Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013 Article ID 560976 5 pageshttpdxdoiorg1011552013560976

Research ArticleOn a Class of Bilinear Pseudodifferential Operators

Aacuterpaacuted Beacutenyi1 and Tadahiro Oh2

1 Department of Mathematics Western Washington University 516 High Street Bellingham WA 98225 USA2Department of Mathematics Princeton University Fine Hall Washington Road Princeton NJ 08544-1000 USA

Correspondence should be addressed to Arpad Benyi arpadbenyiwwuedu

Received 26 September 2012 Accepted 3 December 2012

Academic Editor Baoxiang Wang

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

We provide a direct proof for the boundedness of pseudodifferential operators with symbols in the bilinear Hormander class BS01120575

0 le 120575 lt 1 The proof uses a reduction to bilinear elementary symbols and Littlewood-Paley theory

1 Introduction Main Results and Examples

Coifman andMeyerrsquos ideas onmultilinear operators and theirapplications in partial differential equations (PDEs) havehad a great impact in the future developments and growthwitnessed in the topic of multilinear singular integrals Oneof their classical results [1 Proposition 2 p 154] is about the119871119901times119871119902rarr 119871119903 boundedness of a class of translation invariant

bilinear operators (bilinear multiplier operators) given by

119879120590(119891 119892) (119909) = int

R119899int

R119899120590 (120585 120578) (120585) (120578) 119890

119894119909sdot(120585+120578)119889120585 119889120578 (1)

We have the following

Theorem A If |120597120573120585120597120574

120578120590(120585 120578)| ≲ (1 + |120585| + |120578|)

minus|120573|minus|120574| for all120585 120578 isin R119899 and all multi-indices 120573 120574 then 119879

120590has a bounded

extension from 119871119901 times 119871119902 into 119871119903 for all 1 lt 119901 119902 lt infin such that1119901 + 1119902 = 1119903

In fact Coifman and Meyerrsquos approach yields TheoremA only for 119903 gt 1 The optimal extension of their result tothe range 119903 gt 12 (as implied in the theorem above) canbe obtained using interpolation arguments and an end-pointestimate 1198711 times 1198711 into 11987112infin in the works of Grafakos andTorres [2] and Kenig and Stein [3]

Bilinear pseudodifferential operators are natural non-translation invariant generalizations of the translation invari-ant ones they allow symbols to depend on the space variable

119909 as well Let us then consider bilinear operators a prioridefined from S times S into S1015840 of the form

119879120590(119891 119892) (119909) = int

R119899int

R119899120590 (119909 120585 120578) (120585) (120578) 119890

119894119909sdot(120585+120578)119889120585 119889120578

(2)

Perhaps unsurprisingly we impose then similar conditionson the derivatives of the symbol 120590 with the expectationthat they would yield indeed bounded operators 119879

120590on

appropriate spaces of functions The estimates that we havein mind define the so-called bilinear Hormander classes ofsymbols denoted by BS119898

120588120575 We say that 120590 isin BS119898

120588120575if

100381610038161003816100381610038161003816120597120572

119909120597120573

120585120597120574

120578120590 (119909 120585 120578)

100381610038161003816100381610038161003816≲ (1 +

10038161003816100381610038161205851003816100381610038161003816 +10038161003816100381610038161205781003816100381610038161003816)119898+120575|120572|minus120588(|120573|+|120574|) (3)

for all 119909 120585 120578 isin R119899 and all multi-indices 120572 120573 120574 Note that weneed smoothness in 119909 as in the linear Hormander classes Asusual the notation 119886 ≲ 119887 means that there exists a positiveconstant119870 (independent of 119886 119887) such that 119886 le 119870119887

With this terminology we can restate Theorem A asfollows

If the 119909-independent symbol 120590(120585 120578) belongs to the classBS010 then119879

120590is bounded from 119871119901times119871119902 into 119871119903 for all 1 lt 119901 119902 lt

infin such that 1119901 + 1119902 = 1119903The condition of translation invariance (equivalently the

119909-independence of the symbol) is superfluous Moreover theprevious boundedness result can be shown to hold for thelarger class of symbols BS0

1120575supe BS010 where 0 le 120575 lt 1This is a

known fact that is tightly connected to the bilinear Calderon-Zygmund theory developed by Grafakos and Torres in [2]

2 Journal of Function Spaces and Applications

and the existence of a transposition symbolic calculus provedby Benyi et al [4] Let us briefly give an outline of how thisfollows First we note that the bilinear kernels associated tobilinear operators with symbols in BS0

1120575 0 le 120575 lt 1 are

bilinear Calderon-Zygmund operators in the sense of [2]Second we recall that [2 Corollary 1] which is an applicationof the bilinear 119879(1) theorem therein states the following

Theorem B If 119879 and its transposes 119879lowast1 and 119879

lowast2 havesymbols in BS0

11 then they can be extended as bounded

operators from 119871119901times 119871119902 into 119871119903 for 1 lt 119901 119902 lt infin and

1119901 + 1119902 = 1119903

Third by [4 Theorem 21] we have the following

Theorem C Assume that 0 le 120575 le 120588 le 1 120575 lt 1 and 120590 isinBS119898120588120575 Then for 119895 = 1 2119879lowast119895

120590= 119879120590lowast119895 where 120590lowast119895 isin BS119898

120588120575

Finally since BS01120575

sub BS011 we can directly combine

Theorems B and C to recover the following optimal extensionof the Coifman-Meyer result note that now the symbol isallowed to depend on 119909 while 119903 is still allowed to be in theoptimal interval (12infin)

Theorem 1 If 120590 is a symbol in BS01120575 0 le 120575 lt 1 then 119879

120590has a

bounded extension from 119871119901 times 119871119902 into 119871119903 for all 1 lt 119901 119902 lt infinsuch that 1119901 + 1119902 = 1119903

Once we have the boundedness of the class BS01120575

onproducts of Lebesgue spaces a ldquoreduction methodrdquo allows usto deduce also the boundedness of the class BS119898

1120575on appro-

priate products of Sobolev spaces Moreover our estimatesin this case come in the form of Leibniz-type rules formore on these kinds of properties see the work of Bernicotet al [5] In the particular case when the bilinear operatoris just a differential operator the Leibniz-type rules arereferred to as Kato-Poncersquos commutator estimates and areknown to play a significant role in the study of the Eulerand Navier-Stokes equations see [6] see also Kenig et al[7] for further applications of commutators to nonlinearSchrodinger equations Let 119869119898 = (119868 minus Δ)

1198982 denote thelinear Fourier multiplier operator with symbol ⟨120585⟩119898 where⟨120585⟩ = (1 + |120585|

2)12 By definition we say that 119891 belongs to the

Sobolev space 119871119901119898if 119869119898119891 isin 119871119901 We have the following

Theorem 2 Let 120590 be a symbol in BS1198981120575 0 le 120575 lt 1 119898 ge 0

and let 119879120590be its associated operator Then there exist symbols

1205901and 120590

2in BS01120575

such that for all 119891 119892 isin S

119879120590(119891 119892) = 119879

1205901

(119869119898119891 119892) + 119879

1205902

(119891 119869119898119892) (4)

In particular then one has that 119879120590has a bounded extension

from 119871119901119898times 119871119902

119898into 119871119903 provided that 1119901 + 1119902 = 1119903 1 lt 119901

119902 lt infin Moreover1003817100381710038171003817119879120590 (119891 119892)

1003817100381710038171003817119871119903≲10038171003817100381710038171198911003817100381710038171003817119871119901

119898

10038171003817100381710038171198921003817100381710038171003817119871119902+1003817100381710038171003817119891100381710038171003817100381711987111990110038171003817100381710038171198921003817100381710038171003817119871119902

119898

(5)

Theproof ofTheorem2 follows a similar path as the one inthe work of Benyi et al [8Theorem 27] For the convenience

of the reader we sketch here the main steps in the argumentLet 120601 be a 119862infin-function on R such that 0 le 120601 le 1 supp120601 sub [minus2 2] and 120601(119903) + 120601(1119903) = 1 on [0infin) Then (4) holdsif we let

1205901(119909 120585 120578) = 120590 (119909 120585 120578) 120601(

⟨120578⟩2

⟨120585⟩2)⟨120585⟩

minus119898

1205902(119909 120585 120578) = 120590 (119909 120585 120578) 120601(

⟨120585⟩2

⟨120578⟩2)⟨120578⟩

minus119898

(6)

Now straightforward calculations that take into account thesupport condition on 120601 given that 120590

1and 120590

2belong to BS0

1120575

The Leibniz-type estimate (5) follows now from Theorem 1and (4)

It is also worthwhile to note that we can replace (5) witha more general Leibniz-type rule of the form

1003817100381710038171003817119879120590(119891 119892)1003817100381710038171003817119871119903≲100381710038171003817100381711989110038171003817100381710038171198711199011

119898

100381710038171003817100381711989210038171003817100381710038171198711199021

+100381710038171003817100381711989210038171003817100381710038171198711199012

119898

100381710038171003817100381711989110038171003817100381710038171198711199022 (7)

where 11199011+11199021= 1119901

2+11199022= 1119903 1 lt 119901

11199012 1199021 1199022lt infin

One of the main reasons for the study of the Hormanderclasses of bilinear pseudodifferential operators is the factthat the conditions imposed on the symbols arise naturallyin PDEs In particular the bilinear Hormader classes BS119898

120588120575

model the product of two functions and their derivatives

Example 3 Consider first a bilinear partial differential oper-ator with variable coefficients

119863119896ℓ(119891 119892) = sum

|120573|le119896

sum

|120574|leℓ

119888120573120574(119909)

120597120573119891

120597119909120573

120597120574119892

120597119909120574 (8)

Note that119863119896ℓ= 119879120590119896ℓ

where the bilinear symbol is given by

120590119896ℓ(119909 120585 120578) = (2120587)

minus2119899sum

120573120574

119888120573120574(119909) (119894120585)

120573(119894120578)120574 (9)

Assuming that the coefficients 119888120573120574

have bounded derivativesit is easy to show that 120590

119896ℓisin BS119896+ℓ10

Example 4 The symbol in the previous example is almostequivalent to a multiplier of the form

120590119898(120585 120578) = (1 +

10038161003816100381610038161205851003816100381610038161003816

2+10038161003816100381610038161205781003816100381610038161003816

2)1198982

(10)

Indeed this symbol belongs to BS11989810We can also think of this

symbol as the bilinear counterpart of the multiplier ⟨120585⟩119898 thatdefines the linear operator 119869119898

Example 5 With the notation in Example 4 the multipliers120585120590minus1(120585 120578) and 120578120590

minus1(120585 120578) belong to BS0

10 In general the

multipliers 120590119896+ℓ(120585 120578) = 120585

119896120578ℓ120590minus1(120585 120578) belong to BS119896+ℓ

10

Example 6 One of the recurrent techniques in PDE estimatesis to truncate a given multiplier at the right scale Considernow

120590 (120585 120578) = 120590119898(120585 120578) sum

119886119887isinN

119888119886119887(119909) 120593 (2

minus119886120585) 120594 (2

minus119887120578) (11)

Journal of Function Spaces and Applications 3

where120593 and120594 are smooth ldquocutoffrdquo functions supported in theannulus 12 le |120585| le 2 and the coefficients satisfy derivativeestimates of the form

1003817100381710038171003817120597120572

119909119888119886119887(119909)1003817100381710038171003817119871infin

≲ 2120575|120572|max(119886119887)

(12)

Elementary calculations show that 120590 isin BS1198981120575

Remark 7 Theorems 1 and 2 lead to the natural questionabout the boundedness properties of other Hormanderclasses of bilinear pseudodifferential operators An interest-ing situation arises when we consider the bilinear Calderon-Vaillancourt class BS0

00 A result of Benyi andTorres [9] shows

that in this case the 119871119901 times 119871119902 rarr 119871119903 boundedness fails

One can impose some additional conditions (besides being inBS000) on a symbol to guarantee that the corresponding bilin-

ear pseudodifferential operator is 119871119901 times 119871119902 rarr 119871119903 bounded

see for example [9] and the recent work of Bernicot andShrivastava [10] However there is a nice substitute for theLebesgue space estimates If we consider instead modulationspaces119872119901119902 (see the excellent book byGrochenig [11] for theirdefinition and basic properties) we can show for examplethat if 120590 isin BS0

00then 119879

120590 1198712times 1198712rarr 119872

1infin (which contains1198711) This and other more general boundedness results on

modulation spaces for the class BS000

were obtained by Benyiet al [12] Then this particular boundedness result with thereduction method employed in Theorem 2 allows us to alsoobtain the boundedness of the class BS119898

00from 1198712

119898times 1198712

119898into

1198721infin Interestingly we can also obtain the 119871119901 times 119871119902 rarr 119871

119903

boundedness of the class BS11989800 but we have to require in this

case the order 119898 to depend on the Lebesgue exponents seethe work of Benyi et al [13] also Miyachi and Tomita [14] forthe optimality of the order 119898 and the extension of the resultin [13] below 119903 = 1 The most general case of the classes BS119898

120588120575

is also given in [13]

In the remainder of the paper we will provide an alternateproof ofTheorem 1 that does not use sophisticated tools suchas the symbolic calculus The proof is in the original spiritof the work of Coifman and Meyer that made use of theLittlewood-Paley theory As such we will only be concernedhere with the boundedness into the target space 119871119903 with 119903 gt1 Of course obtaining the full result for 119903 gt 12 is thenpossible because of the bilinear Calderon-Zygmund theorywhich applies to our case We will borrow some of the ideasfrom Benyi and Torres [15] which in turn go back to the niceexposition (in the linear case) by Journe [16] by making useof the so-called bilinear elementary symbols

2 Proof of Theorem 1

We start with two lemmas that provide the anticipateddecomposition of our symbol into bilinear elementary sym-bols Since they are the immediate counterparts of [15Lemma 1 and Lemma 2] to our class BS0

1120575 we will skip their

proofs see also [16 pp 72ndash75] and [1 pp 55ndash57] The firstreduction is as follows

Lemma 8 Fix a symbol 120590 in the class BS01120575 0 le 120575 lt 1 and

an arbitrary large positive integer 119873 Then for any 119891 119892 isin S119879120590(119891 119892) can be written in the form

119879120590(119891 119892) = sum

119896ℓisinZ119899

119889119896ℓ119879120590119896ℓ

(119891 119892) + 119877 (119891 119892) (13)

where 119889119896ℓ is an absolutely convergent sequence of numbers

120590119896ℓ(119909 120585 120578) =

infin

sum

119895=0

120581119895119896ℓ(119909) 120595119896ℓ(2minus119895120585 2minus119895120578) (14)

with each 120595119896ℓ

a 119862infin-function supported on the set 13 le

max (|120585| |120578|) le 1100381610038161003816100381610038161003816120597120573

120585120597120574

120578120595119896ℓ(120585 120578)

100381610038161003816100381610038161003816≲ 1 forall

10038161003816100381610038161205731003816100381610038161003816 10038161003816100381610038161205741003816100381610038161003816 le 119873

10038161003816100381610038161003816120597120572120581119895119896ℓ(119909)10038161003816100381610038161003816≲ 2119895120575|120572|

forall |120572| ge 0

(15)

and 119877 is a bounded operator from 119871119901 times 119871119902 into 119871119903 for 1119901 +1119902 = 1119903 1 lt 119901 119902 119903 lt infin

Now if 120590119896ℓ

is any of the symbols in (14) and we knew apriori that119879

120590119896ℓ

are bounded from119871119901times119871119902 into119871119903 with operatornorms depending only on the implicit constants from (15)the fact that the sequence 119889

119896ℓ is absolutely convergent

immediately implies the 119871119901 times 119871119902 rarr 119871119903 boundedness of 119879

120590

Our first step has thus reduced the study of generic symbolsin the class BS0

1120575to symbols of the form

120590 (119909 120585 120578) =

infin

sum

119895=0

119898119895(119909) 120595 (2

minus119895120585 2minus119895120578) (16)

where ||120597120572119898119895||119871infin ≲ 2

119895120575|120572| and 120595 is supported in 13 le

max(|120585| |120578|) le 1Our second step is to further reduce the simpler looking

symbol given in (16) to a sum of bilinear elementary symbols

Lemma9 Let 120590 be as in (16) One can further reduce the studyto symbols of the form

120590 = 1205901+ 1205902+ 1205903 (17)

where the elementary symbols 120590119896 119896 = 1 2 3 are defined via

120590119896(119909 120585 120578) =

infin

sum

119895=0

119898119895(119909) 120593119896(2minus119895120585) 120594119896(2minus119895120578) (18)

with supp1205931sube 14 le |120585| le 2 supp120594

1sube |120578| le 18

1205933= 1205941 1205943= 1205931 supp120593

2 supp120594

2sube 120 le |120585| le 2 and

||120597120572119898119895||119871infin ≲ 2

119895120575|120572|

Therefore we are now only faced with the question ofboundedness for the two operators 119879

1205901

and 1198791205902

with 1205901and

1205902defined in Lemma 9 boundedness of 119879

1205903

follows from thatof 1198791205901

by symmetry In the following for 119891 119892 isin S we write

119895119896(120585) = 120593

119896(2minus119895120585) (120585)

119895119896(120578) = 120594

119896(2minus119895120578) (120578)

(19)

4 Journal of Function Spaces and Applications

In this case for 119896 = 1 2 we can write

119879120590119896

(119891 119892) (119909) =

infin

sum

119895=0

119898119895 (119909) 119891119895119896 (119909) 119892119895119896 (119909) (20)

Claim 1 1198791205902

is bounded from 119871119901 times 119871119902 into 119871119903

Proof By (20) and the Cauchy-Schwarz inequality we canwrite

100381610038161003816100381610038161198791205902

(119891 119892)10038161003816100381610038161003816le

infin

sum

119895=0

10038161003816100381610038161003816119898119895

10038161003816100381610038161003816

100381610038161003816100381610038161198911198952

10038161003816100381610038161003816

100381610038161003816100381610038161198921198952

10038161003816100381610038161003816

≲ (

infin

sum

119895=0

100381610038161003816100381610038161198911198952

10038161003816100381610038161003816

2

)

12

(

infin

sum

119895=0

100381610038161003816100381610038161198921198952

10038161003816100381610038161003816

2

)

12

(21)

We used here the fact that the coefficients 119898119895are bounded

see Lemma 9 Using now Holderrsquos inequality we have

100381710038171003817100381710038171198791205902

(119891 119892)10038171003817100381710038171003817119871119903≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

100381610038161003816100381610038161198911198952

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

100381610038161003816100381610038161198921198952

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119902

(22)

Finally the conditions on the supports of 1205932and 120594

2allow us

tomake use of the Littlewood-Paley theory and conclude that100381710038171003817100381710038171198791205902

(119891 119892)10038171003817100381710038171003817119871119903≲1003817100381710038171003817119891100381710038171003817100381711987111990110038171003817100381710038171198921003817100381710038171003817119871119902 (23)

Obtaining the boundedness of the operator 1198791205901

is a bitmore delicate due to the support condition on 120594

1 specifically

having supp1205941contained in a disk rather than in an annulus

Nevertheless we still claim the following

Claim 2 1198791205901

is bounded from 119871119901 times 119871119902 into 119871119903

Proof Note first that we can still use the Littlewood-Paleytheory on the 119891

1198951part of the sum that defines 119879

1205901

(119891 119892)(119909) =

suminfin

119895=0119898119895(119909)1198911198951(119909)1198921198951(119909) We have the following inequalities

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

100381610038161003816100381610038161198911198952

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲10038171003817100381710038171198911003817100381710038171003817119871119901

1003817100381710038171003817100381710038171003817100381710038171003817

sup119895ge0

|1198921198951|

1003817100381710038171003817100381710038171003817100381710038171003817119871119902

≲10038171003817100381710038171198921003817100381710038171003817119871119902

(24)

At this point however we must proceed more cautiouslyObserve that

supp11989111989511198921198951sub supp119891

1198951+ supp119892

1198951sub 2119895minus3le10038161003816100381610038161205851003816100381610038161003816 le 2119895+3

(25)

Denoting then ℎ1198951= 11989111989511198921198951 we now have 119879

1205901

(119891 119892)(119909) =

suminfin

119895=0119898119895ℎ1198951 where 120597120572119898

119895119871infin ≲ 2

119895120575|120572| and ℎ1198951

satisfies thesupport condition (25) Assume for the moment that thefollowing inequality holds

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119898119895ℎ1198951

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ1198951

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

(26)

Then the boundedness of the operator1198791205901

can be obtained asfollows

100381710038171003817100381710038171198791205901

(119891 119892)10038171003817100381710038171003817119871119903=

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119898119895ℎ1198951

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ1198951

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sup119895ge0

|1198921198951| sdot (

infin

sum

119895=0

100381610038161003816100381610038161198911198951

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

100381610038161003816100381610038161198911198951

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

1003817100381710038171003817100381710038171003817100381710038171003817

sup119895ge0

100381610038161003816100381610038161198921198951

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817100381710038171003817119871119902

≲1003817100381710038171003817119891100381710038171003817100381711987111990110038171003817100381710038171198921003817100381710038171003817119871119902

(27)

The proof of Claim 2 assumed the estimate (26) Our nextclaim is that (26) is indeed true

Claim 3 Assume that 120597120572119898119895119871infin ≲ 2

119895120575|120572| and supp ℎ119895sub

2119895minus3le |120585| le 2

119895+3 Then for all 119903 gt 1 we have

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119898119895ℎ119895

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

(28)

In our proof of this claim we will make use of Journersquoslemma [16 p 69]

Lemma 10 There exists a constant 119862 gt 0 such that for all119895 ge 0119898

119895= 119892119895+ 119887119895 where 119892

119895119871infin le 119862 119887

119895119871infin le 1198622

(120575minus1)119895 andsupp ℎ

119895119892119895sub 211989572 le |120585| le 9 sdot 2

119895

Proof of Claim 3 First we consider the case 119903 = 2 With thenotation in Lemma 10 it suffices to estimate suminfin

119895=0119887119895ℎ1198951198712

and suminfin119895=0119892119895ℎ1198951198712

The estimate on the ldquobad partrdquo follows from the triangleand Cauchy-Schwarz inequalities and the control 119887

119895119871infin ≲

2(120575minus1)119895 recall that 120575 lt 1

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119887119895ℎ119895

100381710038171003817100381710038171003817100381710038171003817100381710038171198712

le

infin

sum

119895=0

10038171003817100381710038171003817119887119895

10038171003817100381710038171003817119871infin

10038171003817100381710038171003817ℎ119895

100381710038171003817100381710038171198712

le (

infin

sum

119895=0

10038171003817100381710038171003817119887119895

10038171003817100381710038171003817

2

119871infin)

12

(

infin

sum

119895=0

10038171003817100381710038171003817ℎ119895

10038171003817100381710038171003817

2

1198712)

12

≲ (

infin

sum

119895=0

2(2120575minus2)119895

)

12

(

infin

sum

119895=0

10038171003817100381710038171003817ℎ119895

10038171003817100381710038171003817

2

1198712)

12

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

121003817100381710038171003817100381710038171003817100381710038171003817100381710038171198712

(29)

Journal of Function Spaces and Applications 5

Let us now look at the ldquogood partrdquo We start by noticing thatgiven the two summation indices 119895 119896 ge 0 we have

supp119892119895ℎ119895sub

2119895

72le10038161003816100381610038161205851003816100381610038161003816 le 9 sdot 2

119895

supp119892119896ℎ119896sub

2119896

72le10038161003816100381610038161205851003816100381610038161003816 le 9 sdot 2

119896

(30)

Thus for |119895 minus 119896| ge 11 we have supp119892119895ℎ119895cap supp119892

119896ℎ119896= 0

In view of this orthogonality Plancherelrsquos theorem with theestimate 119892

119895119871infin ≲ 1 gives

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119892119895ℎ119895

100381710038171003817100381710038171003817100381710038171003817100381710038171198712

le

11

sum

119894=0

1003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119896=0

119892119894+11119896

ℎ119894+11119896

10038171003817100381710038171003817100381710038171003817100381710038171198712

≲ (

infin

sum

119895=0

10038171003817100381710038171003817119892119895ℎ119895

10038171003817100381710038171003817

2

1198712)

12

le (

infin

sum

119895=0

10038171003817100381710038171003817119892119895

10038171003817100381710038171003817

2

119871infin

10038171003817100381710038171003817ℎ119895

10038171003817100381710038171003817

2

1198712)

12

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

121003817100381710038171003817100381710038171003817100381710038171003817100381710038171198712

(31)

This completes the proof of the case 119903 = 2 In the general case119903 gt 1 we again seek the control of the ldquobadrdquo and ldquogoodrdquo partsThe estimate on the ldquobadrdquo part follows virtually the same asin the case 119903 = 210038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119887119895ℎ119895

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

le

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816119887119895

10038161003816100381610038161003816

2

)

12

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

le

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816119887119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871infin

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

(32)

wherewe usedMinkowskirsquos integral inequality in the last stepFor the ldquogoodrdquo part we can think of 119892

119896ℎ119896as being dyadic

blocks in the Littlewood-Paley decomposition of the sum119878119894= sum119896equiv119894 ( mod 11) 119892119896ℎ119896 Thus it will be enough to control

uniformly (in the 119871119903 norm) the sums 119878119894 0 le 119894 le 11 in order

to obtain the same bound on sum119895ge0119892119895ℎ119895119871119903 The control on

119878119894however follows from the uniform estimate on the 119892

119896rsquos and

an immediate application of Littlewood-Paley theory

Acknowledgments

A Benyirsquos work is partially supported by a Grant fromthe Simons Foundation (no 246024) T Oh acknowledgessupport from an AMS-Simons Travel Grant

References

[1] R R Coifman and Y Meyer Au Dela des Operateurs Pseudo-Differentiels vol 57 of Asterisque 2nd edition 1978

[2] L Grafakos and R H Torres ldquoMultilinear Calderon-Zygmundtheoryrdquo Advances in Mathematics vol 165 no 1 pp 124ndash1642002

[3] C E Kenig and E M Stein ldquoMultilinear estimates and frac-tional integrationrdquo Mathematical Research Letters vol 6 no 1pp 1ndash15 1999

[4] A Benyi D Maldonado V Naibo and R H Torres ldquoOn theHormander classes of bilinear pseudodifferential operatorsrdquoIntegral Equations and Operator Theory vol 67 no 3 pp 341ndash364 2010

[5] F Bernicot D Maldonado K Moen and V Naibo ldquoBilinearSobolev-Poincare inequalities and Leibniz-type rulesrdquo Journalof Geometric Analysis In press httparxivorgpdf11043942pdf

[6] T Kato andG Ponce ldquoCommutator estimates and the Euler andNavier-Stokes equationsrdquoCommunications on Pure and AppliedMathematics vol 41 no 7 pp 891ndash907 1988

[7] C E Kenig G Ponce and L Vega ldquoOn unique continuationfor nonlinear Schrodinger equationsrdquo Communications on Pureand Applied Mathematics vol 56 no 9 pp 1247ndash1262 2003

[8] A Benyi A R Nahmod and R H Torres ldquoSobolev spaceestimates and symbolic calculus for bilinear pseudodifferentialoperatorsrdquo Journal of Geometric Analysis vol 16 no 3 pp 431ndash453 2006

[9] A Benyi and R H Torres ldquoAlmost orthogonality and a class ofbounded bilinear pseudodifferential operatorsrdquo MathematicalResearch Letters vol 11 no 1 pp 1ndash11 2004

[10] F Bernicot and S Shrivastava ldquoBoundedness of smooth bilinearsquare functions and applications to some bilinear pseudo-differential operatorsrdquo Indiana University Mathematics Journalvol 60 no 1 pp 233ndash268 2011

[11] K Grochenig Foundations of Time-Frequency AnalysisBirkhauser Boston Mass USA 2001

[12] A Benyi K Grochenig C Heil and K Okoudjou ldquoModulationspaces and a class of bounded multilinear pseudodifferentialoperatorsrdquo Journal ofOperatorTheory vol 54 pp 301ndash313 2005

[13] A Benyi F Bernicot D Maldonado V Naibo and R H Tor-res ldquoOn the Hormander classes of bilinear pseudodifferentialoperators IIrdquo Indiana University Mathematics Journal In presshttparxivorgpdf11120486pdf

[14] A Miyachi and N Tomita ldquoCalderon-Vaillancourt type theo-rem for bilinear pseudo-differential operatorsrdquo Indiana Univer-sity Mathematics Journal In press

[15] A Benyi and R H Torres ldquoSymbolic calculus and the trans-poses of bilinear pseudodifferential operatorsrdquo Communica-tions in Partial Differential Equations vol 28 no 5-6 pp 1161ndash1181 2003

[16] J-L Journe Calderon-Zygmund Operators Pseudo-DifferentialOperators and the Cauchy Integral of Calderon vol 994 ofLecture Notes in Mathematics Springer Berlin Germany 1983

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

and the existence of a transposition symbolic calculus provedby Benyi et al [4] Let us briefly give an outline of how thisfollows First we note that the bilinear kernels associated tobilinear operators with symbols in BS0

1120575 0 le 120575 lt 1 are

bilinear Calderon-Zygmund operators in the sense of [2]Second we recall that [2 Corollary 1] which is an applicationof the bilinear 119879(1) theorem therein states the following

Theorem B If 119879 and its transposes 119879lowast1 and 119879

lowast2 havesymbols in BS0

11 then they can be extended as bounded

operators from 119871119901times 119871119902 into 119871119903 for 1 lt 119901 119902 lt infin and

1119901 + 1119902 = 1119903

Third by [4 Theorem 21] we have the following

Theorem C Assume that 0 le 120575 le 120588 le 1 120575 lt 1 and 120590 isinBS119898120588120575 Then for 119895 = 1 2119879lowast119895

120590= 119879120590lowast119895 where 120590lowast119895 isin BS119898

120588120575

Finally since BS01120575

sub BS011 we can directly combine

Theorems B and C to recover the following optimal extensionof the Coifman-Meyer result note that now the symbol isallowed to depend on 119909 while 119903 is still allowed to be in theoptimal interval (12infin)

Theorem 1 If 120590 is a symbol in BS01120575 0 le 120575 lt 1 then 119879

120590has a

bounded extension from 119871119901 times 119871119902 into 119871119903 for all 1 lt 119901 119902 lt infinsuch that 1119901 + 1119902 = 1119903

Once we have the boundedness of the class BS01120575

onproducts of Lebesgue spaces a ldquoreduction methodrdquo allows usto deduce also the boundedness of the class BS119898

1120575on appro-

priate products of Sobolev spaces Moreover our estimatesin this case come in the form of Leibniz-type rules formore on these kinds of properties see the work of Bernicotet al [5] In the particular case when the bilinear operatoris just a differential operator the Leibniz-type rules arereferred to as Kato-Poncersquos commutator estimates and areknown to play a significant role in the study of the Eulerand Navier-Stokes equations see [6] see also Kenig et al[7] for further applications of commutators to nonlinearSchrodinger equations Let 119869119898 = (119868 minus Δ)

1198982 denote thelinear Fourier multiplier operator with symbol ⟨120585⟩119898 where⟨120585⟩ = (1 + |120585|

2)12 By definition we say that 119891 belongs to the

Sobolev space 119871119901119898if 119869119898119891 isin 119871119901 We have the following

Theorem 2 Let 120590 be a symbol in BS1198981120575 0 le 120575 lt 1 119898 ge 0

and let 119879120590be its associated operator Then there exist symbols

1205901and 120590

2in BS01120575

such that for all 119891 119892 isin S

119879120590(119891 119892) = 119879

1205901

(119869119898119891 119892) + 119879

1205902

(119891 119869119898119892) (4)

In particular then one has that 119879120590has a bounded extension

from 119871119901119898times 119871119902

119898into 119871119903 provided that 1119901 + 1119902 = 1119903 1 lt 119901

119902 lt infin Moreover1003817100381710038171003817119879120590 (119891 119892)

1003817100381710038171003817119871119903≲10038171003817100381710038171198911003817100381710038171003817119871119901

119898

10038171003817100381710038171198921003817100381710038171003817119871119902+1003817100381710038171003817119891100381710038171003817100381711987111990110038171003817100381710038171198921003817100381710038171003817119871119902

119898

(5)

Theproof ofTheorem2 follows a similar path as the one inthe work of Benyi et al [8Theorem 27] For the convenience

of the reader we sketch here the main steps in the argumentLet 120601 be a 119862infin-function on R such that 0 le 120601 le 1 supp120601 sub [minus2 2] and 120601(119903) + 120601(1119903) = 1 on [0infin) Then (4) holdsif we let

1205901(119909 120585 120578) = 120590 (119909 120585 120578) 120601(

⟨120578⟩2

⟨120585⟩2)⟨120585⟩

minus119898

1205902(119909 120585 120578) = 120590 (119909 120585 120578) 120601(

⟨120585⟩2

⟨120578⟩2)⟨120578⟩

minus119898

(6)

Now straightforward calculations that take into account thesupport condition on 120601 given that 120590

1and 120590

2belong to BS0

1120575

The Leibniz-type estimate (5) follows now from Theorem 1and (4)

It is also worthwhile to note that we can replace (5) witha more general Leibniz-type rule of the form

1003817100381710038171003817119879120590(119891 119892)1003817100381710038171003817119871119903≲100381710038171003817100381711989110038171003817100381710038171198711199011

119898

100381710038171003817100381711989210038171003817100381710038171198711199021

+100381710038171003817100381711989210038171003817100381710038171198711199012

119898

100381710038171003817100381711989110038171003817100381710038171198711199022 (7)

where 11199011+11199021= 1119901

2+11199022= 1119903 1 lt 119901

11199012 1199021 1199022lt infin

One of the main reasons for the study of the Hormanderclasses of bilinear pseudodifferential operators is the factthat the conditions imposed on the symbols arise naturallyin PDEs In particular the bilinear Hormader classes BS119898

120588120575

model the product of two functions and their derivatives

Example 3 Consider first a bilinear partial differential oper-ator with variable coefficients

119863119896ℓ(119891 119892) = sum

|120573|le119896

sum

|120574|leℓ

119888120573120574(119909)

120597120573119891

120597119909120573

120597120574119892

120597119909120574 (8)

Note that119863119896ℓ= 119879120590119896ℓ

where the bilinear symbol is given by

120590119896ℓ(119909 120585 120578) = (2120587)

minus2119899sum

120573120574

119888120573120574(119909) (119894120585)

120573(119894120578)120574 (9)

Assuming that the coefficients 119888120573120574

have bounded derivativesit is easy to show that 120590

119896ℓisin BS119896+ℓ10

Example 4 The symbol in the previous example is almostequivalent to a multiplier of the form

120590119898(120585 120578) = (1 +

10038161003816100381610038161205851003816100381610038161003816

2+10038161003816100381610038161205781003816100381610038161003816

2)1198982

(10)

Indeed this symbol belongs to BS11989810We can also think of this

symbol as the bilinear counterpart of the multiplier ⟨120585⟩119898 thatdefines the linear operator 119869119898

Example 5 With the notation in Example 4 the multipliers120585120590minus1(120585 120578) and 120578120590

minus1(120585 120578) belong to BS0

10 In general the

multipliers 120590119896+ℓ(120585 120578) = 120585

119896120578ℓ120590minus1(120585 120578) belong to BS119896+ℓ

10

Example 6 One of the recurrent techniques in PDE estimatesis to truncate a given multiplier at the right scale Considernow

120590 (120585 120578) = 120590119898(120585 120578) sum

119886119887isinN

119888119886119887(119909) 120593 (2

minus119886120585) 120594 (2

minus119887120578) (11)

Journal of Function Spaces and Applications 3

where120593 and120594 are smooth ldquocutoffrdquo functions supported in theannulus 12 le |120585| le 2 and the coefficients satisfy derivativeestimates of the form

1003817100381710038171003817120597120572

119909119888119886119887(119909)1003817100381710038171003817119871infin

≲ 2120575|120572|max(119886119887)

(12)

Elementary calculations show that 120590 isin BS1198981120575

Remark 7 Theorems 1 and 2 lead to the natural questionabout the boundedness properties of other Hormanderclasses of bilinear pseudodifferential operators An interest-ing situation arises when we consider the bilinear Calderon-Vaillancourt class BS0

00 A result of Benyi andTorres [9] shows

that in this case the 119871119901 times 119871119902 rarr 119871119903 boundedness fails

One can impose some additional conditions (besides being inBS000) on a symbol to guarantee that the corresponding bilin-

ear pseudodifferential operator is 119871119901 times 119871119902 rarr 119871119903 bounded

see for example [9] and the recent work of Bernicot andShrivastava [10] However there is a nice substitute for theLebesgue space estimates If we consider instead modulationspaces119872119901119902 (see the excellent book byGrochenig [11] for theirdefinition and basic properties) we can show for examplethat if 120590 isin BS0

00then 119879

120590 1198712times 1198712rarr 119872

1infin (which contains1198711) This and other more general boundedness results on

modulation spaces for the class BS000

were obtained by Benyiet al [12] Then this particular boundedness result with thereduction method employed in Theorem 2 allows us to alsoobtain the boundedness of the class BS119898

00from 1198712

119898times 1198712

119898into

1198721infin Interestingly we can also obtain the 119871119901 times 119871119902 rarr 119871

119903

boundedness of the class BS11989800 but we have to require in this

case the order 119898 to depend on the Lebesgue exponents seethe work of Benyi et al [13] also Miyachi and Tomita [14] forthe optimality of the order 119898 and the extension of the resultin [13] below 119903 = 1 The most general case of the classes BS119898

120588120575

is also given in [13]

In the remainder of the paper we will provide an alternateproof ofTheorem 1 that does not use sophisticated tools suchas the symbolic calculus The proof is in the original spiritof the work of Coifman and Meyer that made use of theLittlewood-Paley theory As such we will only be concernedhere with the boundedness into the target space 119871119903 with 119903 gt1 Of course obtaining the full result for 119903 gt 12 is thenpossible because of the bilinear Calderon-Zygmund theorywhich applies to our case We will borrow some of the ideasfrom Benyi and Torres [15] which in turn go back to the niceexposition (in the linear case) by Journe [16] by making useof the so-called bilinear elementary symbols

2 Proof of Theorem 1

We start with two lemmas that provide the anticipateddecomposition of our symbol into bilinear elementary sym-bols Since they are the immediate counterparts of [15Lemma 1 and Lemma 2] to our class BS0

1120575 we will skip their

proofs see also [16 pp 72ndash75] and [1 pp 55ndash57] The firstreduction is as follows

Lemma 8 Fix a symbol 120590 in the class BS01120575 0 le 120575 lt 1 and

an arbitrary large positive integer 119873 Then for any 119891 119892 isin S119879120590(119891 119892) can be written in the form

119879120590(119891 119892) = sum

119896ℓisinZ119899

119889119896ℓ119879120590119896ℓ

(119891 119892) + 119877 (119891 119892) (13)

where 119889119896ℓ is an absolutely convergent sequence of numbers

120590119896ℓ(119909 120585 120578) =

infin

sum

119895=0

120581119895119896ℓ(119909) 120595119896ℓ(2minus119895120585 2minus119895120578) (14)

with each 120595119896ℓ

a 119862infin-function supported on the set 13 le

max (|120585| |120578|) le 1100381610038161003816100381610038161003816120597120573

120585120597120574

120578120595119896ℓ(120585 120578)

100381610038161003816100381610038161003816≲ 1 forall

10038161003816100381610038161205731003816100381610038161003816 10038161003816100381610038161205741003816100381610038161003816 le 119873

10038161003816100381610038161003816120597120572120581119895119896ℓ(119909)10038161003816100381610038161003816≲ 2119895120575|120572|

forall |120572| ge 0

(15)

and 119877 is a bounded operator from 119871119901 times 119871119902 into 119871119903 for 1119901 +1119902 = 1119903 1 lt 119901 119902 119903 lt infin

Now if 120590119896ℓ

is any of the symbols in (14) and we knew apriori that119879

120590119896ℓ

are bounded from119871119901times119871119902 into119871119903 with operatornorms depending only on the implicit constants from (15)the fact that the sequence 119889

119896ℓ is absolutely convergent

immediately implies the 119871119901 times 119871119902 rarr 119871119903 boundedness of 119879

120590

Our first step has thus reduced the study of generic symbolsin the class BS0

1120575to symbols of the form

120590 (119909 120585 120578) =

infin

sum

119895=0

119898119895(119909) 120595 (2

minus119895120585 2minus119895120578) (16)

where ||120597120572119898119895||119871infin ≲ 2

119895120575|120572| and 120595 is supported in 13 le

max(|120585| |120578|) le 1Our second step is to further reduce the simpler looking

symbol given in (16) to a sum of bilinear elementary symbols

Lemma9 Let 120590 be as in (16) One can further reduce the studyto symbols of the form

120590 = 1205901+ 1205902+ 1205903 (17)

where the elementary symbols 120590119896 119896 = 1 2 3 are defined via

120590119896(119909 120585 120578) =

infin

sum

119895=0

119898119895(119909) 120593119896(2minus119895120585) 120594119896(2minus119895120578) (18)

with supp1205931sube 14 le |120585| le 2 supp120594

1sube |120578| le 18

1205933= 1205941 1205943= 1205931 supp120593

2 supp120594

2sube 120 le |120585| le 2 and

||120597120572119898119895||119871infin ≲ 2

119895120575|120572|

Therefore we are now only faced with the question ofboundedness for the two operators 119879

1205901

and 1198791205902

with 1205901and

1205902defined in Lemma 9 boundedness of 119879

1205903

follows from thatof 1198791205901

by symmetry In the following for 119891 119892 isin S we write

119895119896(120585) = 120593

119896(2minus119895120585) (120585)

119895119896(120578) = 120594

119896(2minus119895120578) (120578)

(19)

4 Journal of Function Spaces and Applications

In this case for 119896 = 1 2 we can write

119879120590119896

(119891 119892) (119909) =

infin

sum

119895=0

119898119895 (119909) 119891119895119896 (119909) 119892119895119896 (119909) (20)

Claim 1 1198791205902

is bounded from 119871119901 times 119871119902 into 119871119903

Proof By (20) and the Cauchy-Schwarz inequality we canwrite

100381610038161003816100381610038161198791205902

(119891 119892)10038161003816100381610038161003816le

infin

sum

119895=0

10038161003816100381610038161003816119898119895

10038161003816100381610038161003816

100381610038161003816100381610038161198911198952

10038161003816100381610038161003816

100381610038161003816100381610038161198921198952

10038161003816100381610038161003816

≲ (

infin

sum

119895=0

100381610038161003816100381610038161198911198952

10038161003816100381610038161003816

2

)

12

(

infin

sum

119895=0

100381610038161003816100381610038161198921198952

10038161003816100381610038161003816

2

)

12

(21)

We used here the fact that the coefficients 119898119895are bounded

see Lemma 9 Using now Holderrsquos inequality we have

100381710038171003817100381710038171198791205902

(119891 119892)10038171003817100381710038171003817119871119903≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

100381610038161003816100381610038161198911198952

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

100381610038161003816100381610038161198921198952

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119902

(22)

Finally the conditions on the supports of 1205932and 120594

2allow us

tomake use of the Littlewood-Paley theory and conclude that100381710038171003817100381710038171198791205902

(119891 119892)10038171003817100381710038171003817119871119903≲1003817100381710038171003817119891100381710038171003817100381711987111990110038171003817100381710038171198921003817100381710038171003817119871119902 (23)

Obtaining the boundedness of the operator 1198791205901

is a bitmore delicate due to the support condition on 120594

1 specifically

having supp1205941contained in a disk rather than in an annulus

Nevertheless we still claim the following

Claim 2 1198791205901

is bounded from 119871119901 times 119871119902 into 119871119903

Proof Note first that we can still use the Littlewood-Paleytheory on the 119891

1198951part of the sum that defines 119879

1205901

(119891 119892)(119909) =

suminfin

119895=0119898119895(119909)1198911198951(119909)1198921198951(119909) We have the following inequalities

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

100381610038161003816100381610038161198911198952

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲10038171003817100381710038171198911003817100381710038171003817119871119901

1003817100381710038171003817100381710038171003817100381710038171003817

sup119895ge0

|1198921198951|

1003817100381710038171003817100381710038171003817100381710038171003817119871119902

≲10038171003817100381710038171198921003817100381710038171003817119871119902

(24)

At this point however we must proceed more cautiouslyObserve that

supp11989111989511198921198951sub supp119891

1198951+ supp119892

1198951sub 2119895minus3le10038161003816100381610038161205851003816100381610038161003816 le 2119895+3

(25)

Denoting then ℎ1198951= 11989111989511198921198951 we now have 119879

1205901

(119891 119892)(119909) =

suminfin

119895=0119898119895ℎ1198951 where 120597120572119898

119895119871infin ≲ 2

119895120575|120572| and ℎ1198951

satisfies thesupport condition (25) Assume for the moment that thefollowing inequality holds

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119898119895ℎ1198951

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ1198951

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

(26)

Then the boundedness of the operator1198791205901

can be obtained asfollows

100381710038171003817100381710038171198791205901

(119891 119892)10038171003817100381710038171003817119871119903=

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119898119895ℎ1198951

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ1198951

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sup119895ge0

|1198921198951| sdot (

infin

sum

119895=0

100381610038161003816100381610038161198911198951

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

100381610038161003816100381610038161198911198951

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

1003817100381710038171003817100381710038171003817100381710038171003817

sup119895ge0

100381610038161003816100381610038161198921198951

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817100381710038171003817119871119902

≲1003817100381710038171003817119891100381710038171003817100381711987111990110038171003817100381710038171198921003817100381710038171003817119871119902

(27)

The proof of Claim 2 assumed the estimate (26) Our nextclaim is that (26) is indeed true

Claim 3 Assume that 120597120572119898119895119871infin ≲ 2

119895120575|120572| and supp ℎ119895sub

2119895minus3le |120585| le 2

119895+3 Then for all 119903 gt 1 we have

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119898119895ℎ119895

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

(28)

In our proof of this claim we will make use of Journersquoslemma [16 p 69]

Lemma 10 There exists a constant 119862 gt 0 such that for all119895 ge 0119898

119895= 119892119895+ 119887119895 where 119892

119895119871infin le 119862 119887

119895119871infin le 1198622

(120575minus1)119895 andsupp ℎ

119895119892119895sub 211989572 le |120585| le 9 sdot 2

119895

Proof of Claim 3 First we consider the case 119903 = 2 With thenotation in Lemma 10 it suffices to estimate suminfin

119895=0119887119895ℎ1198951198712

and suminfin119895=0119892119895ℎ1198951198712

The estimate on the ldquobad partrdquo follows from the triangleand Cauchy-Schwarz inequalities and the control 119887

119895119871infin ≲

2(120575minus1)119895 recall that 120575 lt 1

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119887119895ℎ119895

100381710038171003817100381710038171003817100381710038171003817100381710038171198712

le

infin

sum

119895=0

10038171003817100381710038171003817119887119895

10038171003817100381710038171003817119871infin

10038171003817100381710038171003817ℎ119895

100381710038171003817100381710038171198712

le (

infin

sum

119895=0

10038171003817100381710038171003817119887119895

10038171003817100381710038171003817

2

119871infin)

12

(

infin

sum

119895=0

10038171003817100381710038171003817ℎ119895

10038171003817100381710038171003817

2

1198712)

12

≲ (

infin

sum

119895=0

2(2120575minus2)119895

)

12

(

infin

sum

119895=0

10038171003817100381710038171003817ℎ119895

10038171003817100381710038171003817

2

1198712)

12

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

121003817100381710038171003817100381710038171003817100381710038171003817100381710038171198712

(29)

Journal of Function Spaces and Applications 5

Let us now look at the ldquogood partrdquo We start by noticing thatgiven the two summation indices 119895 119896 ge 0 we have

supp119892119895ℎ119895sub

2119895

72le10038161003816100381610038161205851003816100381610038161003816 le 9 sdot 2

119895

supp119892119896ℎ119896sub

2119896

72le10038161003816100381610038161205851003816100381610038161003816 le 9 sdot 2

119896

(30)

Thus for |119895 minus 119896| ge 11 we have supp119892119895ℎ119895cap supp119892

119896ℎ119896= 0

In view of this orthogonality Plancherelrsquos theorem with theestimate 119892

119895119871infin ≲ 1 gives

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119892119895ℎ119895

100381710038171003817100381710038171003817100381710038171003817100381710038171198712

le

11

sum

119894=0

1003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119896=0

119892119894+11119896

ℎ119894+11119896

10038171003817100381710038171003817100381710038171003817100381710038171198712

≲ (

infin

sum

119895=0

10038171003817100381710038171003817119892119895ℎ119895

10038171003817100381710038171003817

2

1198712)

12

le (

infin

sum

119895=0

10038171003817100381710038171003817119892119895

10038171003817100381710038171003817

2

119871infin

10038171003817100381710038171003817ℎ119895

10038171003817100381710038171003817

2

1198712)

12

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

121003817100381710038171003817100381710038171003817100381710038171003817100381710038171198712

(31)

This completes the proof of the case 119903 = 2 In the general case119903 gt 1 we again seek the control of the ldquobadrdquo and ldquogoodrdquo partsThe estimate on the ldquobadrdquo part follows virtually the same asin the case 119903 = 210038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119887119895ℎ119895

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

le

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816119887119895

10038161003816100381610038161003816

2

)

12

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

le

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816119887119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871infin

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

(32)

wherewe usedMinkowskirsquos integral inequality in the last stepFor the ldquogoodrdquo part we can think of 119892

119896ℎ119896as being dyadic

blocks in the Littlewood-Paley decomposition of the sum119878119894= sum119896equiv119894 ( mod 11) 119892119896ℎ119896 Thus it will be enough to control

uniformly (in the 119871119903 norm) the sums 119878119894 0 le 119894 le 11 in order

to obtain the same bound on sum119895ge0119892119895ℎ119895119871119903 The control on

119878119894however follows from the uniform estimate on the 119892

119896rsquos and

an immediate application of Littlewood-Paley theory

Acknowledgments

A Benyirsquos work is partially supported by a Grant fromthe Simons Foundation (no 246024) T Oh acknowledgessupport from an AMS-Simons Travel Grant

References

[1] R R Coifman and Y Meyer Au Dela des Operateurs Pseudo-Differentiels vol 57 of Asterisque 2nd edition 1978

[2] L Grafakos and R H Torres ldquoMultilinear Calderon-Zygmundtheoryrdquo Advances in Mathematics vol 165 no 1 pp 124ndash1642002

[3] C E Kenig and E M Stein ldquoMultilinear estimates and frac-tional integrationrdquo Mathematical Research Letters vol 6 no 1pp 1ndash15 1999

[4] A Benyi D Maldonado V Naibo and R H Torres ldquoOn theHormander classes of bilinear pseudodifferential operatorsrdquoIntegral Equations and Operator Theory vol 67 no 3 pp 341ndash364 2010

[5] F Bernicot D Maldonado K Moen and V Naibo ldquoBilinearSobolev-Poincare inequalities and Leibniz-type rulesrdquo Journalof Geometric Analysis In press httparxivorgpdf11043942pdf

[6] T Kato andG Ponce ldquoCommutator estimates and the Euler andNavier-Stokes equationsrdquoCommunications on Pure and AppliedMathematics vol 41 no 7 pp 891ndash907 1988

[7] C E Kenig G Ponce and L Vega ldquoOn unique continuationfor nonlinear Schrodinger equationsrdquo Communications on Pureand Applied Mathematics vol 56 no 9 pp 1247ndash1262 2003

[8] A Benyi A R Nahmod and R H Torres ldquoSobolev spaceestimates and symbolic calculus for bilinear pseudodifferentialoperatorsrdquo Journal of Geometric Analysis vol 16 no 3 pp 431ndash453 2006

[9] A Benyi and R H Torres ldquoAlmost orthogonality and a class ofbounded bilinear pseudodifferential operatorsrdquo MathematicalResearch Letters vol 11 no 1 pp 1ndash11 2004

[10] F Bernicot and S Shrivastava ldquoBoundedness of smooth bilinearsquare functions and applications to some bilinear pseudo-differential operatorsrdquo Indiana University Mathematics Journalvol 60 no 1 pp 233ndash268 2011

[11] K Grochenig Foundations of Time-Frequency AnalysisBirkhauser Boston Mass USA 2001

[12] A Benyi K Grochenig C Heil and K Okoudjou ldquoModulationspaces and a class of bounded multilinear pseudodifferentialoperatorsrdquo Journal ofOperatorTheory vol 54 pp 301ndash313 2005

[13] A Benyi F Bernicot D Maldonado V Naibo and R H Tor-res ldquoOn the Hormander classes of bilinear pseudodifferentialoperators IIrdquo Indiana University Mathematics Journal In presshttparxivorgpdf11120486pdf

[14] A Miyachi and N Tomita ldquoCalderon-Vaillancourt type theo-rem for bilinear pseudo-differential operatorsrdquo Indiana Univer-sity Mathematics Journal In press

[15] A Benyi and R H Torres ldquoSymbolic calculus and the trans-poses of bilinear pseudodifferential operatorsrdquo Communica-tions in Partial Differential Equations vol 28 no 5-6 pp 1161ndash1181 2003

[16] J-L Journe Calderon-Zygmund Operators Pseudo-DifferentialOperators and the Cauchy Integral of Calderon vol 994 ofLecture Notes in Mathematics Springer Berlin Germany 1983

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 3

where120593 and120594 are smooth ldquocutoffrdquo functions supported in theannulus 12 le |120585| le 2 and the coefficients satisfy derivativeestimates of the form

1003817100381710038171003817120597120572

119909119888119886119887(119909)1003817100381710038171003817119871infin

≲ 2120575|120572|max(119886119887)

(12)

Elementary calculations show that 120590 isin BS1198981120575

Remark 7 Theorems 1 and 2 lead to the natural questionabout the boundedness properties of other Hormanderclasses of bilinear pseudodifferential operators An interest-ing situation arises when we consider the bilinear Calderon-Vaillancourt class BS0

00 A result of Benyi andTorres [9] shows

that in this case the 119871119901 times 119871119902 rarr 119871119903 boundedness fails

One can impose some additional conditions (besides being inBS000) on a symbol to guarantee that the corresponding bilin-

ear pseudodifferential operator is 119871119901 times 119871119902 rarr 119871119903 bounded

see for example [9] and the recent work of Bernicot andShrivastava [10] However there is a nice substitute for theLebesgue space estimates If we consider instead modulationspaces119872119901119902 (see the excellent book byGrochenig [11] for theirdefinition and basic properties) we can show for examplethat if 120590 isin BS0

00then 119879

120590 1198712times 1198712rarr 119872

1infin (which contains1198711) This and other more general boundedness results on

modulation spaces for the class BS000

were obtained by Benyiet al [12] Then this particular boundedness result with thereduction method employed in Theorem 2 allows us to alsoobtain the boundedness of the class BS119898

00from 1198712

119898times 1198712

119898into

1198721infin Interestingly we can also obtain the 119871119901 times 119871119902 rarr 119871

119903

boundedness of the class BS11989800 but we have to require in this

case the order 119898 to depend on the Lebesgue exponents seethe work of Benyi et al [13] also Miyachi and Tomita [14] forthe optimality of the order 119898 and the extension of the resultin [13] below 119903 = 1 The most general case of the classes BS119898

120588120575

is also given in [13]

In the remainder of the paper we will provide an alternateproof ofTheorem 1 that does not use sophisticated tools suchas the symbolic calculus The proof is in the original spiritof the work of Coifman and Meyer that made use of theLittlewood-Paley theory As such we will only be concernedhere with the boundedness into the target space 119871119903 with 119903 gt1 Of course obtaining the full result for 119903 gt 12 is thenpossible because of the bilinear Calderon-Zygmund theorywhich applies to our case We will borrow some of the ideasfrom Benyi and Torres [15] which in turn go back to the niceexposition (in the linear case) by Journe [16] by making useof the so-called bilinear elementary symbols

2 Proof of Theorem 1

We start with two lemmas that provide the anticipateddecomposition of our symbol into bilinear elementary sym-bols Since they are the immediate counterparts of [15Lemma 1 and Lemma 2] to our class BS0

1120575 we will skip their

proofs see also [16 pp 72ndash75] and [1 pp 55ndash57] The firstreduction is as follows

Lemma 8 Fix a symbol 120590 in the class BS01120575 0 le 120575 lt 1 and

an arbitrary large positive integer 119873 Then for any 119891 119892 isin S119879120590(119891 119892) can be written in the form

119879120590(119891 119892) = sum

119896ℓisinZ119899

119889119896ℓ119879120590119896ℓ

(119891 119892) + 119877 (119891 119892) (13)

where 119889119896ℓ is an absolutely convergent sequence of numbers

120590119896ℓ(119909 120585 120578) =

infin

sum

119895=0

120581119895119896ℓ(119909) 120595119896ℓ(2minus119895120585 2minus119895120578) (14)

with each 120595119896ℓ

a 119862infin-function supported on the set 13 le

max (|120585| |120578|) le 1100381610038161003816100381610038161003816120597120573

120585120597120574

120578120595119896ℓ(120585 120578)

100381610038161003816100381610038161003816≲ 1 forall

10038161003816100381610038161205731003816100381610038161003816 10038161003816100381610038161205741003816100381610038161003816 le 119873

10038161003816100381610038161003816120597120572120581119895119896ℓ(119909)10038161003816100381610038161003816≲ 2119895120575|120572|

forall |120572| ge 0

(15)

and 119877 is a bounded operator from 119871119901 times 119871119902 into 119871119903 for 1119901 +1119902 = 1119903 1 lt 119901 119902 119903 lt infin

Now if 120590119896ℓ

is any of the symbols in (14) and we knew apriori that119879

120590119896ℓ

are bounded from119871119901times119871119902 into119871119903 with operatornorms depending only on the implicit constants from (15)the fact that the sequence 119889

119896ℓ is absolutely convergent

immediately implies the 119871119901 times 119871119902 rarr 119871119903 boundedness of 119879

120590

Our first step has thus reduced the study of generic symbolsin the class BS0

1120575to symbols of the form

120590 (119909 120585 120578) =

infin

sum

119895=0

119898119895(119909) 120595 (2

minus119895120585 2minus119895120578) (16)

where ||120597120572119898119895||119871infin ≲ 2

119895120575|120572| and 120595 is supported in 13 le

max(|120585| |120578|) le 1Our second step is to further reduce the simpler looking

symbol given in (16) to a sum of bilinear elementary symbols

Lemma9 Let 120590 be as in (16) One can further reduce the studyto symbols of the form

120590 = 1205901+ 1205902+ 1205903 (17)

where the elementary symbols 120590119896 119896 = 1 2 3 are defined via

120590119896(119909 120585 120578) =

infin

sum

119895=0

119898119895(119909) 120593119896(2minus119895120585) 120594119896(2minus119895120578) (18)

with supp1205931sube 14 le |120585| le 2 supp120594

1sube |120578| le 18

1205933= 1205941 1205943= 1205931 supp120593

2 supp120594

2sube 120 le |120585| le 2 and

||120597120572119898119895||119871infin ≲ 2

119895120575|120572|

Therefore we are now only faced with the question ofboundedness for the two operators 119879

1205901

and 1198791205902

with 1205901and

1205902defined in Lemma 9 boundedness of 119879

1205903

follows from thatof 1198791205901

by symmetry In the following for 119891 119892 isin S we write

119895119896(120585) = 120593

119896(2minus119895120585) (120585)

119895119896(120578) = 120594

119896(2minus119895120578) (120578)

(19)

4 Journal of Function Spaces and Applications

In this case for 119896 = 1 2 we can write

119879120590119896

(119891 119892) (119909) =

infin

sum

119895=0

119898119895 (119909) 119891119895119896 (119909) 119892119895119896 (119909) (20)

Claim 1 1198791205902

is bounded from 119871119901 times 119871119902 into 119871119903

Proof By (20) and the Cauchy-Schwarz inequality we canwrite

100381610038161003816100381610038161198791205902

(119891 119892)10038161003816100381610038161003816le

infin

sum

119895=0

10038161003816100381610038161003816119898119895

10038161003816100381610038161003816

100381610038161003816100381610038161198911198952

10038161003816100381610038161003816

100381610038161003816100381610038161198921198952

10038161003816100381610038161003816

≲ (

infin

sum

119895=0

100381610038161003816100381610038161198911198952

10038161003816100381610038161003816

2

)

12

(

infin

sum

119895=0

100381610038161003816100381610038161198921198952

10038161003816100381610038161003816

2

)

12

(21)

We used here the fact that the coefficients 119898119895are bounded

see Lemma 9 Using now Holderrsquos inequality we have

100381710038171003817100381710038171198791205902

(119891 119892)10038171003817100381710038171003817119871119903≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

100381610038161003816100381610038161198911198952

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

100381610038161003816100381610038161198921198952

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119902

(22)

Finally the conditions on the supports of 1205932and 120594

2allow us

tomake use of the Littlewood-Paley theory and conclude that100381710038171003817100381710038171198791205902

(119891 119892)10038171003817100381710038171003817119871119903≲1003817100381710038171003817119891100381710038171003817100381711987111990110038171003817100381710038171198921003817100381710038171003817119871119902 (23)

Obtaining the boundedness of the operator 1198791205901

is a bitmore delicate due to the support condition on 120594

1 specifically

having supp1205941contained in a disk rather than in an annulus

Nevertheless we still claim the following

Claim 2 1198791205901

is bounded from 119871119901 times 119871119902 into 119871119903

Proof Note first that we can still use the Littlewood-Paleytheory on the 119891

1198951part of the sum that defines 119879

1205901

(119891 119892)(119909) =

suminfin

119895=0119898119895(119909)1198911198951(119909)1198921198951(119909) We have the following inequalities

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

100381610038161003816100381610038161198911198952

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲10038171003817100381710038171198911003817100381710038171003817119871119901

1003817100381710038171003817100381710038171003817100381710038171003817

sup119895ge0

|1198921198951|

1003817100381710038171003817100381710038171003817100381710038171003817119871119902

≲10038171003817100381710038171198921003817100381710038171003817119871119902

(24)

At this point however we must proceed more cautiouslyObserve that

supp11989111989511198921198951sub supp119891

1198951+ supp119892

1198951sub 2119895minus3le10038161003816100381610038161205851003816100381610038161003816 le 2119895+3

(25)

Denoting then ℎ1198951= 11989111989511198921198951 we now have 119879

1205901

(119891 119892)(119909) =

suminfin

119895=0119898119895ℎ1198951 where 120597120572119898

119895119871infin ≲ 2

119895120575|120572| and ℎ1198951

satisfies thesupport condition (25) Assume for the moment that thefollowing inequality holds

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119898119895ℎ1198951

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ1198951

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

(26)

Then the boundedness of the operator1198791205901

can be obtained asfollows

100381710038171003817100381710038171198791205901

(119891 119892)10038171003817100381710038171003817119871119903=

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119898119895ℎ1198951

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ1198951

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sup119895ge0

|1198921198951| sdot (

infin

sum

119895=0

100381610038161003816100381610038161198911198951

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

100381610038161003816100381610038161198911198951

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

1003817100381710038171003817100381710038171003817100381710038171003817

sup119895ge0

100381610038161003816100381610038161198921198951

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817100381710038171003817119871119902

≲1003817100381710038171003817119891100381710038171003817100381711987111990110038171003817100381710038171198921003817100381710038171003817119871119902

(27)

The proof of Claim 2 assumed the estimate (26) Our nextclaim is that (26) is indeed true

Claim 3 Assume that 120597120572119898119895119871infin ≲ 2

119895120575|120572| and supp ℎ119895sub

2119895minus3le |120585| le 2

119895+3 Then for all 119903 gt 1 we have

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119898119895ℎ119895

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

(28)

In our proof of this claim we will make use of Journersquoslemma [16 p 69]

Lemma 10 There exists a constant 119862 gt 0 such that for all119895 ge 0119898

119895= 119892119895+ 119887119895 where 119892

119895119871infin le 119862 119887

119895119871infin le 1198622

(120575minus1)119895 andsupp ℎ

119895119892119895sub 211989572 le |120585| le 9 sdot 2

119895

Proof of Claim 3 First we consider the case 119903 = 2 With thenotation in Lemma 10 it suffices to estimate suminfin

119895=0119887119895ℎ1198951198712

and suminfin119895=0119892119895ℎ1198951198712

The estimate on the ldquobad partrdquo follows from the triangleand Cauchy-Schwarz inequalities and the control 119887

119895119871infin ≲

2(120575minus1)119895 recall that 120575 lt 1

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119887119895ℎ119895

100381710038171003817100381710038171003817100381710038171003817100381710038171198712

le

infin

sum

119895=0

10038171003817100381710038171003817119887119895

10038171003817100381710038171003817119871infin

10038171003817100381710038171003817ℎ119895

100381710038171003817100381710038171198712

le (

infin

sum

119895=0

10038171003817100381710038171003817119887119895

10038171003817100381710038171003817

2

119871infin)

12

(

infin

sum

119895=0

10038171003817100381710038171003817ℎ119895

10038171003817100381710038171003817

2

1198712)

12

≲ (

infin

sum

119895=0

2(2120575minus2)119895

)

12

(

infin

sum

119895=0

10038171003817100381710038171003817ℎ119895

10038171003817100381710038171003817

2

1198712)

12

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

121003817100381710038171003817100381710038171003817100381710038171003817100381710038171198712

(29)

Journal of Function Spaces and Applications 5

Let us now look at the ldquogood partrdquo We start by noticing thatgiven the two summation indices 119895 119896 ge 0 we have

supp119892119895ℎ119895sub

2119895

72le10038161003816100381610038161205851003816100381610038161003816 le 9 sdot 2

119895

supp119892119896ℎ119896sub

2119896

72le10038161003816100381610038161205851003816100381610038161003816 le 9 sdot 2

119896

(30)

Thus for |119895 minus 119896| ge 11 we have supp119892119895ℎ119895cap supp119892

119896ℎ119896= 0

In view of this orthogonality Plancherelrsquos theorem with theestimate 119892

119895119871infin ≲ 1 gives

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119892119895ℎ119895

100381710038171003817100381710038171003817100381710038171003817100381710038171198712

le

11

sum

119894=0

1003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119896=0

119892119894+11119896

ℎ119894+11119896

10038171003817100381710038171003817100381710038171003817100381710038171198712

≲ (

infin

sum

119895=0

10038171003817100381710038171003817119892119895ℎ119895

10038171003817100381710038171003817

2

1198712)

12

le (

infin

sum

119895=0

10038171003817100381710038171003817119892119895

10038171003817100381710038171003817

2

119871infin

10038171003817100381710038171003817ℎ119895

10038171003817100381710038171003817

2

1198712)

12

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

121003817100381710038171003817100381710038171003817100381710038171003817100381710038171198712

(31)

This completes the proof of the case 119903 = 2 In the general case119903 gt 1 we again seek the control of the ldquobadrdquo and ldquogoodrdquo partsThe estimate on the ldquobadrdquo part follows virtually the same asin the case 119903 = 210038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119887119895ℎ119895

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

le

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816119887119895

10038161003816100381610038161003816

2

)

12

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

le

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816119887119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871infin

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

(32)

wherewe usedMinkowskirsquos integral inequality in the last stepFor the ldquogoodrdquo part we can think of 119892

119896ℎ119896as being dyadic

blocks in the Littlewood-Paley decomposition of the sum119878119894= sum119896equiv119894 ( mod 11) 119892119896ℎ119896 Thus it will be enough to control

uniformly (in the 119871119903 norm) the sums 119878119894 0 le 119894 le 11 in order

to obtain the same bound on sum119895ge0119892119895ℎ119895119871119903 The control on

119878119894however follows from the uniform estimate on the 119892

119896rsquos and

an immediate application of Littlewood-Paley theory

Acknowledgments

A Benyirsquos work is partially supported by a Grant fromthe Simons Foundation (no 246024) T Oh acknowledgessupport from an AMS-Simons Travel Grant

References

[1] R R Coifman and Y Meyer Au Dela des Operateurs Pseudo-Differentiels vol 57 of Asterisque 2nd edition 1978

[2] L Grafakos and R H Torres ldquoMultilinear Calderon-Zygmundtheoryrdquo Advances in Mathematics vol 165 no 1 pp 124ndash1642002

[3] C E Kenig and E M Stein ldquoMultilinear estimates and frac-tional integrationrdquo Mathematical Research Letters vol 6 no 1pp 1ndash15 1999

[4] A Benyi D Maldonado V Naibo and R H Torres ldquoOn theHormander classes of bilinear pseudodifferential operatorsrdquoIntegral Equations and Operator Theory vol 67 no 3 pp 341ndash364 2010

[5] F Bernicot D Maldonado K Moen and V Naibo ldquoBilinearSobolev-Poincare inequalities and Leibniz-type rulesrdquo Journalof Geometric Analysis In press httparxivorgpdf11043942pdf

[6] T Kato andG Ponce ldquoCommutator estimates and the Euler andNavier-Stokes equationsrdquoCommunications on Pure and AppliedMathematics vol 41 no 7 pp 891ndash907 1988

[7] C E Kenig G Ponce and L Vega ldquoOn unique continuationfor nonlinear Schrodinger equationsrdquo Communications on Pureand Applied Mathematics vol 56 no 9 pp 1247ndash1262 2003

[8] A Benyi A R Nahmod and R H Torres ldquoSobolev spaceestimates and symbolic calculus for bilinear pseudodifferentialoperatorsrdquo Journal of Geometric Analysis vol 16 no 3 pp 431ndash453 2006

[9] A Benyi and R H Torres ldquoAlmost orthogonality and a class ofbounded bilinear pseudodifferential operatorsrdquo MathematicalResearch Letters vol 11 no 1 pp 1ndash11 2004

[10] F Bernicot and S Shrivastava ldquoBoundedness of smooth bilinearsquare functions and applications to some bilinear pseudo-differential operatorsrdquo Indiana University Mathematics Journalvol 60 no 1 pp 233ndash268 2011

[11] K Grochenig Foundations of Time-Frequency AnalysisBirkhauser Boston Mass USA 2001

[12] A Benyi K Grochenig C Heil and K Okoudjou ldquoModulationspaces and a class of bounded multilinear pseudodifferentialoperatorsrdquo Journal ofOperatorTheory vol 54 pp 301ndash313 2005

[13] A Benyi F Bernicot D Maldonado V Naibo and R H Tor-res ldquoOn the Hormander classes of bilinear pseudodifferentialoperators IIrdquo Indiana University Mathematics Journal In presshttparxivorgpdf11120486pdf

[14] A Miyachi and N Tomita ldquoCalderon-Vaillancourt type theo-rem for bilinear pseudo-differential operatorsrdquo Indiana Univer-sity Mathematics Journal In press

[15] A Benyi and R H Torres ldquoSymbolic calculus and the trans-poses of bilinear pseudodifferential operatorsrdquo Communica-tions in Partial Differential Equations vol 28 no 5-6 pp 1161ndash1181 2003

[16] J-L Journe Calderon-Zygmund Operators Pseudo-DifferentialOperators and the Cauchy Integral of Calderon vol 994 ofLecture Notes in Mathematics Springer Berlin Germany 1983

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

In this case for 119896 = 1 2 we can write

119879120590119896

(119891 119892) (119909) =

infin

sum

119895=0

119898119895 (119909) 119891119895119896 (119909) 119892119895119896 (119909) (20)

Claim 1 1198791205902

is bounded from 119871119901 times 119871119902 into 119871119903

Proof By (20) and the Cauchy-Schwarz inequality we canwrite

100381610038161003816100381610038161198791205902

(119891 119892)10038161003816100381610038161003816le

infin

sum

119895=0

10038161003816100381610038161003816119898119895

10038161003816100381610038161003816

100381610038161003816100381610038161198911198952

10038161003816100381610038161003816

100381610038161003816100381610038161198921198952

10038161003816100381610038161003816

≲ (

infin

sum

119895=0

100381610038161003816100381610038161198911198952

10038161003816100381610038161003816

2

)

12

(

infin

sum

119895=0

100381610038161003816100381610038161198921198952

10038161003816100381610038161003816

2

)

12

(21)

We used here the fact that the coefficients 119898119895are bounded

see Lemma 9 Using now Holderrsquos inequality we have

100381710038171003817100381710038171198791205902

(119891 119892)10038171003817100381710038171003817119871119903≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

100381610038161003816100381610038161198911198952

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

100381610038161003816100381610038161198921198952

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119902

(22)

Finally the conditions on the supports of 1205932and 120594

2allow us

tomake use of the Littlewood-Paley theory and conclude that100381710038171003817100381710038171198791205902

(119891 119892)10038171003817100381710038171003817119871119903≲1003817100381710038171003817119891100381710038171003817100381711987111990110038171003817100381710038171198921003817100381710038171003817119871119902 (23)

Obtaining the boundedness of the operator 1198791205901

is a bitmore delicate due to the support condition on 120594

1 specifically

having supp1205941contained in a disk rather than in an annulus

Nevertheless we still claim the following

Claim 2 1198791205901

is bounded from 119871119901 times 119871119902 into 119871119903

Proof Note first that we can still use the Littlewood-Paleytheory on the 119891

1198951part of the sum that defines 119879

1205901

(119891 119892)(119909) =

suminfin

119895=0119898119895(119909)1198911198951(119909)1198921198951(119909) We have the following inequalities

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

100381610038161003816100381610038161198911198952

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

≲10038171003817100381710038171198911003817100381710038171003817119871119901

1003817100381710038171003817100381710038171003817100381710038171003817

sup119895ge0

|1198921198951|

1003817100381710038171003817100381710038171003817100381710038171003817119871119902

≲10038171003817100381710038171198921003817100381710038171003817119871119902

(24)

At this point however we must proceed more cautiouslyObserve that

supp11989111989511198921198951sub supp119891

1198951+ supp119892

1198951sub 2119895minus3le10038161003816100381610038161205851003816100381610038161003816 le 2119895+3

(25)

Denoting then ℎ1198951= 11989111989511198921198951 we now have 119879

1205901

(119891 119892)(119909) =

suminfin

119895=0119898119895ℎ1198951 where 120597120572119898

119895119871infin ≲ 2

119895120575|120572| and ℎ1198951

satisfies thesupport condition (25) Assume for the moment that thefollowing inequality holds

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119898119895ℎ1198951

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ1198951

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

(26)

Then the boundedness of the operator1198791205901

can be obtained asfollows

100381710038171003817100381710038171198791205901

(119891 119892)10038171003817100381710038171003817119871119903=

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119898119895ℎ1198951

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ1198951

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

sup119895ge0

|1198921198951| sdot (

infin

sum

119895=0

100381610038161003816100381610038161198911198951

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

100381610038161003816100381610038161198911198951

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119901

1003817100381710038171003817100381710038171003817100381710038171003817

sup119895ge0

100381610038161003816100381610038161198921198951

10038161003816100381610038161003816

1003817100381710038171003817100381710038171003817100381710038171003817119871119902

≲1003817100381710038171003817119891100381710038171003817100381711987111990110038171003817100381710038171198921003817100381710038171003817119871119902

(27)

The proof of Claim 2 assumed the estimate (26) Our nextclaim is that (26) is indeed true

Claim 3 Assume that 120597120572119898119895119871infin ≲ 2

119895120575|120572| and supp ℎ119895sub

2119895minus3le |120585| le 2

119895+3 Then for all 119903 gt 1 we have

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119898119895ℎ119895

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

(28)

In our proof of this claim we will make use of Journersquoslemma [16 p 69]

Lemma 10 There exists a constant 119862 gt 0 such that for all119895 ge 0119898

119895= 119892119895+ 119887119895 where 119892

119895119871infin le 119862 119887

119895119871infin le 1198622

(120575minus1)119895 andsupp ℎ

119895119892119895sub 211989572 le |120585| le 9 sdot 2

119895

Proof of Claim 3 First we consider the case 119903 = 2 With thenotation in Lemma 10 it suffices to estimate suminfin

119895=0119887119895ℎ1198951198712

and suminfin119895=0119892119895ℎ1198951198712

The estimate on the ldquobad partrdquo follows from the triangleand Cauchy-Schwarz inequalities and the control 119887

119895119871infin ≲

2(120575minus1)119895 recall that 120575 lt 1

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119887119895ℎ119895

100381710038171003817100381710038171003817100381710038171003817100381710038171198712

le

infin

sum

119895=0

10038171003817100381710038171003817119887119895

10038171003817100381710038171003817119871infin

10038171003817100381710038171003817ℎ119895

100381710038171003817100381710038171198712

le (

infin

sum

119895=0

10038171003817100381710038171003817119887119895

10038171003817100381710038171003817

2

119871infin)

12

(

infin

sum

119895=0

10038171003817100381710038171003817ℎ119895

10038171003817100381710038171003817

2

1198712)

12

≲ (

infin

sum

119895=0

2(2120575minus2)119895

)

12

(

infin

sum

119895=0

10038171003817100381710038171003817ℎ119895

10038171003817100381710038171003817

2

1198712)

12

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

121003817100381710038171003817100381710038171003817100381710038171003817100381710038171198712

(29)

Journal of Function Spaces and Applications 5

Let us now look at the ldquogood partrdquo We start by noticing thatgiven the two summation indices 119895 119896 ge 0 we have

supp119892119895ℎ119895sub

2119895

72le10038161003816100381610038161205851003816100381610038161003816 le 9 sdot 2

119895

supp119892119896ℎ119896sub

2119896

72le10038161003816100381610038161205851003816100381610038161003816 le 9 sdot 2

119896

(30)

Thus for |119895 minus 119896| ge 11 we have supp119892119895ℎ119895cap supp119892

119896ℎ119896= 0

In view of this orthogonality Plancherelrsquos theorem with theestimate 119892

119895119871infin ≲ 1 gives

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119892119895ℎ119895

100381710038171003817100381710038171003817100381710038171003817100381710038171198712

le

11

sum

119894=0

1003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119896=0

119892119894+11119896

ℎ119894+11119896

10038171003817100381710038171003817100381710038171003817100381710038171198712

≲ (

infin

sum

119895=0

10038171003817100381710038171003817119892119895ℎ119895

10038171003817100381710038171003817

2

1198712)

12

le (

infin

sum

119895=0

10038171003817100381710038171003817119892119895

10038171003817100381710038171003817

2

119871infin

10038171003817100381710038171003817ℎ119895

10038171003817100381710038171003817

2

1198712)

12

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

121003817100381710038171003817100381710038171003817100381710038171003817100381710038171198712

(31)

This completes the proof of the case 119903 = 2 In the general case119903 gt 1 we again seek the control of the ldquobadrdquo and ldquogoodrdquo partsThe estimate on the ldquobadrdquo part follows virtually the same asin the case 119903 = 210038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119887119895ℎ119895

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

le

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816119887119895

10038161003816100381610038161003816

2

)

12

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

le

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816119887119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871infin

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

(32)

wherewe usedMinkowskirsquos integral inequality in the last stepFor the ldquogoodrdquo part we can think of 119892

119896ℎ119896as being dyadic

blocks in the Littlewood-Paley decomposition of the sum119878119894= sum119896equiv119894 ( mod 11) 119892119896ℎ119896 Thus it will be enough to control

uniformly (in the 119871119903 norm) the sums 119878119894 0 le 119894 le 11 in order

to obtain the same bound on sum119895ge0119892119895ℎ119895119871119903 The control on

119878119894however follows from the uniform estimate on the 119892

119896rsquos and

an immediate application of Littlewood-Paley theory

Acknowledgments

A Benyirsquos work is partially supported by a Grant fromthe Simons Foundation (no 246024) T Oh acknowledgessupport from an AMS-Simons Travel Grant

References

[1] R R Coifman and Y Meyer Au Dela des Operateurs Pseudo-Differentiels vol 57 of Asterisque 2nd edition 1978

[2] L Grafakos and R H Torres ldquoMultilinear Calderon-Zygmundtheoryrdquo Advances in Mathematics vol 165 no 1 pp 124ndash1642002

[3] C E Kenig and E M Stein ldquoMultilinear estimates and frac-tional integrationrdquo Mathematical Research Letters vol 6 no 1pp 1ndash15 1999

[4] A Benyi D Maldonado V Naibo and R H Torres ldquoOn theHormander classes of bilinear pseudodifferential operatorsrdquoIntegral Equations and Operator Theory vol 67 no 3 pp 341ndash364 2010

[5] F Bernicot D Maldonado K Moen and V Naibo ldquoBilinearSobolev-Poincare inequalities and Leibniz-type rulesrdquo Journalof Geometric Analysis In press httparxivorgpdf11043942pdf

[6] T Kato andG Ponce ldquoCommutator estimates and the Euler andNavier-Stokes equationsrdquoCommunications on Pure and AppliedMathematics vol 41 no 7 pp 891ndash907 1988

[7] C E Kenig G Ponce and L Vega ldquoOn unique continuationfor nonlinear Schrodinger equationsrdquo Communications on Pureand Applied Mathematics vol 56 no 9 pp 1247ndash1262 2003

[8] A Benyi A R Nahmod and R H Torres ldquoSobolev spaceestimates and symbolic calculus for bilinear pseudodifferentialoperatorsrdquo Journal of Geometric Analysis vol 16 no 3 pp 431ndash453 2006

[9] A Benyi and R H Torres ldquoAlmost orthogonality and a class ofbounded bilinear pseudodifferential operatorsrdquo MathematicalResearch Letters vol 11 no 1 pp 1ndash11 2004

[10] F Bernicot and S Shrivastava ldquoBoundedness of smooth bilinearsquare functions and applications to some bilinear pseudo-differential operatorsrdquo Indiana University Mathematics Journalvol 60 no 1 pp 233ndash268 2011

[11] K Grochenig Foundations of Time-Frequency AnalysisBirkhauser Boston Mass USA 2001

[12] A Benyi K Grochenig C Heil and K Okoudjou ldquoModulationspaces and a class of bounded multilinear pseudodifferentialoperatorsrdquo Journal ofOperatorTheory vol 54 pp 301ndash313 2005

[13] A Benyi F Bernicot D Maldonado V Naibo and R H Tor-res ldquoOn the Hormander classes of bilinear pseudodifferentialoperators IIrdquo Indiana University Mathematics Journal In presshttparxivorgpdf11120486pdf

[14] A Miyachi and N Tomita ldquoCalderon-Vaillancourt type theo-rem for bilinear pseudo-differential operatorsrdquo Indiana Univer-sity Mathematics Journal In press

[15] A Benyi and R H Torres ldquoSymbolic calculus and the trans-poses of bilinear pseudodifferential operatorsrdquo Communica-tions in Partial Differential Equations vol 28 no 5-6 pp 1161ndash1181 2003

[16] J-L Journe Calderon-Zygmund Operators Pseudo-DifferentialOperators and the Cauchy Integral of Calderon vol 994 ofLecture Notes in Mathematics Springer Berlin Germany 1983

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

Let us now look at the ldquogood partrdquo We start by noticing thatgiven the two summation indices 119895 119896 ge 0 we have

supp119892119895ℎ119895sub

2119895

72le10038161003816100381610038161205851003816100381610038161003816 le 9 sdot 2

119895

supp119892119896ℎ119896sub

2119896

72le10038161003816100381610038161205851003816100381610038161003816 le 9 sdot 2

119896

(30)

Thus for |119895 minus 119896| ge 11 we have supp119892119895ℎ119895cap supp119892

119896ℎ119896= 0

In view of this orthogonality Plancherelrsquos theorem with theestimate 119892

119895119871infin ≲ 1 gives

10038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119892119895ℎ119895

100381710038171003817100381710038171003817100381710038171003817100381710038171198712

le

11

sum

119894=0

1003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119896=0

119892119894+11119896

ℎ119894+11119896

10038171003817100381710038171003817100381710038171003817100381710038171198712

≲ (

infin

sum

119895=0

10038171003817100381710038171003817119892119895ℎ119895

10038171003817100381710038171003817

2

1198712)

12

le (

infin

sum

119895=0

10038171003817100381710038171003817119892119895

10038171003817100381710038171003817

2

119871infin

10038171003817100381710038171003817ℎ119895

10038171003817100381710038171003817

2

1198712)

12

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

121003817100381710038171003817100381710038171003817100381710038171003817100381710038171198712

(31)

This completes the proof of the case 119903 = 2 In the general case119903 gt 1 we again seek the control of the ldquobadrdquo and ldquogoodrdquo partsThe estimate on the ldquobadrdquo part follows virtually the same asin the case 119903 = 210038171003817100381710038171003817100381710038171003817100381710038171003817

infin

sum

119895=0

119887119895ℎ119895

10038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

le

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816119887119895

10038161003816100381610038161003816

2

)

12

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

le

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816119887119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871infin

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

≲

100381710038171003817100381710038171003817100381710038171003817100381710038171003817

(

infin

sum

119895=0

10038161003816100381610038161003816ℎ119895

10038161003816100381610038161003816

2

)

12100381710038171003817100381710038171003817100381710038171003817100381710038171003817119871119903

(32)

wherewe usedMinkowskirsquos integral inequality in the last stepFor the ldquogoodrdquo part we can think of 119892

119896ℎ119896as being dyadic

blocks in the Littlewood-Paley decomposition of the sum119878119894= sum119896equiv119894 ( mod 11) 119892119896ℎ119896 Thus it will be enough to control

uniformly (in the 119871119903 norm) the sums 119878119894 0 le 119894 le 11 in order

to obtain the same bound on sum119895ge0119892119895ℎ119895119871119903 The control on

119878119894however follows from the uniform estimate on the 119892

119896rsquos and

an immediate application of Littlewood-Paley theory

Acknowledgments

A Benyirsquos work is partially supported by a Grant fromthe Simons Foundation (no 246024) T Oh acknowledgessupport from an AMS-Simons Travel Grant

References

[1] R R Coifman and Y Meyer Au Dela des Operateurs Pseudo-Differentiels vol 57 of Asterisque 2nd edition 1978

[2] L Grafakos and R H Torres ldquoMultilinear Calderon-Zygmundtheoryrdquo Advances in Mathematics vol 165 no 1 pp 124ndash1642002

[3] C E Kenig and E M Stein ldquoMultilinear estimates and frac-tional integrationrdquo Mathematical Research Letters vol 6 no 1pp 1ndash15 1999

[4] A Benyi D Maldonado V Naibo and R H Torres ldquoOn theHormander classes of bilinear pseudodifferential operatorsrdquoIntegral Equations and Operator Theory vol 67 no 3 pp 341ndash364 2010

[5] F Bernicot D Maldonado K Moen and V Naibo ldquoBilinearSobolev-Poincare inequalities and Leibniz-type rulesrdquo Journalof Geometric Analysis In press httparxivorgpdf11043942pdf

[6] T Kato andG Ponce ldquoCommutator estimates and the Euler andNavier-Stokes equationsrdquoCommunications on Pure and AppliedMathematics vol 41 no 7 pp 891ndash907 1988

[7] C E Kenig G Ponce and L Vega ldquoOn unique continuationfor nonlinear Schrodinger equationsrdquo Communications on Pureand Applied Mathematics vol 56 no 9 pp 1247ndash1262 2003

[8] A Benyi A R Nahmod and R H Torres ldquoSobolev spaceestimates and symbolic calculus for bilinear pseudodifferentialoperatorsrdquo Journal of Geometric Analysis vol 16 no 3 pp 431ndash453 2006

[9] A Benyi and R H Torres ldquoAlmost orthogonality and a class ofbounded bilinear pseudodifferential operatorsrdquo MathematicalResearch Letters vol 11 no 1 pp 1ndash11 2004

[10] F Bernicot and S Shrivastava ldquoBoundedness of smooth bilinearsquare functions and applications to some bilinear pseudo-differential operatorsrdquo Indiana University Mathematics Journalvol 60 no 1 pp 233ndash268 2011

[11] K Grochenig Foundations of Time-Frequency AnalysisBirkhauser Boston Mass USA 2001

[12] A Benyi K Grochenig C Heil and K Okoudjou ldquoModulationspaces and a class of bounded multilinear pseudodifferentialoperatorsrdquo Journal ofOperatorTheory vol 54 pp 301ndash313 2005

[13] A Benyi F Bernicot D Maldonado V Naibo and R H Tor-res ldquoOn the Hormander classes of bilinear pseudodifferentialoperators IIrdquo Indiana University Mathematics Journal In presshttparxivorgpdf11120486pdf

[14] A Miyachi and N Tomita ldquoCalderon-Vaillancourt type theo-rem for bilinear pseudo-differential operatorsrdquo Indiana Univer-sity Mathematics Journal In press

[15] A Benyi and R H Torres ldquoSymbolic calculus and the trans-poses of bilinear pseudodifferential operatorsrdquo Communica-tions in Partial Differential Equations vol 28 no 5-6 pp 1161ndash1181 2003

[16] J-L Journe Calderon-Zygmund Operators Pseudo-DifferentialOperators and the Cauchy Integral of Calderon vol 994 ofLecture Notes in Mathematics Springer Berlin Germany 1983

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