Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013 Article ID 946435 5 pageshttpdxdoiorg1011552013946435

Research ArticleBoundedness of Oscillatory Integrals with VariableCalderoacuten-Zygmund Kernel on Weighted Morrey Spaces

Yali Pan1 Changwen Li1 and Xinsong Wang2

1 School of Mathematical Sciences Huaibei Normal University Huaibei Anhui 235000 China2 School of Science Tianjin Chengjian University Tianjin 300384 China

Correspondence should be addressed to Changwen Li cwli2008163com

Received 17 August 2013 Accepted 11 October 2013

Academic Editor Yoshihiro Sawano

Copyright copy 2013 Yali Pan et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Oscillatory integral operators play a key role in harmonic analysis In this paper the authors investigate the boundedness of theoscillatory singular integrals with variable Calderon-Zygmund kernel on the weighted Morrey spaces 119871

119901119896(120596) Meanwhile the

corresponding results for the oscillatory singular integrals with standard Calderon-Zygmund kernel are established

1 Introduction and Main Results

Suppose that 119896 is the standard Calderon-Zygmund kernelThat is 119896 isin 119862

infin(R119899

0) is homogeneous of degree 119899 andintΣ

119896(119909)119889120590119909 = 0 where Σ = 119909 isin R119899 |119909| = 1 The oscillatory

integral operator 119879120582 is defined by

119879120582119891 (119909) = 119901 sdot V sdot intR119899

119890119894120582Φ(119909119910)

119896 (119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910 (1)

where 120582 isin R 120593 isin 119862infin

0(R119899

times R119899) where 119862

infin

0(R119899

times R119899) is

the space of infinitely differentiable functions on R119899times R119899

with compact supports and Φ is a real-analytic function or areal-119862infin

(R119899times R119899

) function satisfying that for any (1199090 1199100) isin

supp120593 there exists (1198950 1198960) 1 le 1198950 1198960 le 119899 such that1205972Φ(1199090 1199100)120597119909119895

0

1205971199101198960

does not vanish up to infinite orderThese operators have arisen in the study of singular integralssupported on lower dimensional varieties and the singularRadon transform In [1] Pan proved that 119879120582 are uniform in120582 bounded on 119871

119901(R119899

) (1 lt 119901 lt infin) Lu et al [2] proved theweighted 119871

119901 boundedness of 119879120582 defined by (1)Let 119896(119909 119910) be a variable Calderon-Zygmund kernel That

means for ae 119909 isin R119899 119896(119909 sdot) is a standard Calderon-

Zygmund kernel and

max|119895|le2119899119895isinZ

100381710038171003817100381710038171003817100381710038171003817

120597|119895|

119896

120597119910119895

100381710038171003817100381710038171003817100381710038171003817119871infin(R119899timesΣ)

= 119860 lt infin (2)

Define the oscillatory integral operator with variable Cal-deron-Zygmund kernel 119879

lowast

120582by

119879lowast

120582119891 (119909) = 119901 sdot V sdot int

R119899119890119894120582Φ(119909119910)

119896 (119909 119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

(3)

where 120582 120593 and Φ satisfy the same assumptions as those inthe operator defined by (1)

Lu et al [2] investigated the 119871119901 and weighted 119871

119901 bound-edness about this class of oscillatory integral operators

The classical Morrey space 119871119901120582 was first introduced by

Morrey in [3] to study the local behavior of solutions tosecond order elliptic partial differential equations In 2009Komori and Shirai [4] first defined the weighted Morreyspaces 119871

119901120581(120596) which could be viewed as an extension of

weighted Lebesgue spaces They studied the boundedness ofthe fractional integral operator the Hardy-Littlewood max-imal operator and the Calderon-Zygmund singular integraloperator on the space The boundedness results about someoperators on these spaces can be see in ([5ndash17]) Recently Shiet al [18] obtained the boundedness of a class of oscillatoryintegrals with Calderon-Zygmund kernel and polynomialphase on weighted Morrey spaces Their results are stated asfollows

2 Journal of Function Spaces and Applications

Let 119875(119909 119910) be a real valued polynomial defined on R119899times

R119899 and let 119896 satisfy the following hypotheses

1003816100381610038161003816119896 (119909 119910)1003816100381610038161003816 le

119862

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899 forall119909 = 119910

1003816100381610038161003816nabla119909119896 (119909 119910)1003816100381610038161003816 +

10038161003816100381610038161003816nabla119910119896 (119909 119910)

10038161003816100381610038161003816le

119862

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899+1 forall119909 = 119910

(4)

We define

119878119891 (119909) = 119901 sdot V sdot intR119899

119896 (119909 119910) 119891 (119910) 119889119910

119877119891 (119909) = 119901 sdot V sdot intR119899

119890119894119875(119909119910)

119896 (119909 119910) 119891 (119910) 119889119910

(5)

Theorem A (see [18]) Let 1 lt 119901 lt infin 0 lt 120581 lt 1 and120596 isin 119860119901 If 119878 is of type (119871

2 119871

2) then for any real polynomial

119875(119909 119910) there exists a constant 119862 gt 0 such that1003817100381710038171003817119877119891

1003817100381710038171003817 119871119901120581(120596) le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901120581(120596) (6)

Thepurpose of this paper is to generalize the above resultsto the case with real-119862infin or analytic phase functions Ourmain results in this paper are formulated as follows

Theorem 1 Let 120582 isin R 120593 isin 119862infin

0(R119899

times R119899) and Φ a real-

119862infin

(R119899times R119899

) function satisfying that for any (1199090 1199100) isin

supp120593 there exists (1198950 1198960) 1 le 1198950 1198960 le 119899 such that1205972Φ(1199090 1199100)120597119909119895

0

1205971199101198960

does not vanish up to infinite orderAssume that 119896 is a standard Calderon-Zygmund kernel and 119879120582

is defined as in (1) Then for any 1 lt 119901 lt infin 0 lt 120581 lt 1 and120596 isin 119860119901 119879120582 is bounded on 119871

119901120581(120596)

Theorem 2 Let 120582 isin R 120593 isin 119862infin

0(R119899

times R119899) and Φ a real-

119862infin

(R119899times R119899

) function satisfying that for any (1199090 1199100) isin

supp120593 there exists (1198950 1198960) 1 le 1198950 1198960 le 119899 such that1205972Φ(1199090 1199100)120597119909119895

0

1205971199101198960

does not vanish up to infinite orderAssume that 119896 is a variable Calderon-Zygmund kernel and 119879

lowast

120582

is defined as in (3) Then for any 1 lt 119901 lt infin 0 lt 120581 lt 1 and120596 isin 119860119901 119879

lowast

120582is bounded on 119871

119901120581(120596)

2 Notations and Preliminary Lemmas

Let 119861 = 119861(1199090 119903) be the ball with the center 1199090 and radius 119903Given a ball 119861 and 120582 gt 0 120582119861 denotes the ball with the samecenter as 119861 whose radius is 120582 times that of 119861

The classical 119860119901 weighted theory was first introducedby Muckenhoupt in [19] A weight 120596 is a locally integrablefunction on R119899 which takes values in (0 infin) ae For a givenweight function 120596 we denote the Lebesgue measure of 119861 by|119861| and the weighted measure of 119864 by 120596(119864) that is 120596(119864) =

int119864

120596(119909)119889119909 Given a weight 120596 we say that 120596 satisfies thedoubling condition if there exists a constant 119863 gt 0 such thatfor any ball 119861 we have 120596(2119861) le 119863120596(119861)

We say 120596 isin 119860119901 with 1 lt 119901 lt infin if there exists a constant119862 gt 0 such that

(1

|119861|int119861

120596 (119909) 119889119909) (1

|119861|int119861

120596(119909)minus1(119901minus1)

119889119909)

119901minus1

le 119862 (7)

for every ball 119861 sube R119899 When 119901 = 1 120596 isin 1198601 if there exists119862 gt 0 such that

1

|119861|int119861

120596 (119909) 119889119909 le 119862 ess inf119909isin119861

120596 (119909) (8)

for almost every 119909 isin R119899 We define 119860infin = ⋃119901ge1 119860119901 A weightfunction 120596 is said to belong to the reverse Holder class 119877119867119903

if there exist two constants 119903 gt 0 and 119862 gt 0 such that thefollowing reverse Holder inequality holds

(1

|119861|int119861

120596(119909)119903119889119909)

1119903

le 119862 (1

|119861|int119861

120596 (119909) 119889119909) (9)

for every ball 119861 sube R119899It is well known that if 120596 isin 119860119901 with 1 le 119901 lt infin then

there exists 119903 gt 1 such that 120596 isin 119877119867119903

Lemma 3 (see [20]) Let 120596 isin 119860119901 119901 ge 1 and 119903 gt 0 Then forany ball 119861 and 120582 gt 1

120596 (2119861) le 119862120596 (119861)

120596 (120582119861) le 119862120582119899119901

120596 (119861)

(10)

where 119862 does not depend on 119861 nor on 120582

Lemma4 (see [21]) Let120596 isin 119877119867119903 with 119903 gt 1Then there existsa constant 119862 such that

120596 (119864)

120596 (119861)le 119862(

|119864|

|119861|)

(119903minus1)119903

(11)

for any measurable subset 119864 of a ball 119861

The weighted Morrey spaces were defined as follows

Definition 5 (see [4]) Let 1 le 119901 lt infin 0 lt 120581 lt 1 and 120596 aweight function Then the weighted Morrey space is definedby

119871119901120581

(120596) = 119891 isin 119871119901

loc (120596) 1003817100381710038171003817119891

1003817100381710038171003817 119871119901120581(120596) lt infin (12)

where

10038171003817100381710038171198911003817100381710038171003817119871119901120581(120596)

= sup119861

(1

120596(119861)120581 int

119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909)

1119901

(13)

and the supremum is taken over all balls 119861 in R119899 The space119871119901

loc(120596) is defined by

119871119901

loc (120596) = 119891 119891120594119870 isin 119871119901

(120596)

for every compact set 119870 sube R119899

(14)

Our argument is based heavily on the following results

Lemma 6 (see [2]) Assume that 119879120582 is defined as in (1) Thenfor any 1 lt 119901 lt infin and 120596 isin 119860119901 one has

10038171003817100381710038171198791205821198911003817100381710038171003817 119871119901(120596) le 119862 (119899 119901 Φ 120593 119862119901120596) 119861

10038171003817100381710038171198911003817100381710038171003817 119871119901(120596) (15)

where119862(119899 119901 Φ 120593 119862119901120596) is independent of 120582 119896 and119891 and 119861 =

1198961198621(Σ)

Journal of Function Spaces and Applications 3

Lemma 7 (see [2]) Assume that 119879lowast

120582is defined as in (3) Then

for any 1 lt 119901 lt infin and 120596 isin 119860119901 one has1003817100381710038171003817119879

lowast

120582119891

1003817100381710038171003817 119871119901(120596) le 119862 (119899 119901 Φ 120593 119862119901120596) 1198601003817100381710038171003817119891

1003817100381710038171003817 119871119901(120596) (16)

where 119862(119899 119901 Φ 120593 119862119901120596) is independent of 120582 119896 and 119891 119860 isdefined in (2)

Definition 8 (see [4]) The Hardy-Littlewood maximal oper-ator 119872 is defined by

119872119891 (119909) = sup119861ni119909

1

|119861|int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 119891 isin 119871 loc (R

119899) (17)

Lemma 9 (see [4]) If 1 lt 119901 lt infin 0 lt 120581 lt 1 and 120596 isin 119860119901

then the Hardy-Littlewoodmaximal operator119872 is bounded on119871119901120581

(120596)

Lemma 10 (see [22]) Denote by H119898 the spaces of sphericalharmonic functions of degree 119898 Then

(a) 1198712(Σ) = oplus

infin

119898=0H119898 and 119892119898 = dimH119898 le 119862(119899)119898

119899minus2 forany 119898 isin N

(b) for any 119898 = 0 1 2 there exists an orthogo-nal system 119884119895119898

119892119898

119895=1of H119898 such that 119884119895119898

119871infin(Σ)le

119862(119899)1198981198992minus1 119884119895119898 = (minus119898)

minus119899(119898 + 119899 minus 2)

minus119899Λ

119899119884119895119898 119895 =

1 119892119898 and Λ is the Beltrami-Laplace operator onΣ

In the following the letter 119862 will denote a constant whichmay vary at each occurrence

3 Proof of Theorems

Proof of Theorem 1 It is sufficient to prove that there exists aconstant 119862 gt 0 such that

1

120596(119861)120581 int

119861

1003816100381610038161003816119879120582119891 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909 le 119862

10038171003817100381710038171198911003817100381710038171003817

119901

119871119901120581(120596) (18)

Fix a ball 119861 = 119861(1199090 119903119861) and decompose 119891 = 1198911 + 1198912 with1198911 = 1198911205942119861 Then we have

1

120596(119861)120581 int

119861

1003816100381610038161003816119879120582119891 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909

le 119862 1

120596(119861)120581 int

119861

10038161003816100381610038161198791205821198911 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909

+1

120596(119861)120581 int

119861

10038161003816100381610038161198791205821198912 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909

= 119862 1198681 + 1198682

(19)

Using Lemmas 3 and 6 we get

1198681 le 1198621

120596(119861)120581 int

2119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909

le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596)sdot

120596(2119861)120581

120596(119861)120581

le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596)

(20)

We now estimate 1198682 We can write

10038161003816100381610038161198791205821198912 (119909)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

int(2119861)119888

119890119894120582Φ(119909119910)

119896 (119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

10038161003816100381610038161003816100381610038161003816

(21)

Now by an argument similar to the proof of Lemma 6 in[2] we choose 1206011 isin 119862

infin

0(R119899

) such that 1206011(119909) equiv 1 when |119909| le

1 and 1206011(119909) equiv 0 when |119909| gt 2 Let 1206012 = 1 minus 1206011 and 119873 isin N

which is large enough and will be determined later Write

119896 (119909) = 1198961

120582(119909) + 119896

2

120582(119909) (22)

where

119896119895

120582(119909) = 119896 (119909) 120601119895 (120582

1119873119909) 119895 = 1 2 (23)

Then

1198791205821198912 (119909) = 119901 sdot V sdot int(2119861)119888

119890119894120582Φ(119909119910)

1198961

120582(119909 minus 119910)

times 120593 (119909 119910) 119891 (119910) 119889119910

+ 119901 sdot V sdot int(2119861)119888

119890119894120582Φ(119909119910)

1198962

120582(119909 minus 119910)

times 120593 (119909 119910) 119891 (119910) 119889119910

= 1198791

1205821198912 (119909) + 119879

2

1205821198912 (119909)

(24)

Let us first estimate 1198791

1205821198912(119909) To do so using Taylorrsquos

expansion and the compactness of supp120593 we write

Φ (119909 119910) = Φ (119909 119909) + 119875 (119909 119910) + 119903119873 (119909 119910) (25)

for(119909 119910) isin supp120593 where 119875(119909 119910) is a polynomial with deg119875 lt 119873 and |119903119873(119909 119910)| le 119862|119909 minus 119910|

119873 with 119862 independent of 119909

and 119910 Define

119877119891 (119909) = 119901 sdot V sdot int(2119861)119888

119890119894120582119875(119909119910)

1198961

120582(119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

(26)

Therefore

119890minus119894120582Φ(119909119909)

1198791

1205821198912 (119909) minus 119877119891 (119909)

= int|119909minus119910|le2120582minus1119873

119890119894120582119875(119909119910)

[119890119894120582119903119873(119909119910)

minus 1]

times 1198961

120582(119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

=

infin

sum

119895=0

int2minus119895120582minus1119873lt|119909minus119910|le2minus119895+1120582minus1119873

119890119894120582119875(119909119910)

[119890119894120582119903119873(119909119910)

minus 1]

times 1198961

120582(119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

equiv

infin

sum

119895=0

1198791

1205821198951198912 (119909)

(27)

4 Journal of Function Spaces and Applications

On 1198791

1205821198951198912(119909) by the properties of 119903119873 and 119896 we have

10038161003816100381610038161003816119879

1

1205821198951198912 (119909)

10038161003816100381610038161003816le 1198622

minus119895119873119872119891 (119909) (28)

So we have10038161003816100381610038161003816119879

1

1205821198912 (119909)

10038161003816100381610038161003816le 119862

infin

sum

119895=0

2minus119895119873 1003816100381610038161003816119872119891 (119909)

1003816100381610038161003816 + 1198621003816100381610038161003816119877119891 (119909)

1003816100381610038161003816 (29)

ByTheorem A and Lemma 9 we have10038171003817100381710038171003817119879

1

1205821198912

10038171003817100381710038171003817119871119901120581(120596)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(120596)

(30)

Now let us turn to estimate 1198792

1205821198912(119909) We consider the

following two cases

Case 1 (120582 le 1) Similar to that estimate of 1198792

120582in Lemma 6 in

[2] we have10038161003816100381610038161003816119879

2

1205821198912 (119909)

10038161003816100381610038161003816le 119862119872 (119891) (119909) (31)

By Lemma 9 we have10038171003817100381710038171003817119879

2

1205821198912

10038171003817100381710038171003817119871119901120581(120596)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(120596)

(32)

Case 2 (120582 gt 1) We choose 1205930 isin 119862infin

0(R119899

) such that

supp1205930 sube 119909 isin R119899

1 lt |119909| le 2

1206012 (119909) =

infin

sum

119895=0

1205930 (2minus119895

119909)

(33)

Let

1198962

120582119895(119909) = 119896 (119909) 1205930 (2

minus1198951205821119873

119909) (34)

Then

1198792

1205821198912 (119909) = int

(2119861)119888

119890119894120582Φ(119909119910)

1198962

120582(119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

infin

sum

119895=0

int(2119861)119888

119890119894120582Φ(119909119910)

1198962

120582119895(119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

equiv

infin

sum

119895=0

1198792

1205821198951198912 (119909)

(35)

For 1198792

120582119895 by its definition we can get

10038161003816100381610038161003816119879

2

1205821198951198912 (119909)

10038161003816100381610038161003816le 119862 int

2119895120582minus1119873lt|119909minus119910|le2119895+1120582minus1119873

times 1198881

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 le 119862119872 (119891) (119909)

(36)

The inequality (36) also can be seen in [2] we omit the detailshere

By Lemma 9 we have10038171003817100381710038171003817119879

2

1205821198912

10038171003817100381710038171003817119871119901120581(120596)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(120596)

(37)

Therefore

1198682 le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596) (38)

This finishes the proof of Theorem 1

Proof of Theorem 2 It is sufficient to prove that there exists aconstant 119862 gt 0 such that

1

120596(119861)120581 int

119861

1003816100381610038161003816119879lowast

120582119891 (119909)

1003816100381610038161003816

119901120596 (119909) 119889119909 le 119862

10038171003817100381710038171198911003817100381710038171003817

119901

119871119901120581(120596) (39)

Fix a ball 119861 = 119861(1199090 119903119861) and decompose 119891 = 1198911 + 1198912 with1198911 = 1198911205942119861 Then we have

1

120596(119861)120581 int

119861

1003816100381610038161003816119879lowast

120582119891 (119909)

1003816100381610038161003816

119901120596 (119909) 119889119909

le 119862 1

120596(119861)120581 int

119861

1003816100381610038161003816119879lowast

1205821198911 (119909)

1003816100381610038161003816

119901120596 (119909) 119889119909

+1

120596(119861)120581 int

119861

1003816100381610038161003816119879lowast

1205821198912 (119909)

1003816100381610038161003816

119901120596 (119909) 119889119909

= 119862 1198691 + 1198692

(40)

Using Lemmas 3 and 7 we get

1198691 le 1198621

120596(119861)120581 int

2119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909

le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596)sdot

120596(2119861)120581

120596(119861)120581

le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596)

(41)

We now estimate 1198692For each 119898 isin N and 119895 = 1 119892119898 we get

119886119895119898 (119909) = intΣ

Ω (119909 119911) 119884119895119898 (119911) 119889120590119911 (42)

where Ω(119909 119911) = |119911|119899119896(119909 119911) Then for ae 119909 isin R119899

Ω (119909 119911) =

infin

sum

119898=1

119892119898

sum

119895=1

119886119895119898 (119909) 119884119895119898 (1199111015840) (43)

where 1199111015840

= 119911|119911| for any 119911 isin R119899 0 By Lemma 10 we have

that for any 119909 isin R119899

10038161003816100381610038161003816119886119895119898 (119909)

10038161003816100381610038161003816= 119898

minus119899(119898 + 119899 minus 2)

minus1198991003816100381610038161003816100381610038161003816intΣ

Ω (119909 119911) Λ119899119884119895119898 (119911) 119889120590119911

1003816100381610038161003816100381610038161003816

= 119898minus119899

(119898 + 119899 minus 2)minus119899

1003816100381610038161003816100381610038161003816intΣ

Λ119899Ω (119909 119911) 119884119895119898 (119911) 119889120590119911

1003816100381610038161003816100381610038161003816

le 119862 (119899) 119860119898minus2119899

(44)

By Lemma 10 again we can verify that for any 120598 gt 0 119873 isin

N and ae 119909 isin R119899 if |119910 minus 119909| ge 120598 then

10038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119898=1

119892119898

sum

119895=1

119890119894120582Φ(119909119910)

119886119895119898 (119909) 119884119895119898 ((119909 minus 119910)1015840)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899 120593 (119909 119910) 1198912 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862 (120598) 11986010038161003816100381610038161198912 (119910)

1003816100381610038161003816

(45)

Journal of Function Spaces and Applications 5

Therefore from (43) (45) and the Lebesgue dominatedconvergence theorem it follows that

119879lowast

1205821198912 (119909)

= lim120598rarr0

int|119909minus119910|ge120598

119890119894120582Φ(119909119910)

119896 (119909 119909 minus 119910) 120593 (119909 119910) 1198912 (119910) 119889119910

= lim120598rarr0

infin

sum

119898=1

119892119898

sum

119895=1

int|119909minus119910|ge120598

119890119894120582Φ(119909119910)

119886119895119898 (119909) 119884119895119898 ((119909 minus 119910)1015840)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899

times 120593 (119909 119910) 1198912 (119910) 119889119910

= lim120598rarr0

infin

sum

119898=1

119892119898

sum

119895=1

119886119895119898 (119909) int|119909minus119910|ge120598

119890119894120582Φ(119909119910)

119884119895119898 ((119909 minus 119910)1015840)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899

times 120593 (119909 119910) 1198912 (119910) 119889119910

(46)

We write

1198771198951198981198912 (119909)

= int|119909minus119910|ge120598

119890119894120582Φ(119909119910)

119884119895119898 ((119909 minus 119910)1015840)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899 120593 (119909 119910) 1198912 (119910) 119889119910

(47)

It is easy to see that 1198771198951198981198912(119909) is the oscillatory integraloperator defined by (1) By Theorem 1 we have that 119877119895119898 isbounded on weighted Morrey spaces Therefore by (44) andthe above discussion we have

1198692 le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596) (48)

This finishes the proof of Theorem 2

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant no 11001001) Natural Sci-ence Foundation from the Education Department of AnhuiProvince (nos KJ2012B166 KJ2013A235)

References

[1] Y Pan ldquoUniform estimates for oscillatory integral operatorsrdquoJournal of Functional Analysis vol 100 no 1 pp 207ndash220 1991

[2] S Lu D Yang and Z Zhou ldquoOn local oscillatory integralswith variable Calderon-Zygmund kernelsrdquo Integral Equationsand Operator Theory vol 33 no 4 pp 456ndash470 1999

[3] C B Morrey Jr ldquoOn the solutions of quasi-linear ellipticpartial differential equationsrdquo Transactions of the AmericanMathematical Society vol 43 no 1 pp 126ndash166 1938

[4] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009

[5] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987

[6] J Peetre ldquoOn the theory of 119871119901120582 spacesrdquo Journal of Functional

Analysis vol 4 no 1 pp 71ndash87 1969[7] S Z Lu Y Ding and D Y Yan Singular Integrals and Related

Topics World Scientific Publishing River Edge NJ USA 2007[8] E Nakai ldquoHardy-Littlewood maximal operator singular inte-

gral operators and the Riesz potentials on generalized MorreyspacesrdquoMathematische Nachrichten vol 166 pp 95ndash103 1994

[9] Y Sawano and H Tanaka ldquoMorrey spaces for non-doublingmeasuresrdquo Acta Mathematica Sinica vol 21 no 6 pp 1535ndash1544 2005

[10] H Wang and H P Liu ldquoSome estimates for Bochner-Rieszoperators on the weighted Morrey spacesrdquo Acta MathematicaSinica vol 55 no 3 pp 551ndash560 2012

[11] H Wang and H P Liu ldquoWeak type estimates of intrinsicsquare functions on the weighted Hardy spacesrdquo Archiv derMathematik vol 97 no 1 pp 49ndash59 2011

[12] R Ch Mustafayev ldquoOn boundedness of sublinear operators inweighted Morrey spacesrdquo Azerbaijan Journal of Mathematicsvol 2 no 1 pp 66ndash79 2012

[13] X F Ye and X S Zhu ldquoEstimates of singular integrals andmultilinear commutators in weighted Morrey spacesrdquo Journalof Inequalities and Applications vol 2012 article 302 2012

[14] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica Chinese Series vol 55 no 4 pp 589ndash600 2012 (Chinese)

[15] H Wang ldquoThe boundedness of fractional integral operatorswith rough kernels on the weighted Morrey spacesrdquo ActaMathematica Sinica Chinese Series vol 56 no 2 pp 175ndash1862013 (Chinese)

[16] S He ldquoThe boundedness of some multilinear operator withrough kernel on the weighted Morrey spacesrdquo submittedhttparxivorgabs11115463

[17] H Wang ldquoIntrinsic square functions on the weighted Morreyspacesrdquo Journal of Mathematical Analysis and Applications vol396 no 1 pp 302ndash314 2012

[18] S G Shi Z W Fu and S Z Lu ldquoBoundedness of oscillatoryintegral operators and their commutators on weighted Morreyspacesrdquo Scientia Sinica Mathematica vol 43 pp 147ndash158 2013(Chinese)

[19] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972

[20] J Garcıa-Cuerva and J L Rubio de Francia Weighted NormInequalities and Related Topics vol 116 of North-Holland Math-ematics Studies North-Holland Amsterdam The Netherlands1985

[21] R F Gundy and R LWheeden ldquoWeighted integral inequalitiesfor the nontangential maximal function Lusin area integraland Walsh-Paley seriesrdquo Studia Mathematica vol 49 pp 107ndash124 1974

[22] E M Stein and G Weiss Introduction to Fourier Analysis onEuclidean Spaces Princeton University Press Princeton NJUSA 1971

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

2 Journal of Function Spaces and Applications

Let 119875(119909 119910) be a real valued polynomial defined on R119899times

R119899 and let 119896 satisfy the following hypotheses

1003816100381610038161003816119896 (119909 119910)1003816100381610038161003816 le

119862

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899 forall119909 = 119910

1003816100381610038161003816nabla119909119896 (119909 119910)1003816100381610038161003816 +

10038161003816100381610038161003816nabla119910119896 (119909 119910)

10038161003816100381610038161003816le

119862

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899+1 forall119909 = 119910

(4)

We define

119878119891 (119909) = 119901 sdot V sdot intR119899

119896 (119909 119910) 119891 (119910) 119889119910

119877119891 (119909) = 119901 sdot V sdot intR119899

119890119894119875(119909119910)

119896 (119909 119910) 119891 (119910) 119889119910

(5)

Theorem A (see [18]) Let 1 lt 119901 lt infin 0 lt 120581 lt 1 and120596 isin 119860119901 If 119878 is of type (119871

2 119871

2) then for any real polynomial

119875(119909 119910) there exists a constant 119862 gt 0 such that1003817100381710038171003817119877119891

1003817100381710038171003817 119871119901120581(120596) le 1198621003817100381710038171003817119891

1003817100381710038171003817119871119901120581(120596) (6)

Thepurpose of this paper is to generalize the above resultsto the case with real-119862infin or analytic phase functions Ourmain results in this paper are formulated as follows

Theorem 1 Let 120582 isin R 120593 isin 119862infin

0(R119899

times R119899) and Φ a real-

119862infin

(R119899times R119899

) function satisfying that for any (1199090 1199100) isin

supp120593 there exists (1198950 1198960) 1 le 1198950 1198960 le 119899 such that1205972Φ(1199090 1199100)120597119909119895

0

1205971199101198960

does not vanish up to infinite orderAssume that 119896 is a standard Calderon-Zygmund kernel and 119879120582

is defined as in (1) Then for any 1 lt 119901 lt infin 0 lt 120581 lt 1 and120596 isin 119860119901 119879120582 is bounded on 119871

119901120581(120596)

Theorem 2 Let 120582 isin R 120593 isin 119862infin

0(R119899

times R119899) and Φ a real-

119862infin

(R119899times R119899

) function satisfying that for any (1199090 1199100) isin

supp120593 there exists (1198950 1198960) 1 le 1198950 1198960 le 119899 such that1205972Φ(1199090 1199100)120597119909119895

0

1205971199101198960

does not vanish up to infinite orderAssume that 119896 is a variable Calderon-Zygmund kernel and 119879

lowast

120582

is defined as in (3) Then for any 1 lt 119901 lt infin 0 lt 120581 lt 1 and120596 isin 119860119901 119879

lowast

120582is bounded on 119871

119901120581(120596)

2 Notations and Preliminary Lemmas

Let 119861 = 119861(1199090 119903) be the ball with the center 1199090 and radius 119903Given a ball 119861 and 120582 gt 0 120582119861 denotes the ball with the samecenter as 119861 whose radius is 120582 times that of 119861

The classical 119860119901 weighted theory was first introducedby Muckenhoupt in [19] A weight 120596 is a locally integrablefunction on R119899 which takes values in (0 infin) ae For a givenweight function 120596 we denote the Lebesgue measure of 119861 by|119861| and the weighted measure of 119864 by 120596(119864) that is 120596(119864) =

int119864

120596(119909)119889119909 Given a weight 120596 we say that 120596 satisfies thedoubling condition if there exists a constant 119863 gt 0 such thatfor any ball 119861 we have 120596(2119861) le 119863120596(119861)

We say 120596 isin 119860119901 with 1 lt 119901 lt infin if there exists a constant119862 gt 0 such that

(1

|119861|int119861

120596 (119909) 119889119909) (1

|119861|int119861

120596(119909)minus1(119901minus1)

119889119909)

119901minus1

le 119862 (7)

for every ball 119861 sube R119899 When 119901 = 1 120596 isin 1198601 if there exists119862 gt 0 such that

1

|119861|int119861

120596 (119909) 119889119909 le 119862 ess inf119909isin119861

120596 (119909) (8)

for almost every 119909 isin R119899 We define 119860infin = ⋃119901ge1 119860119901 A weightfunction 120596 is said to belong to the reverse Holder class 119877119867119903

if there exist two constants 119903 gt 0 and 119862 gt 0 such that thefollowing reverse Holder inequality holds

(1

|119861|int119861

120596(119909)119903119889119909)

1119903

le 119862 (1

|119861|int119861

120596 (119909) 119889119909) (9)

for every ball 119861 sube R119899It is well known that if 120596 isin 119860119901 with 1 le 119901 lt infin then

there exists 119903 gt 1 such that 120596 isin 119877119867119903

Lemma 3 (see [20]) Let 120596 isin 119860119901 119901 ge 1 and 119903 gt 0 Then forany ball 119861 and 120582 gt 1

120596 (2119861) le 119862120596 (119861)

120596 (120582119861) le 119862120582119899119901

120596 (119861)

(10)

where 119862 does not depend on 119861 nor on 120582

Lemma4 (see [21]) Let120596 isin 119877119867119903 with 119903 gt 1Then there existsa constant 119862 such that

120596 (119864)

120596 (119861)le 119862(

|119864|

|119861|)

(119903minus1)119903

(11)

for any measurable subset 119864 of a ball 119861

The weighted Morrey spaces were defined as follows

Definition 5 (see [4]) Let 1 le 119901 lt infin 0 lt 120581 lt 1 and 120596 aweight function Then the weighted Morrey space is definedby

119871119901120581

(120596) = 119891 isin 119871119901

loc (120596) 1003817100381710038171003817119891

1003817100381710038171003817 119871119901120581(120596) lt infin (12)

where

10038171003817100381710038171198911003817100381710038171003817119871119901120581(120596)

= sup119861

(1

120596(119861)120581 int

119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909)

1119901

(13)

and the supremum is taken over all balls 119861 in R119899 The space119871119901

loc(120596) is defined by

119871119901

loc (120596) = 119891 119891120594119870 isin 119871119901

(120596)

for every compact set 119870 sube R119899

(14)

Our argument is based heavily on the following results

Lemma 6 (see [2]) Assume that 119879120582 is defined as in (1) Thenfor any 1 lt 119901 lt infin and 120596 isin 119860119901 one has

10038171003817100381710038171198791205821198911003817100381710038171003817 119871119901(120596) le 119862 (119899 119901 Φ 120593 119862119901120596) 119861

10038171003817100381710038171198911003817100381710038171003817 119871119901(120596) (15)

where119862(119899 119901 Φ 120593 119862119901120596) is independent of 120582 119896 and119891 and 119861 =

1198961198621(Σ)

Journal of Function Spaces and Applications 3

Lemma 7 (see [2]) Assume that 119879lowast

120582is defined as in (3) Then

for any 1 lt 119901 lt infin and 120596 isin 119860119901 one has1003817100381710038171003817119879

lowast

120582119891

1003817100381710038171003817 119871119901(120596) le 119862 (119899 119901 Φ 120593 119862119901120596) 1198601003817100381710038171003817119891

1003817100381710038171003817 119871119901(120596) (16)

where 119862(119899 119901 Φ 120593 119862119901120596) is independent of 120582 119896 and 119891 119860 isdefined in (2)

Definition 8 (see [4]) The Hardy-Littlewood maximal oper-ator 119872 is defined by

119872119891 (119909) = sup119861ni119909

1

|119861|int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 119891 isin 119871 loc (R

119899) (17)

Lemma 9 (see [4]) If 1 lt 119901 lt infin 0 lt 120581 lt 1 and 120596 isin 119860119901

then the Hardy-Littlewoodmaximal operator119872 is bounded on119871119901120581

(120596)

Lemma 10 (see [22]) Denote by H119898 the spaces of sphericalharmonic functions of degree 119898 Then

(a) 1198712(Σ) = oplus

infin

119898=0H119898 and 119892119898 = dimH119898 le 119862(119899)119898

119899minus2 forany 119898 isin N

(b) for any 119898 = 0 1 2 there exists an orthogo-nal system 119884119895119898

119892119898

119895=1of H119898 such that 119884119895119898

119871infin(Σ)le

119862(119899)1198981198992minus1 119884119895119898 = (minus119898)

minus119899(119898 + 119899 minus 2)

minus119899Λ

119899119884119895119898 119895 =

1 119892119898 and Λ is the Beltrami-Laplace operator onΣ

In the following the letter 119862 will denote a constant whichmay vary at each occurrence

3 Proof of Theorems

Proof of Theorem 1 It is sufficient to prove that there exists aconstant 119862 gt 0 such that

1

120596(119861)120581 int

119861

1003816100381610038161003816119879120582119891 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909 le 119862

10038171003817100381710038171198911003817100381710038171003817

119901

119871119901120581(120596) (18)

Fix a ball 119861 = 119861(1199090 119903119861) and decompose 119891 = 1198911 + 1198912 with1198911 = 1198911205942119861 Then we have

1

120596(119861)120581 int

119861

1003816100381610038161003816119879120582119891 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909

le 119862 1

120596(119861)120581 int

119861

10038161003816100381610038161198791205821198911 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909

+1

120596(119861)120581 int

119861

10038161003816100381610038161198791205821198912 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909

= 119862 1198681 + 1198682

(19)

Using Lemmas 3 and 6 we get

1198681 le 1198621

120596(119861)120581 int

2119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909

le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596)sdot

120596(2119861)120581

120596(119861)120581

le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596)

(20)

We now estimate 1198682 We can write

10038161003816100381610038161198791205821198912 (119909)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

int(2119861)119888

119890119894120582Φ(119909119910)

119896 (119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

10038161003816100381610038161003816100381610038161003816

(21)

Now by an argument similar to the proof of Lemma 6 in[2] we choose 1206011 isin 119862

infin

0(R119899

) such that 1206011(119909) equiv 1 when |119909| le

1 and 1206011(119909) equiv 0 when |119909| gt 2 Let 1206012 = 1 minus 1206011 and 119873 isin N

which is large enough and will be determined later Write

119896 (119909) = 1198961

120582(119909) + 119896

2

120582(119909) (22)

where

119896119895

120582(119909) = 119896 (119909) 120601119895 (120582

1119873119909) 119895 = 1 2 (23)

Then

1198791205821198912 (119909) = 119901 sdot V sdot int(2119861)119888

119890119894120582Φ(119909119910)

1198961

120582(119909 minus 119910)

times 120593 (119909 119910) 119891 (119910) 119889119910

+ 119901 sdot V sdot int(2119861)119888

119890119894120582Φ(119909119910)

1198962

120582(119909 minus 119910)

times 120593 (119909 119910) 119891 (119910) 119889119910

= 1198791

1205821198912 (119909) + 119879

2

1205821198912 (119909)

(24)

Let us first estimate 1198791

1205821198912(119909) To do so using Taylorrsquos

expansion and the compactness of supp120593 we write

Φ (119909 119910) = Φ (119909 119909) + 119875 (119909 119910) + 119903119873 (119909 119910) (25)

for(119909 119910) isin supp120593 where 119875(119909 119910) is a polynomial with deg119875 lt 119873 and |119903119873(119909 119910)| le 119862|119909 minus 119910|

119873 with 119862 independent of 119909

and 119910 Define

119877119891 (119909) = 119901 sdot V sdot int(2119861)119888

119890119894120582119875(119909119910)

1198961

120582(119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

(26)

Therefore

119890minus119894120582Φ(119909119909)

1198791

1205821198912 (119909) minus 119877119891 (119909)

= int|119909minus119910|le2120582minus1119873

119890119894120582119875(119909119910)

[119890119894120582119903119873(119909119910)

minus 1]

times 1198961

120582(119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

=

infin

sum

119895=0

int2minus119895120582minus1119873lt|119909minus119910|le2minus119895+1120582minus1119873

119890119894120582119875(119909119910)

[119890119894120582119903119873(119909119910)

minus 1]

times 1198961

120582(119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

equiv

infin

sum

119895=0

1198791

1205821198951198912 (119909)

(27)

4 Journal of Function Spaces and Applications

On 1198791

1205821198951198912(119909) by the properties of 119903119873 and 119896 we have

10038161003816100381610038161003816119879

1

1205821198951198912 (119909)

10038161003816100381610038161003816le 1198622

minus119895119873119872119891 (119909) (28)

So we have10038161003816100381610038161003816119879

1

1205821198912 (119909)

10038161003816100381610038161003816le 119862

infin

sum

119895=0

2minus119895119873 1003816100381610038161003816119872119891 (119909)

1003816100381610038161003816 + 1198621003816100381610038161003816119877119891 (119909)

1003816100381610038161003816 (29)

ByTheorem A and Lemma 9 we have10038171003817100381710038171003817119879

1

1205821198912

10038171003817100381710038171003817119871119901120581(120596)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(120596)

(30)

Now let us turn to estimate 1198792

1205821198912(119909) We consider the

following two cases

Case 1 (120582 le 1) Similar to that estimate of 1198792

120582in Lemma 6 in

[2] we have10038161003816100381610038161003816119879

2

1205821198912 (119909)

10038161003816100381610038161003816le 119862119872 (119891) (119909) (31)

By Lemma 9 we have10038171003817100381710038171003817119879

2

1205821198912

10038171003817100381710038171003817119871119901120581(120596)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(120596)

(32)

Case 2 (120582 gt 1) We choose 1205930 isin 119862infin

0(R119899

) such that

supp1205930 sube 119909 isin R119899

1 lt |119909| le 2

1206012 (119909) =

infin

sum

119895=0

1205930 (2minus119895

119909)

(33)

Let

1198962

120582119895(119909) = 119896 (119909) 1205930 (2

minus1198951205821119873

119909) (34)

Then

1198792

1205821198912 (119909) = int

(2119861)119888

119890119894120582Φ(119909119910)

1198962

120582(119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

infin

sum

119895=0

int(2119861)119888

119890119894120582Φ(119909119910)

1198962

120582119895(119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

equiv

infin

sum

119895=0

1198792

1205821198951198912 (119909)

(35)

For 1198792

120582119895 by its definition we can get

10038161003816100381610038161003816119879

2

1205821198951198912 (119909)

10038161003816100381610038161003816le 119862 int

2119895120582minus1119873lt|119909minus119910|le2119895+1120582minus1119873

times 1198881

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 le 119862119872 (119891) (119909)

(36)

The inequality (36) also can be seen in [2] we omit the detailshere

By Lemma 9 we have10038171003817100381710038171003817119879

2

1205821198912

10038171003817100381710038171003817119871119901120581(120596)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(120596)

(37)

Therefore

1198682 le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596) (38)

This finishes the proof of Theorem 1

Proof of Theorem 2 It is sufficient to prove that there exists aconstant 119862 gt 0 such that

1

120596(119861)120581 int

119861

1003816100381610038161003816119879lowast

120582119891 (119909)

1003816100381610038161003816

119901120596 (119909) 119889119909 le 119862

10038171003817100381710038171198911003817100381710038171003817

119901

119871119901120581(120596) (39)

Fix a ball 119861 = 119861(1199090 119903119861) and decompose 119891 = 1198911 + 1198912 with1198911 = 1198911205942119861 Then we have

1

120596(119861)120581 int

119861

1003816100381610038161003816119879lowast

120582119891 (119909)

1003816100381610038161003816

119901120596 (119909) 119889119909

le 119862 1

120596(119861)120581 int

119861

1003816100381610038161003816119879lowast

1205821198911 (119909)

1003816100381610038161003816

119901120596 (119909) 119889119909

+1

120596(119861)120581 int

119861

1003816100381610038161003816119879lowast

1205821198912 (119909)

1003816100381610038161003816

119901120596 (119909) 119889119909

= 119862 1198691 + 1198692

(40)

Using Lemmas 3 and 7 we get

1198691 le 1198621

120596(119861)120581 int

2119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909

le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596)sdot

120596(2119861)120581

120596(119861)120581

le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596)

(41)

We now estimate 1198692For each 119898 isin N and 119895 = 1 119892119898 we get

119886119895119898 (119909) = intΣ

Ω (119909 119911) 119884119895119898 (119911) 119889120590119911 (42)

where Ω(119909 119911) = |119911|119899119896(119909 119911) Then for ae 119909 isin R119899

Ω (119909 119911) =

infin

sum

119898=1

119892119898

sum

119895=1

119886119895119898 (119909) 119884119895119898 (1199111015840) (43)

where 1199111015840

= 119911|119911| for any 119911 isin R119899 0 By Lemma 10 we have

that for any 119909 isin R119899

10038161003816100381610038161003816119886119895119898 (119909)

10038161003816100381610038161003816= 119898

minus119899(119898 + 119899 minus 2)

minus1198991003816100381610038161003816100381610038161003816intΣ

Ω (119909 119911) Λ119899119884119895119898 (119911) 119889120590119911

1003816100381610038161003816100381610038161003816

= 119898minus119899

(119898 + 119899 minus 2)minus119899

1003816100381610038161003816100381610038161003816intΣ

Λ119899Ω (119909 119911) 119884119895119898 (119911) 119889120590119911

1003816100381610038161003816100381610038161003816

le 119862 (119899) 119860119898minus2119899

(44)

By Lemma 10 again we can verify that for any 120598 gt 0 119873 isin

N and ae 119909 isin R119899 if |119910 minus 119909| ge 120598 then

10038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119898=1

119892119898

sum

119895=1

119890119894120582Φ(119909119910)

119886119895119898 (119909) 119884119895119898 ((119909 minus 119910)1015840)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899 120593 (119909 119910) 1198912 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862 (120598) 11986010038161003816100381610038161198912 (119910)

1003816100381610038161003816

(45)

Journal of Function Spaces and Applications 5

Therefore from (43) (45) and the Lebesgue dominatedconvergence theorem it follows that

119879lowast

1205821198912 (119909)

= lim120598rarr0

int|119909minus119910|ge120598

119890119894120582Φ(119909119910)

119896 (119909 119909 minus 119910) 120593 (119909 119910) 1198912 (119910) 119889119910

= lim120598rarr0

infin

sum

119898=1

119892119898

sum

119895=1

int|119909minus119910|ge120598

119890119894120582Φ(119909119910)

119886119895119898 (119909) 119884119895119898 ((119909 minus 119910)1015840)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899

times 120593 (119909 119910) 1198912 (119910) 119889119910

= lim120598rarr0

infin

sum

119898=1

119892119898

sum

119895=1

119886119895119898 (119909) int|119909minus119910|ge120598

119890119894120582Φ(119909119910)

119884119895119898 ((119909 minus 119910)1015840)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899

times 120593 (119909 119910) 1198912 (119910) 119889119910

(46)

We write

1198771198951198981198912 (119909)

= int|119909minus119910|ge120598

119890119894120582Φ(119909119910)

119884119895119898 ((119909 minus 119910)1015840)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899 120593 (119909 119910) 1198912 (119910) 119889119910

(47)

It is easy to see that 1198771198951198981198912(119909) is the oscillatory integraloperator defined by (1) By Theorem 1 we have that 119877119895119898 isbounded on weighted Morrey spaces Therefore by (44) andthe above discussion we have

1198692 le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596) (48)

This finishes the proof of Theorem 2

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant no 11001001) Natural Sci-ence Foundation from the Education Department of AnhuiProvince (nos KJ2012B166 KJ2013A235)

References

[1] Y Pan ldquoUniform estimates for oscillatory integral operatorsrdquoJournal of Functional Analysis vol 100 no 1 pp 207ndash220 1991

[2] S Lu D Yang and Z Zhou ldquoOn local oscillatory integralswith variable Calderon-Zygmund kernelsrdquo Integral Equationsand Operator Theory vol 33 no 4 pp 456ndash470 1999

[3] C B Morrey Jr ldquoOn the solutions of quasi-linear ellipticpartial differential equationsrdquo Transactions of the AmericanMathematical Society vol 43 no 1 pp 126ndash166 1938

[4] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009

[5] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987

[6] J Peetre ldquoOn the theory of 119871119901120582 spacesrdquo Journal of Functional

Analysis vol 4 no 1 pp 71ndash87 1969[7] S Z Lu Y Ding and D Y Yan Singular Integrals and Related

Topics World Scientific Publishing River Edge NJ USA 2007[8] E Nakai ldquoHardy-Littlewood maximal operator singular inte-

gral operators and the Riesz potentials on generalized MorreyspacesrdquoMathematische Nachrichten vol 166 pp 95ndash103 1994

[9] Y Sawano and H Tanaka ldquoMorrey spaces for non-doublingmeasuresrdquo Acta Mathematica Sinica vol 21 no 6 pp 1535ndash1544 2005

[10] H Wang and H P Liu ldquoSome estimates for Bochner-Rieszoperators on the weighted Morrey spacesrdquo Acta MathematicaSinica vol 55 no 3 pp 551ndash560 2012

[11] H Wang and H P Liu ldquoWeak type estimates of intrinsicsquare functions on the weighted Hardy spacesrdquo Archiv derMathematik vol 97 no 1 pp 49ndash59 2011

[12] R Ch Mustafayev ldquoOn boundedness of sublinear operators inweighted Morrey spacesrdquo Azerbaijan Journal of Mathematicsvol 2 no 1 pp 66ndash79 2012

[13] X F Ye and X S Zhu ldquoEstimates of singular integrals andmultilinear commutators in weighted Morrey spacesrdquo Journalof Inequalities and Applications vol 2012 article 302 2012

[14] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica Chinese Series vol 55 no 4 pp 589ndash600 2012 (Chinese)

[15] H Wang ldquoThe boundedness of fractional integral operatorswith rough kernels on the weighted Morrey spacesrdquo ActaMathematica Sinica Chinese Series vol 56 no 2 pp 175ndash1862013 (Chinese)

[16] S He ldquoThe boundedness of some multilinear operator withrough kernel on the weighted Morrey spacesrdquo submittedhttparxivorgabs11115463

[17] H Wang ldquoIntrinsic square functions on the weighted Morreyspacesrdquo Journal of Mathematical Analysis and Applications vol396 no 1 pp 302ndash314 2012

[18] S G Shi Z W Fu and S Z Lu ldquoBoundedness of oscillatoryintegral operators and their commutators on weighted Morreyspacesrdquo Scientia Sinica Mathematica vol 43 pp 147ndash158 2013(Chinese)

[19] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972

[20] J Garcıa-Cuerva and J L Rubio de Francia Weighted NormInequalities and Related Topics vol 116 of North-Holland Math-ematics Studies North-Holland Amsterdam The Netherlands1985

[21] R F Gundy and R LWheeden ldquoWeighted integral inequalitiesfor the nontangential maximal function Lusin area integraland Walsh-Paley seriesrdquo Studia Mathematica vol 49 pp 107ndash124 1974

[22] E M Stein and G Weiss Introduction to Fourier Analysis onEuclidean Spaces Princeton University Press Princeton NJUSA 1971

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 3

Lemma 7 (see [2]) Assume that 119879lowast

120582is defined as in (3) Then

for any 1 lt 119901 lt infin and 120596 isin 119860119901 one has1003817100381710038171003817119879

lowast

120582119891

1003817100381710038171003817 119871119901(120596) le 119862 (119899 119901 Φ 120593 119862119901120596) 1198601003817100381710038171003817119891

1003817100381710038171003817 119871119901(120596) (16)

where 119862(119899 119901 Φ 120593 119862119901120596) is independent of 120582 119896 and 119891 119860 isdefined in (2)

Definition 8 (see [4]) The Hardy-Littlewood maximal oper-ator 119872 is defined by

119872119891 (119909) = sup119861ni119909

1

|119861|int119861

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 119891 isin 119871 loc (R

119899) (17)

Lemma 9 (see [4]) If 1 lt 119901 lt infin 0 lt 120581 lt 1 and 120596 isin 119860119901

then the Hardy-Littlewoodmaximal operator119872 is bounded on119871119901120581

(120596)

Lemma 10 (see [22]) Denote by H119898 the spaces of sphericalharmonic functions of degree 119898 Then

(a) 1198712(Σ) = oplus

infin

119898=0H119898 and 119892119898 = dimH119898 le 119862(119899)119898

119899minus2 forany 119898 isin N

(b) for any 119898 = 0 1 2 there exists an orthogo-nal system 119884119895119898

119892119898

119895=1of H119898 such that 119884119895119898

119871infin(Σ)le

119862(119899)1198981198992minus1 119884119895119898 = (minus119898)

minus119899(119898 + 119899 minus 2)

minus119899Λ

119899119884119895119898 119895 =

1 119892119898 and Λ is the Beltrami-Laplace operator onΣ

In the following the letter 119862 will denote a constant whichmay vary at each occurrence

3 Proof of Theorems

Proof of Theorem 1 It is sufficient to prove that there exists aconstant 119862 gt 0 such that

1

120596(119861)120581 int

119861

1003816100381610038161003816119879120582119891 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909 le 119862

10038171003817100381710038171198911003817100381710038171003817

119901

119871119901120581(120596) (18)

Fix a ball 119861 = 119861(1199090 119903119861) and decompose 119891 = 1198911 + 1198912 with1198911 = 1198911205942119861 Then we have

1

120596(119861)120581 int

119861

1003816100381610038161003816119879120582119891 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909

le 119862 1

120596(119861)120581 int

119861

10038161003816100381610038161198791205821198911 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909

+1

120596(119861)120581 int

119861

10038161003816100381610038161198791205821198912 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909

= 119862 1198681 + 1198682

(19)

Using Lemmas 3 and 6 we get

1198681 le 1198621

120596(119861)120581 int

2119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909

le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596)sdot

120596(2119861)120581

120596(119861)120581

le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596)

(20)

We now estimate 1198682 We can write

10038161003816100381610038161198791205821198912 (119909)1003816100381610038161003816 =

10038161003816100381610038161003816100381610038161003816

int(2119861)119888

119890119894120582Φ(119909119910)

119896 (119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

10038161003816100381610038161003816100381610038161003816

(21)

Now by an argument similar to the proof of Lemma 6 in[2] we choose 1206011 isin 119862

infin

0(R119899

) such that 1206011(119909) equiv 1 when |119909| le

1 and 1206011(119909) equiv 0 when |119909| gt 2 Let 1206012 = 1 minus 1206011 and 119873 isin N

which is large enough and will be determined later Write

119896 (119909) = 1198961

120582(119909) + 119896

2

120582(119909) (22)

where

119896119895

120582(119909) = 119896 (119909) 120601119895 (120582

1119873119909) 119895 = 1 2 (23)

Then

1198791205821198912 (119909) = 119901 sdot V sdot int(2119861)119888

119890119894120582Φ(119909119910)

1198961

120582(119909 minus 119910)

times 120593 (119909 119910) 119891 (119910) 119889119910

+ 119901 sdot V sdot int(2119861)119888

119890119894120582Φ(119909119910)

1198962

120582(119909 minus 119910)

times 120593 (119909 119910) 119891 (119910) 119889119910

= 1198791

1205821198912 (119909) + 119879

2

1205821198912 (119909)

(24)

Let us first estimate 1198791

1205821198912(119909) To do so using Taylorrsquos

expansion and the compactness of supp120593 we write

Φ (119909 119910) = Φ (119909 119909) + 119875 (119909 119910) + 119903119873 (119909 119910) (25)

for(119909 119910) isin supp120593 where 119875(119909 119910) is a polynomial with deg119875 lt 119873 and |119903119873(119909 119910)| le 119862|119909 minus 119910|

119873 with 119862 independent of 119909

and 119910 Define

119877119891 (119909) = 119901 sdot V sdot int(2119861)119888

119890119894120582119875(119909119910)

1198961

120582(119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

(26)

Therefore

119890minus119894120582Φ(119909119909)

1198791

1205821198912 (119909) minus 119877119891 (119909)

= int|119909minus119910|le2120582minus1119873

119890119894120582119875(119909119910)

[119890119894120582119903119873(119909119910)

minus 1]

times 1198961

120582(119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

=

infin

sum

119895=0

int2minus119895120582minus1119873lt|119909minus119910|le2minus119895+1120582minus1119873

119890119894120582119875(119909119910)

[119890119894120582119903119873(119909119910)

minus 1]

times 1198961

120582(119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

equiv

infin

sum

119895=0

1198791

1205821198951198912 (119909)

(27)

4 Journal of Function Spaces and Applications

On 1198791

1205821198951198912(119909) by the properties of 119903119873 and 119896 we have

10038161003816100381610038161003816119879

1

1205821198951198912 (119909)

10038161003816100381610038161003816le 1198622

minus119895119873119872119891 (119909) (28)

So we have10038161003816100381610038161003816119879

1

1205821198912 (119909)

10038161003816100381610038161003816le 119862

infin

sum

119895=0

2minus119895119873 1003816100381610038161003816119872119891 (119909)

1003816100381610038161003816 + 1198621003816100381610038161003816119877119891 (119909)

1003816100381610038161003816 (29)

ByTheorem A and Lemma 9 we have10038171003817100381710038171003817119879

1

1205821198912

10038171003817100381710038171003817119871119901120581(120596)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(120596)

(30)

Now let us turn to estimate 1198792

1205821198912(119909) We consider the

following two cases

Case 1 (120582 le 1) Similar to that estimate of 1198792

120582in Lemma 6 in

[2] we have10038161003816100381610038161003816119879

2

1205821198912 (119909)

10038161003816100381610038161003816le 119862119872 (119891) (119909) (31)

By Lemma 9 we have10038171003817100381710038171003817119879

2

1205821198912

10038171003817100381710038171003817119871119901120581(120596)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(120596)

(32)

Case 2 (120582 gt 1) We choose 1205930 isin 119862infin

0(R119899

) such that

supp1205930 sube 119909 isin R119899

1 lt |119909| le 2

1206012 (119909) =

infin

sum

119895=0

1205930 (2minus119895

119909)

(33)

Let

1198962

120582119895(119909) = 119896 (119909) 1205930 (2

minus1198951205821119873

119909) (34)

Then

1198792

1205821198912 (119909) = int

(2119861)119888

119890119894120582Φ(119909119910)

1198962

120582(119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

infin

sum

119895=0

int(2119861)119888

119890119894120582Φ(119909119910)

1198962

120582119895(119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

equiv

infin

sum

119895=0

1198792

1205821198951198912 (119909)

(35)

For 1198792

120582119895 by its definition we can get

10038161003816100381610038161003816119879

2

1205821198951198912 (119909)

10038161003816100381610038161003816le 119862 int

2119895120582minus1119873lt|119909minus119910|le2119895+1120582minus1119873

times 1198881

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 le 119862119872 (119891) (119909)

(36)

The inequality (36) also can be seen in [2] we omit the detailshere

By Lemma 9 we have10038171003817100381710038171003817119879

2

1205821198912

10038171003817100381710038171003817119871119901120581(120596)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(120596)

(37)

Therefore

1198682 le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596) (38)

This finishes the proof of Theorem 1

Proof of Theorem 2 It is sufficient to prove that there exists aconstant 119862 gt 0 such that

1

120596(119861)120581 int

119861

1003816100381610038161003816119879lowast

120582119891 (119909)

1003816100381610038161003816

119901120596 (119909) 119889119909 le 119862

10038171003817100381710038171198911003817100381710038171003817

119901

119871119901120581(120596) (39)

Fix a ball 119861 = 119861(1199090 119903119861) and decompose 119891 = 1198911 + 1198912 with1198911 = 1198911205942119861 Then we have

1

120596(119861)120581 int

119861

1003816100381610038161003816119879lowast

120582119891 (119909)

1003816100381610038161003816

119901120596 (119909) 119889119909

le 119862 1

120596(119861)120581 int

119861

1003816100381610038161003816119879lowast

1205821198911 (119909)

1003816100381610038161003816

119901120596 (119909) 119889119909

+1

120596(119861)120581 int

119861

1003816100381610038161003816119879lowast

1205821198912 (119909)

1003816100381610038161003816

119901120596 (119909) 119889119909

= 119862 1198691 + 1198692

(40)

Using Lemmas 3 and 7 we get

1198691 le 1198621

120596(119861)120581 int

2119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909

le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596)sdot

120596(2119861)120581

120596(119861)120581

le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596)

(41)

We now estimate 1198692For each 119898 isin N and 119895 = 1 119892119898 we get

119886119895119898 (119909) = intΣ

Ω (119909 119911) 119884119895119898 (119911) 119889120590119911 (42)

where Ω(119909 119911) = |119911|119899119896(119909 119911) Then for ae 119909 isin R119899

Ω (119909 119911) =

infin

sum

119898=1

119892119898

sum

119895=1

119886119895119898 (119909) 119884119895119898 (1199111015840) (43)

where 1199111015840

= 119911|119911| for any 119911 isin R119899 0 By Lemma 10 we have

that for any 119909 isin R119899

10038161003816100381610038161003816119886119895119898 (119909)

10038161003816100381610038161003816= 119898

minus119899(119898 + 119899 minus 2)

minus1198991003816100381610038161003816100381610038161003816intΣ

Ω (119909 119911) Λ119899119884119895119898 (119911) 119889120590119911

1003816100381610038161003816100381610038161003816

= 119898minus119899

(119898 + 119899 minus 2)minus119899

1003816100381610038161003816100381610038161003816intΣ

Λ119899Ω (119909 119911) 119884119895119898 (119911) 119889120590119911

1003816100381610038161003816100381610038161003816

le 119862 (119899) 119860119898minus2119899

(44)

By Lemma 10 again we can verify that for any 120598 gt 0 119873 isin

N and ae 119909 isin R119899 if |119910 minus 119909| ge 120598 then

10038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119898=1

119892119898

sum

119895=1

119890119894120582Φ(119909119910)

119886119895119898 (119909) 119884119895119898 ((119909 minus 119910)1015840)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899 120593 (119909 119910) 1198912 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862 (120598) 11986010038161003816100381610038161198912 (119910)

1003816100381610038161003816

(45)

Journal of Function Spaces and Applications 5

Therefore from (43) (45) and the Lebesgue dominatedconvergence theorem it follows that

119879lowast

1205821198912 (119909)

= lim120598rarr0

int|119909minus119910|ge120598

119890119894120582Φ(119909119910)

119896 (119909 119909 minus 119910) 120593 (119909 119910) 1198912 (119910) 119889119910

= lim120598rarr0

infin

sum

119898=1

119892119898

sum

119895=1

int|119909minus119910|ge120598

119890119894120582Φ(119909119910)

119886119895119898 (119909) 119884119895119898 ((119909 minus 119910)1015840)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899

times 120593 (119909 119910) 1198912 (119910) 119889119910

= lim120598rarr0

infin

sum

119898=1

119892119898

sum

119895=1

119886119895119898 (119909) int|119909minus119910|ge120598

119890119894120582Φ(119909119910)

119884119895119898 ((119909 minus 119910)1015840)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899

times 120593 (119909 119910) 1198912 (119910) 119889119910

(46)

We write

1198771198951198981198912 (119909)

= int|119909minus119910|ge120598

119890119894120582Φ(119909119910)

119884119895119898 ((119909 minus 119910)1015840)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899 120593 (119909 119910) 1198912 (119910) 119889119910

(47)

It is easy to see that 1198771198951198981198912(119909) is the oscillatory integraloperator defined by (1) By Theorem 1 we have that 119877119895119898 isbounded on weighted Morrey spaces Therefore by (44) andthe above discussion we have

1198692 le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596) (48)

This finishes the proof of Theorem 2

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant no 11001001) Natural Sci-ence Foundation from the Education Department of AnhuiProvince (nos KJ2012B166 KJ2013A235)

References

[1] Y Pan ldquoUniform estimates for oscillatory integral operatorsrdquoJournal of Functional Analysis vol 100 no 1 pp 207ndash220 1991

[2] S Lu D Yang and Z Zhou ldquoOn local oscillatory integralswith variable Calderon-Zygmund kernelsrdquo Integral Equationsand Operator Theory vol 33 no 4 pp 456ndash470 1999

[3] C B Morrey Jr ldquoOn the solutions of quasi-linear ellipticpartial differential equationsrdquo Transactions of the AmericanMathematical Society vol 43 no 1 pp 126ndash166 1938

[4] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009

[5] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987

[6] J Peetre ldquoOn the theory of 119871119901120582 spacesrdquo Journal of Functional

Analysis vol 4 no 1 pp 71ndash87 1969[7] S Z Lu Y Ding and D Y Yan Singular Integrals and Related

Topics World Scientific Publishing River Edge NJ USA 2007[8] E Nakai ldquoHardy-Littlewood maximal operator singular inte-

gral operators and the Riesz potentials on generalized MorreyspacesrdquoMathematische Nachrichten vol 166 pp 95ndash103 1994

[9] Y Sawano and H Tanaka ldquoMorrey spaces for non-doublingmeasuresrdquo Acta Mathematica Sinica vol 21 no 6 pp 1535ndash1544 2005

[10] H Wang and H P Liu ldquoSome estimates for Bochner-Rieszoperators on the weighted Morrey spacesrdquo Acta MathematicaSinica vol 55 no 3 pp 551ndash560 2012

[11] H Wang and H P Liu ldquoWeak type estimates of intrinsicsquare functions on the weighted Hardy spacesrdquo Archiv derMathematik vol 97 no 1 pp 49ndash59 2011

[12] R Ch Mustafayev ldquoOn boundedness of sublinear operators inweighted Morrey spacesrdquo Azerbaijan Journal of Mathematicsvol 2 no 1 pp 66ndash79 2012

[13] X F Ye and X S Zhu ldquoEstimates of singular integrals andmultilinear commutators in weighted Morrey spacesrdquo Journalof Inequalities and Applications vol 2012 article 302 2012

[14] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica Chinese Series vol 55 no 4 pp 589ndash600 2012 (Chinese)

[15] H Wang ldquoThe boundedness of fractional integral operatorswith rough kernels on the weighted Morrey spacesrdquo ActaMathematica Sinica Chinese Series vol 56 no 2 pp 175ndash1862013 (Chinese)

[16] S He ldquoThe boundedness of some multilinear operator withrough kernel on the weighted Morrey spacesrdquo submittedhttparxivorgabs11115463

[17] H Wang ldquoIntrinsic square functions on the weighted Morreyspacesrdquo Journal of Mathematical Analysis and Applications vol396 no 1 pp 302ndash314 2012

[18] S G Shi Z W Fu and S Z Lu ldquoBoundedness of oscillatoryintegral operators and their commutators on weighted Morreyspacesrdquo Scientia Sinica Mathematica vol 43 pp 147ndash158 2013(Chinese)

[19] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972

[20] J Garcıa-Cuerva and J L Rubio de Francia Weighted NormInequalities and Related Topics vol 116 of North-Holland Math-ematics Studies North-Holland Amsterdam The Netherlands1985

[21] R F Gundy and R LWheeden ldquoWeighted integral inequalitiesfor the nontangential maximal function Lusin area integraland Walsh-Paley seriesrdquo Studia Mathematica vol 49 pp 107ndash124 1974

[22] E M Stein and G Weiss Introduction to Fourier Analysis onEuclidean Spaces Princeton University Press Princeton NJUSA 1971

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

4 Journal of Function Spaces and Applications

On 1198791

1205821198951198912(119909) by the properties of 119903119873 and 119896 we have

10038161003816100381610038161003816119879

1

1205821198951198912 (119909)

10038161003816100381610038161003816le 1198622

minus119895119873119872119891 (119909) (28)

So we have10038161003816100381610038161003816119879

1

1205821198912 (119909)

10038161003816100381610038161003816le 119862

infin

sum

119895=0

2minus119895119873 1003816100381610038161003816119872119891 (119909)

1003816100381610038161003816 + 1198621003816100381610038161003816119877119891 (119909)

1003816100381610038161003816 (29)

ByTheorem A and Lemma 9 we have10038171003817100381710038171003817119879

1

1205821198912

10038171003817100381710038171003817119871119901120581(120596)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(120596)

(30)

Now let us turn to estimate 1198792

1205821198912(119909) We consider the

following two cases

Case 1 (120582 le 1) Similar to that estimate of 1198792

120582in Lemma 6 in

[2] we have10038161003816100381610038161003816119879

2

1205821198912 (119909)

10038161003816100381610038161003816le 119862119872 (119891) (119909) (31)

By Lemma 9 we have10038171003817100381710038171003817119879

2

1205821198912

10038171003817100381710038171003817119871119901120581(120596)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(120596)

(32)

Case 2 (120582 gt 1) We choose 1205930 isin 119862infin

0(R119899

) such that

supp1205930 sube 119909 isin R119899

1 lt |119909| le 2

1206012 (119909) =

infin

sum

119895=0

1205930 (2minus119895

119909)

(33)

Let

1198962

120582119895(119909) = 119896 (119909) 1205930 (2

minus1198951205821119873

119909) (34)

Then

1198792

1205821198912 (119909) = int

(2119861)119888

119890119894120582Φ(119909119910)

1198962

120582(119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

infin

sum

119895=0

int(2119861)119888

119890119894120582Φ(119909119910)

1198962

120582119895(119909 minus 119910) 120593 (119909 119910) 119891 (119910) 119889119910

equiv

infin

sum

119895=0

1198792

1205821198951198912 (119909)

(35)

For 1198792

120582119895 by its definition we can get

10038161003816100381610038161003816119879

2

1205821198951198912 (119909)

10038161003816100381610038161003816le 119862 int

2119895120582minus1119873lt|119909minus119910|le2119895+1120582minus1119873

times 1198881

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899

1003816100381610038161003816119891 (119910)1003816100381610038161003816 119889119910 le 119862119872 (119891) (119909)

(36)

The inequality (36) also can be seen in [2] we omit the detailshere

By Lemma 9 we have10038171003817100381710038171003817119879

2

1205821198912

10038171003817100381710038171003817119871119901120581(120596)le 119862

10038171003817100381710038171198911003817100381710038171003817119871119901120581(120596)

(37)

Therefore

1198682 le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596) (38)

This finishes the proof of Theorem 1

Proof of Theorem 2 It is sufficient to prove that there exists aconstant 119862 gt 0 such that

1

120596(119861)120581 int

119861

1003816100381610038161003816119879lowast

120582119891 (119909)

1003816100381610038161003816

119901120596 (119909) 119889119909 le 119862

10038171003817100381710038171198911003817100381710038171003817

119901

119871119901120581(120596) (39)

Fix a ball 119861 = 119861(1199090 119903119861) and decompose 119891 = 1198911 + 1198912 with1198911 = 1198911205942119861 Then we have

1

120596(119861)120581 int

119861

1003816100381610038161003816119879lowast

120582119891 (119909)

1003816100381610038161003816

119901120596 (119909) 119889119909

le 119862 1

120596(119861)120581 int

119861

1003816100381610038161003816119879lowast

1205821198911 (119909)

1003816100381610038161003816

119901120596 (119909) 119889119909

+1

120596(119861)120581 int

119861

1003816100381610038161003816119879lowast

1205821198912 (119909)

1003816100381610038161003816

119901120596 (119909) 119889119909

= 119862 1198691 + 1198692

(40)

Using Lemmas 3 and 7 we get

1198691 le 1198621

120596(119861)120581 int

2119861

1003816100381610038161003816119891 (119909)1003816100381610038161003816

119901120596 (119909) 119889119909

le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596)sdot

120596(2119861)120581

120596(119861)120581

le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596)

(41)

We now estimate 1198692For each 119898 isin N and 119895 = 1 119892119898 we get

119886119895119898 (119909) = intΣ

Ω (119909 119911) 119884119895119898 (119911) 119889120590119911 (42)

where Ω(119909 119911) = |119911|119899119896(119909 119911) Then for ae 119909 isin R119899

Ω (119909 119911) =

infin

sum

119898=1

119892119898

sum

119895=1

119886119895119898 (119909) 119884119895119898 (1199111015840) (43)

where 1199111015840

= 119911|119911| for any 119911 isin R119899 0 By Lemma 10 we have

that for any 119909 isin R119899

10038161003816100381610038161003816119886119895119898 (119909)

10038161003816100381610038161003816= 119898

minus119899(119898 + 119899 minus 2)

minus1198991003816100381610038161003816100381610038161003816intΣ

Ω (119909 119911) Λ119899119884119895119898 (119911) 119889120590119911

1003816100381610038161003816100381610038161003816

= 119898minus119899

(119898 + 119899 minus 2)minus119899

1003816100381610038161003816100381610038161003816intΣ

Λ119899Ω (119909 119911) 119884119895119898 (119911) 119889120590119911

1003816100381610038161003816100381610038161003816

le 119862 (119899) 119860119898minus2119899

(44)

By Lemma 10 again we can verify that for any 120598 gt 0 119873 isin

N and ae 119909 isin R119899 if |119910 minus 119909| ge 120598 then

10038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119898=1

119892119898

sum

119895=1

119890119894120582Φ(119909119910)

119886119895119898 (119909) 119884119895119898 ((119909 minus 119910)1015840)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899 120593 (119909 119910) 1198912 (119910)

10038161003816100381610038161003816100381610038161003816100381610038161003816

le 119862 (120598) 11986010038161003816100381610038161198912 (119910)

1003816100381610038161003816

(45)

Journal of Function Spaces and Applications 5

Therefore from (43) (45) and the Lebesgue dominatedconvergence theorem it follows that

119879lowast

1205821198912 (119909)

= lim120598rarr0

int|119909minus119910|ge120598

119890119894120582Φ(119909119910)

119896 (119909 119909 minus 119910) 120593 (119909 119910) 1198912 (119910) 119889119910

= lim120598rarr0

infin

sum

119898=1

119892119898

sum

119895=1

int|119909minus119910|ge120598

119890119894120582Φ(119909119910)

119886119895119898 (119909) 119884119895119898 ((119909 minus 119910)1015840)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899

times 120593 (119909 119910) 1198912 (119910) 119889119910

= lim120598rarr0

infin

sum

119898=1

119892119898

sum

119895=1

119886119895119898 (119909) int|119909minus119910|ge120598

119890119894120582Φ(119909119910)

119884119895119898 ((119909 minus 119910)1015840)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899

times 120593 (119909 119910) 1198912 (119910) 119889119910

(46)

We write

1198771198951198981198912 (119909)

= int|119909minus119910|ge120598

119890119894120582Φ(119909119910)

119884119895119898 ((119909 minus 119910)1015840)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899 120593 (119909 119910) 1198912 (119910) 119889119910

(47)

It is easy to see that 1198771198951198981198912(119909) is the oscillatory integraloperator defined by (1) By Theorem 1 we have that 119877119895119898 isbounded on weighted Morrey spaces Therefore by (44) andthe above discussion we have

1198692 le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596) (48)

This finishes the proof of Theorem 2

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant no 11001001) Natural Sci-ence Foundation from the Education Department of AnhuiProvince (nos KJ2012B166 KJ2013A235)

References

[1] Y Pan ldquoUniform estimates for oscillatory integral operatorsrdquoJournal of Functional Analysis vol 100 no 1 pp 207ndash220 1991

[2] S Lu D Yang and Z Zhou ldquoOn local oscillatory integralswith variable Calderon-Zygmund kernelsrdquo Integral Equationsand Operator Theory vol 33 no 4 pp 456ndash470 1999

[3] C B Morrey Jr ldquoOn the solutions of quasi-linear ellipticpartial differential equationsrdquo Transactions of the AmericanMathematical Society vol 43 no 1 pp 126ndash166 1938

[4] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009

[5] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987

[6] J Peetre ldquoOn the theory of 119871119901120582 spacesrdquo Journal of Functional

Analysis vol 4 no 1 pp 71ndash87 1969[7] S Z Lu Y Ding and D Y Yan Singular Integrals and Related

Topics World Scientific Publishing River Edge NJ USA 2007[8] E Nakai ldquoHardy-Littlewood maximal operator singular inte-

gral operators and the Riesz potentials on generalized MorreyspacesrdquoMathematische Nachrichten vol 166 pp 95ndash103 1994

[9] Y Sawano and H Tanaka ldquoMorrey spaces for non-doublingmeasuresrdquo Acta Mathematica Sinica vol 21 no 6 pp 1535ndash1544 2005

[10] H Wang and H P Liu ldquoSome estimates for Bochner-Rieszoperators on the weighted Morrey spacesrdquo Acta MathematicaSinica vol 55 no 3 pp 551ndash560 2012

[11] H Wang and H P Liu ldquoWeak type estimates of intrinsicsquare functions on the weighted Hardy spacesrdquo Archiv derMathematik vol 97 no 1 pp 49ndash59 2011

[12] R Ch Mustafayev ldquoOn boundedness of sublinear operators inweighted Morrey spacesrdquo Azerbaijan Journal of Mathematicsvol 2 no 1 pp 66ndash79 2012

[13] X F Ye and X S Zhu ldquoEstimates of singular integrals andmultilinear commutators in weighted Morrey spacesrdquo Journalof Inequalities and Applications vol 2012 article 302 2012

[14] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica Chinese Series vol 55 no 4 pp 589ndash600 2012 (Chinese)

[15] H Wang ldquoThe boundedness of fractional integral operatorswith rough kernels on the weighted Morrey spacesrdquo ActaMathematica Sinica Chinese Series vol 56 no 2 pp 175ndash1862013 (Chinese)

[16] S He ldquoThe boundedness of some multilinear operator withrough kernel on the weighted Morrey spacesrdquo submittedhttparxivorgabs11115463

[17] H Wang ldquoIntrinsic square functions on the weighted Morreyspacesrdquo Journal of Mathematical Analysis and Applications vol396 no 1 pp 302ndash314 2012

[18] S G Shi Z W Fu and S Z Lu ldquoBoundedness of oscillatoryintegral operators and their commutators on weighted Morreyspacesrdquo Scientia Sinica Mathematica vol 43 pp 147ndash158 2013(Chinese)

[19] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972

[20] J Garcıa-Cuerva and J L Rubio de Francia Weighted NormInequalities and Related Topics vol 116 of North-Holland Math-ematics Studies North-Holland Amsterdam The Netherlands1985

[21] R F Gundy and R LWheeden ldquoWeighted integral inequalitiesfor the nontangential maximal function Lusin area integraland Walsh-Paley seriesrdquo Studia Mathematica vol 49 pp 107ndash124 1974

[22] E M Stein and G Weiss Introduction to Fourier Analysis onEuclidean Spaces Princeton University Press Princeton NJUSA 1971

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 5

Therefore from (43) (45) and the Lebesgue dominatedconvergence theorem it follows that

119879lowast

1205821198912 (119909)

= lim120598rarr0

int|119909minus119910|ge120598

119890119894120582Φ(119909119910)

119896 (119909 119909 minus 119910) 120593 (119909 119910) 1198912 (119910) 119889119910

= lim120598rarr0

infin

sum

119898=1

119892119898

sum

119895=1

int|119909minus119910|ge120598

119890119894120582Φ(119909119910)

119886119895119898 (119909) 119884119895119898 ((119909 minus 119910)1015840)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899

times 120593 (119909 119910) 1198912 (119910) 119889119910

= lim120598rarr0

infin

sum

119898=1

119892119898

sum

119895=1

119886119895119898 (119909) int|119909minus119910|ge120598

119890119894120582Φ(119909119910)

119884119895119898 ((119909 minus 119910)1015840)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899

times 120593 (119909 119910) 1198912 (119910) 119889119910

(46)

We write

1198771198951198981198912 (119909)

= int|119909minus119910|ge120598

119890119894120582Φ(119909119910)

119884119895119898 ((119909 minus 119910)1015840)

1003816100381610038161003816119909 minus 1199101003816100381610038161003816

119899 120593 (119909 119910) 1198912 (119910) 119889119910

(47)

It is easy to see that 1198771198951198981198912(119909) is the oscillatory integraloperator defined by (1) By Theorem 1 we have that 119877119895119898 isbounded on weighted Morrey spaces Therefore by (44) andthe above discussion we have

1198692 le 1198621003817100381710038171003817119891

1003817100381710038171003817

119901

119871119901120581(120596) (48)

This finishes the proof of Theorem 2

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant no 11001001) Natural Sci-ence Foundation from the Education Department of AnhuiProvince (nos KJ2012B166 KJ2013A235)

References

[1] Y Pan ldquoUniform estimates for oscillatory integral operatorsrdquoJournal of Functional Analysis vol 100 no 1 pp 207ndash220 1991

[2] S Lu D Yang and Z Zhou ldquoOn local oscillatory integralswith variable Calderon-Zygmund kernelsrdquo Integral Equationsand Operator Theory vol 33 no 4 pp 456ndash470 1999

[3] C B Morrey Jr ldquoOn the solutions of quasi-linear ellipticpartial differential equationsrdquo Transactions of the AmericanMathematical Society vol 43 no 1 pp 126ndash166 1938

[4] Y Komori and S Shirai ldquoWeighted Morrey spaces and asingular integral operatorrdquo Mathematische Nachrichten vol282 no 2 pp 219ndash231 2009

[5] F Chiarenza and M Frasca ldquoMorrey spaces and Hardy-Littlewoodmaximal functionrdquo Rendiconti di Matematica e dellesue Applicazioni vol 7 no 3-4 pp 273ndash279 1987

[6] J Peetre ldquoOn the theory of 119871119901120582 spacesrdquo Journal of Functional

Analysis vol 4 no 1 pp 71ndash87 1969[7] S Z Lu Y Ding and D Y Yan Singular Integrals and Related

Topics World Scientific Publishing River Edge NJ USA 2007[8] E Nakai ldquoHardy-Littlewood maximal operator singular inte-

gral operators and the Riesz potentials on generalized MorreyspacesrdquoMathematische Nachrichten vol 166 pp 95ndash103 1994

[9] Y Sawano and H Tanaka ldquoMorrey spaces for non-doublingmeasuresrdquo Acta Mathematica Sinica vol 21 no 6 pp 1535ndash1544 2005

[10] H Wang and H P Liu ldquoSome estimates for Bochner-Rieszoperators on the weighted Morrey spacesrdquo Acta MathematicaSinica vol 55 no 3 pp 551ndash560 2012

[11] H Wang and H P Liu ldquoWeak type estimates of intrinsicsquare functions on the weighted Hardy spacesrdquo Archiv derMathematik vol 97 no 1 pp 49ndash59 2011

[12] R Ch Mustafayev ldquoOn boundedness of sublinear operators inweighted Morrey spacesrdquo Azerbaijan Journal of Mathematicsvol 2 no 1 pp 66ndash79 2012

[13] X F Ye and X S Zhu ldquoEstimates of singular integrals andmultilinear commutators in weighted Morrey spacesrdquo Journalof Inequalities and Applications vol 2012 article 302 2012

[14] H Wang ldquoThe boundedness of some operators with roughkernel on the weighted Morrey spacesrdquo Acta MathematicaSinica Chinese Series vol 55 no 4 pp 589ndash600 2012 (Chinese)

[15] H Wang ldquoThe boundedness of fractional integral operatorswith rough kernels on the weighted Morrey spacesrdquo ActaMathematica Sinica Chinese Series vol 56 no 2 pp 175ndash1862013 (Chinese)

[16] S He ldquoThe boundedness of some multilinear operator withrough kernel on the weighted Morrey spacesrdquo submittedhttparxivorgabs11115463

[17] H Wang ldquoIntrinsic square functions on the weighted Morreyspacesrdquo Journal of Mathematical Analysis and Applications vol396 no 1 pp 302ndash314 2012

[18] S G Shi Z W Fu and S Z Lu ldquoBoundedness of oscillatoryintegral operators and their commutators on weighted Morreyspacesrdquo Scientia Sinica Mathematica vol 43 pp 147ndash158 2013(Chinese)

[19] B Muckenhoupt ldquoWeighted norm inequalities for the Hardymaximal functionrdquo Transactions of the American MathematicalSociety vol 165 pp 207ndash226 1972

[20] J Garcıa-Cuerva and J L Rubio de Francia Weighted NormInequalities and Related Topics vol 116 of North-Holland Math-ematics Studies North-Holland Amsterdam The Netherlands1985

[21] R F Gundy and R LWheeden ldquoWeighted integral inequalitiesfor the nontangential maximal function Lusin area integraland Walsh-Paley seriesrdquo Studia Mathematica vol 49 pp 107ndash124 1974

[22] E M Stein and G Weiss Introduction to Fourier Analysis onEuclidean Spaces Princeton University Press Princeton NJUSA 1971

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