

120579

H1199041

119901120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573120573

infininfin(R119889+1

+) (131)

where 120579 = 119901119902 119904 = 1205791199041 minus (1 minus 120579)120573 with 120573 gt 0 minus120573 lt 119904 lt 1199041

Proof It suffices to prove that1003817100381710038171003817100381710038171003817

(minusΔ 120573)

(119904minus1199041)2

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le 11986210038171003817100381710038171198911003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(132)

Journal of Function Spaces and Applications 11

Indeed we use the following identity (which may be easilyproven by taking the Weinstein transform in 119909 of both sides)

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

infin

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905 (133)

with 120575 = 1199041 minus 119904 gt 0We decompose the integral in two parts as follows

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

119879

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

+

1

Γ (1205752)

int

infin

119879

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

(134)

where 119879 is a constant to be fixed laterWe proceed as in [8] we obtain

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119872120573 (119891) (119909) (135)

On the other hand we use Theorem 29 and the fact that 119891belongs to Bminus120573minus119904

1120573

infininfin(R119889+1

+) to deduce that

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119905

(minus120573minus1199041)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

) (136)

Thus by applying the preceding estimates on the right part of(134) we obtain

1003816100381610038161003816100381610038161003816

(minusΔ 120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621

Γ (1205752)

1198791205752119872120573 (119891) (119909)

+

1198622

Γ (1205752)

119879(120575minus120573minus119904

1)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

(137)

We fix now

119879 = (

10038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

119872120573 (119891) (119909)

)

2(120573+1199041)

(138)

We obtain

1003816100381610038161003816100381610038161003816

(minusΔ120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621 + 1198622

Γ (1205752)

(119872120573(119891)(119909))

12057910038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(139)

Thus we deduce that1003817100381710038171003817100381710038171003817

(minusΔ120573)

minus1205752

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le

1198621 + 1198622

Γ (1205752)

10038171003817100381710038171003817119872120573 (119891)

10038171003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(140)

To conclude we used the fact that the maximal function119872120573

is bounded of 119871119902120573(R119889+1

+) into itself for 119902 gt 1

43 Estimates in Generalized Besov Spaces For any interval 119868ofR (bounded or unbounded) and a normed space 119865(R119889+1

+)

we define the mixed space-time 119871119901(119868 119865(R119889+1

+)) space of

(classes of) measurable functions 119906 119868 rarr 119865(R119889+1

+) such that

||119906||119871119901(119868119865(R119889+1+

)) lt infin with

119906119871119901(119868119865(R119889+1+

)) = (int

119868

119906 (119905 sdot)119901

119865(R119889+1+

)119889119905)

1119901

if 1 le 119901 lt infin

119906119871infin(119868119865(R119889+1+

)) = ess sup119905isin119868

119906 (119905 sdot)119865(R119889+1+

)

(141)

For any interval 119868 of R (bounded or unbounded) anda Banach space 119883 we define the mixed space-time 119862(119868119883)space of continuous functions 119868 rarr 119883 When 119868 is bounded119862(119868119883) is a Banach space with the norm of 119871infin(119868 119883)

Theorem 33 Let 119904 isin R and 1 le 119901 119902 119903 le infin Let 119879 gt 0 119892 isin

B119904120573

119901119903(R119889+1

+) and119891 in 119871119902((0 119879) B

119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76)

has a unique solution

119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903(R

119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R

119889+1

+))

(142)

and there exists a constant 119862 such that for all 1199021 isin [119902infin] onehas

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862(10038171003817100381710038171198921003817100381710038171003817B119904120573

119901119903(R119889+1+

)+10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)B

119904minus2+(2119902)120573

119901119903(R119889+1+

)))

(143)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] B119904120573

119901119903(R119889+1

+))

Proof Since 119892 and 119891 are temperate distributions (76) has aunique solution 119906 in S1015840

((0 119879) timesR119889+1

+) which satisfies

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(144)

Next we notice that applying Δ 119895 to (76) and using formula(81) yield

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(145)

Therefore

10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(146)

12 Journal of Function Spaces and Applications

By virtue of Lemma 30 we thus have for some 120581 gt 0

10038171003817100381710038171003817Δ 119895119906(119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

119890minus12058122119895(119905minus120591) 10038171003817

100381710038171003817Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(147)

Applying convolution inequalities we get

10038171003817100381710038171003817Δ 119895119906

100381710038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862[

[

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

)

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

]

]

(148)

with 11199022 = 1+ 11199021 minus1119902 Finally taking the 119897119903(Z) norm we

conclude that (with the usual convention if 119903 = infin)

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

1199031199021

)(211989511990410038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

]

]

1119903

+ 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

1199031199022

)

times (2119895(119904minus2+2119902)10038171003817

100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902(0119879119871

119901

120573(R119889+1+

))

119903

]

]

1119903

(149)

which insures that 119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903 (R119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R119889+1

+)) and yields the desired inequality

Since 119906 belongs to 119862([0 119879]B119904120573

119901119903(R119889+1

+)) in the case where

119903 is finite may be easily deduced from the density ofSlowast(R

119889+1)⋂

B119904120573

119901119903(R119889+1

+) in B

119904120573

119901119903(R)

Theorem 34 Let 119904 isin R 119879 gt 0 and 1 le 119901 119902 119903 le infinOne supposes that 119892 isin 119861

119904120573

119901119903(R119889+1

+) and 119891 isin 119871

119902((0 119879)

119861119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76) has a unique solution 119906 belong-

ing to

119871119902((0 119879) 119861

119904+(2119902)120573

119901119903(R

119889+1

+))⋂119871

infin((0 119879) 119861

119904120573

119901119903(R

119889+1

+))

(150)

and there exists a constant 119862 such that for all 119902 le 1199021 le infin

1199061198711199021 ((0119879)119861

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862 [ (1 + 11987911199021)10038171003817100381710038171198921003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (1 + 1198791+1119902

1minus1119902

)10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)119861

119904minus2+(2119902)120573

119901119903(R119889+1+

))]

(151)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] 119861119904120573119901119903(R119889+1

+))

Proof Since 119892 119891 are tempered (76) has a unique solution 119906in S1015840

((0 119879) timesR119889+1

+) satisfying

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(152)

Hence applying Δ 119895 119895 ge 0 to (81) we see that

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(153)

and thus by Lemma 30 we can deduce that10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+int

119905

0

119890minus12058122119895(119905minus120591)10038171003817

100381710038171003817Δ 119895119891(120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(154)

Then it follows from convolution inequalities thatΔ 1198951199061198711199021 ((0119879)119871

119901

120573(R119889+1+

))is dominated by

(

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

(155)

with 11199022 = 1 + 11199021 minus 1119902 Moreover similarly as above wecan obtain that

1003817100381710038171003817Δminus1119906(119905 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)le1003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

1003817100381710038171003817Δminus1119891 (120591 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(156)

and thus if 1 le 119902 le 1199021 le infin1003817100381710038171003817Δminus1119906

10038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862(119879111990211003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)+ 119879

111990221003817100381710038171003817Δminus1119891

1003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

)))

(157)

Journal of Function Spaces and Applications 13

Finally taking the 119897119903-normwith respect to 119895 in (155) and (157)with the usual convention if 119903 = infin we can deduce the desiredestimate

Acknowledgment

Theauthor gratefully acknowledges theDeanship of ScientificResearch at the University of Taibah The author is deeplyindebted to the referee for providing constructive commentsand help in improving the contents of this paper

References

[1] Z Ben Nahia and N Ben Salem ldquoSpherical harmonics andapplications associated with the Weinstein operatorrdquo in Pro-ceedings of the International Conference on PotentialTheory heldin Kouty Czech Republic (ICPT rsquo94) pp 235ndash241 1996

[2] Z Ben Nahia and N Ben Salem ldquoOn a mean value propertyassociated with the Weinstein operatorrdquo in Proceedings of theInternational Conference on Potential Theory held in KoutyCzech Republic (ICPT rsquo94) pp 243ndash253 1996

[3] M Brelot ldquoEquation de Weinstein et potentiels de MarcelRieszrdquo in Seminaire de Theorie de Potentiel Paris No 3 vol 681of Lecture Notes in Mathematics pp 18ndash38 Springer BerlinGermany 1978

[4] H Mejjaoli and M Salhi ldquoUncertainty principles for theweinstein transformrdquo Czechoslovak Mathematical Journal vol61 no 4 pp 941ndash974 2011

[5] H Mejjaoli and A Ould Ahmed Salem ldquoWeinstein Gabortransform and applicationsrdquo Advanced Studies in Pure Mathe-matics vol 2 no 3 pp 203ndash210 2012

[6] H Mejjaoli ldquoBesov spaces associated withthe Weinstein opera-tor and applicationsrdquo In press

[7] T Kawazoe and H Mejjaoli ldquoGeneralized Besov spaces andtheir applicationsrdquo Tokyo Journal of Mathematics vol 35 no 2pp 297ndash320 2012

[8] H Mejjaoli ldquoLittlewood-Paley decomposition associated withthe Dunkl operators and paraproduct operatorsrdquo Journal ofInequalities in Pure and Applied Mathematics vol 9 no 4 pp1ndash25 2008

[9] H Mejjaoli and N Sraeib ldquoGeneralized sobolev spaces inquantum calculus and applicationsrdquo Journal of Inequalities andSpecial Functions vol 1 no 4 pp 43ndash64 2012

[10] H Mejjaoli ldquoGeneralized homogeneous Besov spaces and theirapplicationsrdquo Serdica Mathematical Journal vol 38 no 4 pp575ndash614 2012

[11] H Triebel Interpolation Theory Functions Spaces DifferentialOperators North-Holland AmsterdamThe Netherlands 1978

[12] V S Guliev ldquoOn maximal function and fractional integralassociated with the Bessel differential operatorrdquo MathematicalInequalities and Applications vol 6 no 2 pp 317ndash330 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 11

Indeed we use the following identity (which may be easilyproven by taking the Weinstein transform in 119909 of both sides)

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

infin

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905 (133)

with 120575 = 1199041 minus 119904 gt 0We decompose the integral in two parts as follows

(minusΔ 120573)

minus1205752

119891 (119909) =

1

Γ (1205752)

int

119879

0

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

+

1

Γ (1205752)

int

infin

119879

1199051205752minus1

119867120573 (119905) 119891 (119909) 119889119905

(134)

where 119879 is a constant to be fixed laterWe proceed as in [8] we obtain

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119872120573 (119891) (119909) (135)

On the other hand we use Theorem 29 and the fact that 119891belongs to Bminus120573minus119904

1120573

infininfin(R119889+1

+) to deduce that

10038161003816100381610038161003816119867120573 (119905) 119891 (119909)

10038161003816100381610038161003816le 119862119905

(minus120573minus1199041)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

) (136)

Thus by applying the preceding estimates on the right part of(134) we obtain

1003816100381610038161003816100381610038161003816

(minusΔ 120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621

Γ (1205752)

1198791205752119872120573 (119891) (119909)

+

1198622

Γ (1205752)

119879(120575minus120573minus119904

1)210038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

(137)

We fix now

119879 = (

10038171003817100381710038171198911003817100381710038171003817Bminus120573minus1199041120573

infininfin (R119889+1+

)

119872120573 (119891) (119909)

)

2(120573+1199041)

(138)

We obtain

1003816100381610038161003816100381610038161003816

(minusΔ120573)

minus1205752

119891 (119909)

1003816100381610038161003816100381610038161003816

le

1198621 + 1198622

Γ (1205752)

(119872120573(119891)(119909))

12057910038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(139)

Thus we deduce that1003817100381710038171003817100381710038171003817

(minusΔ120573)

minus1205752

119891

1003817100381710038171003817100381710038171003817119871119902

120573(R119889+1+

)

le

1198621 + 1198622

Γ (1205752)

10038171003817100381710038171003817119872120573 (119891)

10038171003817100381710038171003817

120579

119871119901

120573(R119889+1+

)

10038171003817100381710038171198911003817100381710038171003817

1minus120579

Bminus120573minus1199041120573

infininfin (R119889+1+

)

(140)

To conclude we used the fact that the maximal function119872120573

is bounded of 119871119902120573(R119889+1

+) into itself for 119902 gt 1

43 Estimates in Generalized Besov Spaces For any interval 119868ofR (bounded or unbounded) and a normed space 119865(R119889+1

+)

we define the mixed space-time 119871119901(119868 119865(R119889+1

+)) space of

(classes of) measurable functions 119906 119868 rarr 119865(R119889+1

+) such that

||119906||119871119901(119868119865(R119889+1+

)) lt infin with

119906119871119901(119868119865(R119889+1+

)) = (int

119868

119906 (119905 sdot)119901

119865(R119889+1+

)119889119905)

1119901

if 1 le 119901 lt infin

119906119871infin(119868119865(R119889+1+

)) = ess sup119905isin119868

119906 (119905 sdot)119865(R119889+1+

)

(141)

For any interval 119868 of R (bounded or unbounded) anda Banach space 119883 we define the mixed space-time 119862(119868119883)space of continuous functions 119868 rarr 119883 When 119868 is bounded119862(119868119883) is a Banach space with the norm of 119871infin(119868 119883)

Theorem 33 Let 119904 isin R and 1 le 119901 119902 119903 le infin Let 119879 gt 0 119892 isin

B119904120573

119901119903(R119889+1

+) and119891 in 119871119902((0 119879) B

119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76)

has a unique solution

119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903(R

119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R

119889+1

+))

(142)

and there exists a constant 119862 such that for all 1199021 isin [119902infin] onehas

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862(10038171003817100381710038171198921003817100381710038171003817B119904120573

119901119903(R119889+1+

)+10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)B

119904minus2+(2119902)120573

119901119903(R119889+1+

)))

(143)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] B119904120573

119901119903(R119889+1

+))

Proof Since 119892 and 119891 are temperate distributions (76) has aunique solution 119906 in S1015840

((0 119879) timesR119889+1

+) which satisfies

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(144)

Next we notice that applying Δ 119895 to (76) and using formula(81) yield

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(145)

Therefore

10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(146)

12 Journal of Function Spaces and Applications

By virtue of Lemma 30 we thus have for some 120581 gt 0

10038171003817100381710038171003817Δ 119895119906(119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

119890minus12058122119895(119905minus120591) 10038171003817

100381710038171003817Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(147)

Applying convolution inequalities we get

10038171003817100381710038171003817Δ 119895119906

100381710038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862[

[

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

)

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

]

]

(148)

with 11199022 = 1+ 11199021 minus1119902 Finally taking the 119897119903(Z) norm we

conclude that (with the usual convention if 119903 = infin)

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

1199031199021

)(211989511990410038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

]

]

1119903

+ 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

1199031199022

)

times (2119895(119904minus2+2119902)10038171003817

100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902(0119879119871

119901

120573(R119889+1+

))

119903

]

]

1119903

(149)

which insures that 119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903 (R119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R119889+1

+)) and yields the desired inequality

Since 119906 belongs to 119862([0 119879]B119904120573

119901119903(R119889+1

+)) in the case where

119903 is finite may be easily deduced from the density ofSlowast(R

119889+1)⋂

B119904120573

119901119903(R119889+1

+) in B

119904120573

119901119903(R)

Theorem 34 Let 119904 isin R 119879 gt 0 and 1 le 119901 119902 119903 le infinOne supposes that 119892 isin 119861

119904120573

119901119903(R119889+1

+) and 119891 isin 119871

119902((0 119879)

119861119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76) has a unique solution 119906 belong-

ing to

119871119902((0 119879) 119861

119904+(2119902)120573

119901119903(R

119889+1

+))⋂119871

infin((0 119879) 119861

119904120573

119901119903(R

119889+1

+))

(150)

and there exists a constant 119862 such that for all 119902 le 1199021 le infin

1199061198711199021 ((0119879)119861

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862 [ (1 + 11987911199021)10038171003817100381710038171198921003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (1 + 1198791+1119902

1minus1119902

)10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)119861

119904minus2+(2119902)120573

119901119903(R119889+1+

))]

(151)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] 119861119904120573119901119903(R119889+1

+))

Proof Since 119892 119891 are tempered (76) has a unique solution 119906in S1015840

((0 119879) timesR119889+1

+) satisfying

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(152)

Hence applying Δ 119895 119895 ge 0 to (81) we see that

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(153)

and thus by Lemma 30 we can deduce that10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+int

119905

0

119890minus12058122119895(119905minus120591)10038171003817

100381710038171003817Δ 119895119891(120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(154)

Then it follows from convolution inequalities thatΔ 1198951199061198711199021 ((0119879)119871

119901

120573(R119889+1+

))is dominated by

(

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

(155)

with 11199022 = 1 + 11199021 minus 1119902 Moreover similarly as above wecan obtain that

1003817100381710038171003817Δminus1119906(119905 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)le1003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

1003817100381710038171003817Δminus1119891 (120591 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(156)

and thus if 1 le 119902 le 1199021 le infin1003817100381710038171003817Δminus1119906

10038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862(119879111990211003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)+ 119879

111990221003817100381710038171003817Δminus1119891

1003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

)))

(157)

Journal of Function Spaces and Applications 13

Finally taking the 119897119903-normwith respect to 119895 in (155) and (157)with the usual convention if 119903 = infin we can deduce the desiredestimate

Acknowledgment

Theauthor gratefully acknowledges theDeanship of ScientificResearch at the University of Taibah The author is deeplyindebted to the referee for providing constructive commentsand help in improving the contents of this paper

References

[1] Z Ben Nahia and N Ben Salem ldquoSpherical harmonics andapplications associated with the Weinstein operatorrdquo in Pro-ceedings of the International Conference on PotentialTheory heldin Kouty Czech Republic (ICPT rsquo94) pp 235ndash241 1996

[2] Z Ben Nahia and N Ben Salem ldquoOn a mean value propertyassociated with the Weinstein operatorrdquo in Proceedings of theInternational Conference on Potential Theory held in KoutyCzech Republic (ICPT rsquo94) pp 243ndash253 1996

[3] M Brelot ldquoEquation de Weinstein et potentiels de MarcelRieszrdquo in Seminaire de Theorie de Potentiel Paris No 3 vol 681of Lecture Notes in Mathematics pp 18ndash38 Springer BerlinGermany 1978

[4] H Mejjaoli and M Salhi ldquoUncertainty principles for theweinstein transformrdquo Czechoslovak Mathematical Journal vol61 no 4 pp 941ndash974 2011

[5] H Mejjaoli and A Ould Ahmed Salem ldquoWeinstein Gabortransform and applicationsrdquo Advanced Studies in Pure Mathe-matics vol 2 no 3 pp 203ndash210 2012

[6] H Mejjaoli ldquoBesov spaces associated withthe Weinstein opera-tor and applicationsrdquo In press

[7] T Kawazoe and H Mejjaoli ldquoGeneralized Besov spaces andtheir applicationsrdquo Tokyo Journal of Mathematics vol 35 no 2pp 297ndash320 2012

[8] H Mejjaoli ldquoLittlewood-Paley decomposition associated withthe Dunkl operators and paraproduct operatorsrdquo Journal ofInequalities in Pure and Applied Mathematics vol 9 no 4 pp1ndash25 2008

[9] H Mejjaoli and N Sraeib ldquoGeneralized sobolev spaces inquantum calculus and applicationsrdquo Journal of Inequalities andSpecial Functions vol 1 no 4 pp 43ndash64 2012

[10] H Mejjaoli ldquoGeneralized homogeneous Besov spaces and theirapplicationsrdquo Serdica Mathematical Journal vol 38 no 4 pp575ndash614 2012

[11] H Triebel Interpolation Theory Functions Spaces DifferentialOperators North-Holland AmsterdamThe Netherlands 1978

[12] V S Guliev ldquoOn maximal function and fractional integralassociated with the Bessel differential operatorrdquo MathematicalInequalities and Applications vol 6 no 2 pp 317ndash330 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Journal of Function Spaces and Applications

By virtue of Lemma 30 we thus have for some 120581 gt 0

10038171003817100381710038171003817Δ 119895119906(119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

119890minus12058122119895(119905minus120591) 10038171003817

100381710038171003817Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(147)

Applying convolution inequalities we get

10038171003817100381710038171003817Δ 119895119906

100381710038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862[

[

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

)

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

)

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

]

]

(148)

with 11199022 = 1+ 11199021 minus1119902 Finally taking the 119897119903(Z) norm we

conclude that (with the usual convention if 119903 = infin)

1199061198711199021 ((0119879)B

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

1199031199021

)(211989511990410038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

))

119903

]

]

1119903

+ 119862[

[

sum

119895isinZ

((

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

1199031199022

)

times (2119895(119904minus2+2119902)10038171003817

100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902(0119879119871

119901

120573(R119889+1+

))

119903

]

]

1119903

(149)

which insures that 119906 isin 119871119902((0 119879)

B119904+(2119902)120573

119901119903 (R119889+1

+))

⋂119871infin((0 119879)

B119904120573

119901119903(R119889+1

+)) and yields the desired inequality

Since 119906 belongs to 119862([0 119879]B119904120573

119901119903(R119889+1

+)) in the case where

119903 is finite may be easily deduced from the density ofSlowast(R

119889+1)⋂

B119904120573

119901119903(R119889+1

+) in B

119904120573

119901119903(R)

Theorem 34 Let 119904 isin R 119879 gt 0 and 1 le 119901 119902 119903 le infinOne supposes that 119892 isin 119861

119904120573

119901119903(R119889+1

+) and 119891 isin 119871

119902((0 119879)

119861119904minus2+(2119902)120573

119901119903 (R119889+1

+)) Then (76) has a unique solution 119906 belong-

ing to

119871119902((0 119879) 119861

119904+(2119902)120573

119901119903(R

119889+1

+))⋂119871

infin((0 119879) 119861

119904120573

119901119903(R

119889+1

+))

(150)

and there exists a constant 119862 such that for all 119902 le 1199021 le infin

1199061198711199021 ((0119879)119861

119904+(21199021)120573

119901119903(R119889+1+

))

le 119862 [ (1 + 11987911199021)10038171003817100381710038171198921003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (1 + 1198791+1119902

1minus1119902

)10038171003817100381710038171198911003817100381710038171003817119871119902((0119879)119861

119904minus2+(2119902)120573

119901119903(R119889+1+

))]

(151)

If in addition 119903 lt infin then 119906 isin 119862([0 119879] 119861119904120573119901119903(R119889+1

+))

Proof Since 119892 119891 are tempered (76) has a unique solution 119906in S1015840

((0 119879) timesR119889+1

+) satisfying

F119882 (119906) (119905 120585) = 119890minus1199051205852

F119882 (119892) (120585)

+ int

119905

0

119890(120591minus119905)120585

2

F119882 (119891) (120591 120585) 119889120591

(152)

Hence applying Δ 119895 119895 ge 0 to (81) we see that

Δ 119895119906 (119905 sdot) = 119867120573 (119905) Δ 119895119892 + int

119905

0

119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot) 119889120591(153)

and thus by Lemma 30 we can deduce that10038171003817100381710038171003817Δ 119895119906 (119905 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

le

10038171003817100381710038171003817119867120573(119905)Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)+ int

119905

0

10038171003817100381710038171003817119867120573 (119905 minus 120591) Δ 119895119891 (120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

le 119862[119890minus1205812211989511990510038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119871119901

120573(R119889+1+

)

+int

119905

0

119890minus12058122119895(119905minus120591)10038171003817

100381710038171003817Δ 119895119891(120591 sdot)

10038171003817100381710038171003817119871119901

120573(R119889+1+

)119889120591]

(154)

Then it follows from convolution inequalities thatΔ 1198951199061198711199021 ((0119879)119871

119901

120573(R119889+1+

))is dominated by

(

1 minus 119890minus120581119879119902122119895

120581119902122119895

)

11199021

10038171003817100381710038171003817Δ 119895119892

10038171003817100381710038171003817119861119904120573

119901119903(R119889+1+

)

+ (

1 minus 119890minus120581119879119902222119895

120581119902222119895

)

11199022

10038171003817100381710038171003817Δ 119895119891

10038171003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

))

(155)

with 11199022 = 1 + 11199021 minus 1119902 Moreover similarly as above wecan obtain that

1003817100381710038171003817Δminus1119906(119905 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)le1003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)

+ int

119905

0

1003817100381710038171003817Δminus1119891 (120591 sdot)

1003817100381710038171003817119871119901

120573(R119889+1+

)119889120591

(156)

and thus if 1 le 119902 le 1199021 le infin1003817100381710038171003817Δminus1119906

10038171003817100381710038171198711199021 ((0119879)119871

119901

120573(R119889+1+

))

le 119862(119879111990211003817100381710038171003817Δminus1119892

1003817100381710038171003817119871119901

120573(R119889+1+

)+ 119879

111990221003817100381710038171003817Δminus1119891

1003817100381710038171003817119871119902((0119879)119871

119901

120573(R119889+1+

)))

(157)

Journal of Function Spaces and Applications 13

Finally taking the 119897119903-normwith respect to 119895 in (155) and (157)with the usual convention if 119903 = infin we can deduce the desiredestimate

Acknowledgment

Theauthor gratefully acknowledges theDeanship of ScientificResearch at the University of Taibah The author is deeplyindebted to the referee for providing constructive commentsand help in improving the contents of this paper

References

[1] Z Ben Nahia and N Ben Salem ldquoSpherical harmonics andapplications associated with the Weinstein operatorrdquo in Pro-ceedings of the International Conference on PotentialTheory heldin Kouty Czech Republic (ICPT rsquo94) pp 235ndash241 1996

[2] Z Ben Nahia and N Ben Salem ldquoOn a mean value propertyassociated with the Weinstein operatorrdquo in Proceedings of theInternational Conference on Potential Theory held in KoutyCzech Republic (ICPT rsquo94) pp 243ndash253 1996

[3] M Brelot ldquoEquation de Weinstein et potentiels de MarcelRieszrdquo in Seminaire de Theorie de Potentiel Paris No 3 vol 681of Lecture Notes in Mathematics pp 18ndash38 Springer BerlinGermany 1978

[4] H Mejjaoli and M Salhi ldquoUncertainty principles for theweinstein transformrdquo Czechoslovak Mathematical Journal vol61 no 4 pp 941ndash974 2011

[5] H Mejjaoli and A Ould Ahmed Salem ldquoWeinstein Gabortransform and applicationsrdquo Advanced Studies in Pure Mathe-matics vol 2 no 3 pp 203ndash210 2012

[6] H Mejjaoli ldquoBesov spaces associated withthe Weinstein opera-tor and applicationsrdquo In press

[7] T Kawazoe and H Mejjaoli ldquoGeneralized Besov spaces andtheir applicationsrdquo Tokyo Journal of Mathematics vol 35 no 2pp 297ndash320 2012

[8] H Mejjaoli ldquoLittlewood-Paley decomposition associated withthe Dunkl operators and paraproduct operatorsrdquo Journal ofInequalities in Pure and Applied Mathematics vol 9 no 4 pp1ndash25 2008

[9] H Mejjaoli and N Sraeib ldquoGeneralized sobolev spaces inquantum calculus and applicationsrdquo Journal of Inequalities andSpecial Functions vol 1 no 4 pp 43ndash64 2012

[10] H Mejjaoli ldquoGeneralized homogeneous Besov spaces and theirapplicationsrdquo Serdica Mathematical Journal vol 38 no 4 pp575ndash614 2012

[11] H Triebel Interpolation Theory Functions Spaces DifferentialOperators North-Holland AmsterdamThe Netherlands 1978

[12] V S Guliev ldquoOn maximal function and fractional integralassociated with the Bessel differential operatorrdquo MathematicalInequalities and Applications vol 6 no 2 pp 317ndash330 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 13

Finally taking the 119897119903-normwith respect to 119895 in (155) and (157)with the usual convention if 119903 = infin we can deduce the desiredestimate

Acknowledgment

Theauthor gratefully acknowledges theDeanship of ScientificResearch at the University of Taibah The author is deeplyindebted to the referee for providing constructive commentsand help in improving the contents of this paper

References

[1] Z Ben Nahia and N Ben Salem ldquoSpherical harmonics andapplications associated with the Weinstein operatorrdquo in Pro-ceedings of the International Conference on PotentialTheory heldin Kouty Czech Republic (ICPT rsquo94) pp 235ndash241 1996

[2] Z Ben Nahia and N Ben Salem ldquoOn a mean value propertyassociated with the Weinstein operatorrdquo in Proceedings of theInternational Conference on Potential Theory held in KoutyCzech Republic (ICPT rsquo94) pp 243ndash253 1996

[3] M Brelot ldquoEquation de Weinstein et potentiels de MarcelRieszrdquo in Seminaire de Theorie de Potentiel Paris No 3 vol 681of Lecture Notes in Mathematics pp 18ndash38 Springer BerlinGermany 1978

[4] H Mejjaoli and M Salhi ldquoUncertainty principles for theweinstein transformrdquo Czechoslovak Mathematical Journal vol61 no 4 pp 941ndash974 2011

[5] H Mejjaoli and A Ould Ahmed Salem ldquoWeinstein Gabortransform and applicationsrdquo Advanced Studies in Pure Mathe-matics vol 2 no 3 pp 203ndash210 2012

[6] H Mejjaoli ldquoBesov spaces associated withthe Weinstein opera-tor and applicationsrdquo In press

[7] T Kawazoe and H Mejjaoli ldquoGeneralized Besov spaces andtheir applicationsrdquo Tokyo Journal of Mathematics vol 35 no 2pp 297ndash320 2012

[8] H Mejjaoli ldquoLittlewood-Paley decomposition associated withthe Dunkl operators and paraproduct operatorsrdquo Journal ofInequalities in Pure and Applied Mathematics vol 9 no 4 pp1ndash25 2008

[9] H Mejjaoli and N Sraeib ldquoGeneralized sobolev spaces inquantum calculus and applicationsrdquo Journal of Inequalities andSpecial Functions vol 1 no 4 pp 43ndash64 2012

[10] H Mejjaoli ldquoGeneralized homogeneous Besov spaces and theirapplicationsrdquo Serdica Mathematical Journal vol 38 no 4 pp575ndash614 2012

[11] H Triebel Interpolation Theory Functions Spaces DifferentialOperators North-Holland AmsterdamThe Netherlands 1978

[12] V S Guliev ldquoOn maximal function and fractional integralassociated with the Bessel differential operatorrdquo MathematicalInequalities and Applications vol 6 no 2 pp 317ndash330 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of