Weighted Compact Commutator of Bilinear Fourier Multiplier Operator?

Guoen HU

1 Introduction

As it is well known,the study of bilinear Fourier multiplier operator was origined by Coifman and Meyer.Let σ ∈ L∞(R2n).De fine the bilinear Fourier multiplier operator Tσby

for f1,f2∈S(Rn),where and in the following,Ff denotes the Fourier transform of f.Coifman and Meyer[6]proved that if σ∈Cs(R2n{0})satis fies

for all|α1|+|α2|≤ s with s ≥ 4n+1,then Tσis bounded from Lp1(Rn)× Lp2(Rn)to Lp(Rn)for all 1

For κ∈Z,set

and

Tomita[21]proved that if σ satis fies the Sobolev regularity that

for some s∈ (n,2n],then Tσis bounded from Lp1(Rn)×Lp2(Rn)to Lp(Rn)provided thatGrafakos and Si[11]considered the mapping properties fromfor Tσwhen σ satis fies(1.5)andthen T is bounded fromMiyachi and Tomita[20]considered the problem to find minimal smoothness condition for bilinear Fourier multiplier.Let

whereMiyachi and Tomita[20]proved that if

for somethen Tσis bounded fromfor anyandMoreover,they also gave minimal smoothness condition for which Tσis bounded from Hp1(Rn)×Hp2(Rn)to Lp(Rn).

The weighted estimates for the operator Tσare also of great interest.As it is well known,when σ satis fies(1.2)for some s ≥ 2n+1,then Tσis a standard bilinear Calder′on-Zygmund operator,and then by the weighted estimates with multiple weights for bilinear Calder′on-Zygmund operators,which was established by Lerner et al.[19],we know that for any p1,p2∈[1,∞)and p ∈ (0,∞)withand weights w1,w2such that(for the de finition ofsee De finition 1.1 below),

where and in the following,for indices p1,p2,we setand p ∈ (0,∞)such thatBy developing the ideas used in[19],Bui and Duong[4]established the weighted estimates with multiple weights for Tσwhen σ satis fies(1.2)for some s ∈ (n,2n].To consider the weighted estimates for Tσwhen σ satis fies(1.5),Jiao[17]introduced the following class of multiple weights.

De finition 1.1Let m ≥ 1 be an integer,w1,···,wmbe weights,p1,···,pm,p ∈ (0,∞)with

where and in the following,whenis understood as

Whenis just the weight classintroduced by Lerner et al.[19].By some kernel estimates of the operator Tσ,Jiao proved that for t1,t2∈ [1,2)such thatwith k=1,2,and w1,w2such thatthen Tσis bounded fromFor the weighted estimates with Apweights when σ satis fies the regularity(1.6)(see[8,15]),here and in the following,fordenotes the weight function class Muckenhoupt,and

The commutator of the multiplier operator Tσhas been considered by many authors.Let Tσbe the multiplier operator de fined by(1.1),b1,b2∈ BMO(Rn)and=(b1,b2).De fine the commutator ofand Tσby

with

and

Bui and Duong[4]established the weighted estimates with multiple weights forwhen σ satis fies(1.2)for s∈(n,2n].Hu and Yi[16]considered the behavior onwhen σ satis fies(1.6)forand showed thatenjoys the samemapping properties as that of the operator Tσ.Fairly recently,Hu[14]considered the compactness ofand proved that if b1,b2∈ CMO(Rn),σ satis fies(1.6)fora compact operator fromwhere and in the following,CMO(Rn)denotes the closure ofin the BMO(Rn)topology,which coincide with the space of functions of vanishing mean oscillation(see[3,7]for details).Zhou and Li[22]considered the weighted compactness with Apweights forBy combining the ideas used inand Li showed that ifand σ satis fies(1.6)for somecompact operator from

The main purpose of this paper is to consider the weighted compactness ofwith multiple weights.We will show that if σ satis fies(1.5)and b1,b2∈ CMO(Rn),then for appropriatethis paper can be stated as follows.

Theorem 1.1 Let σ be a multiplier satisfying(1.5)for some s ∈ (n,2n]and Tσ be the operator de fined by(1.1).Let t1,t2∈[1,2)such thatb1,b2∈CMO(Rn).Then for pk∈ (tk,∞)with k=1,2,p∈ (1,∞)withand weights w1,w2such that

Remark 1.1 It is well known that,the classwithis really large than the weight classand the weighted estimates with multiple weightsare more interesting and more re fined than the weighted estimates with×for the bilinear Calder′on-Zygmund operators(see[19]).To prove Theorem 1.1,we will employ the idea used in[2,14].However,the idea that controllingbywhich was used in[14,22](even if the functionwithintroduced by[17])does not work.To overcome this difficulty,we establish some new estimates for the kernel of Tσ,and introduce a new subtle bi(sub)linear maximal operator to control

Throughout the article,C always denotes a positive constant that may vary from line to line but remains independent of the main variables.We use the symbol A.B to denote that there exists a positive constant C such that A≤CB.For any set E?Rn,χEdenotes its characteristic function.We use B(x,R)to denote a ball centered at x with radius R and C(x,R)=B(x,R)Bx,.For a ball B ? Rnand λ >0,we use λB to denote the ball concentric with B whose radius is λ times of B’s.For any γ ∈ [1,∞],we useto denote the dual exponent of γ,namely,=1.For a locally integrable function f,Mf denotes the Hardy-Littlewood maximal function of f,and for τ∈ (0,∞),

Let M?be the Fefferman-Stein sharp maximal operator.For ?>0,denotes the operator de fined by

2 A New Maximal Operator

To control the multilinear Calder′on-Zygmund operators via the Fefferman-Stein sharp maximal operator,Lerner et al.[19]introduced the bi(sub)linear maximal operator M by

For r1,r2∈ (0,∞),Jiao[17]generalized the operator M,de fined the maximal operatorby

and established the weighted norm inequalities with multiple weightsforLet δ∈R and r1,r2∈[1,∞).De fine the bi(sub)linear maximal operatorsandby

and

respectively.It is obvious that for any δ<0,x ∈ Rnand k=1,2,

For the case of δ=0 and r1=r2=1,these operators were introduced by Grafakos et al.in[9].Although we do not know if the operator Mrcan be applied to prove Theorem 1.1,as the operator M do in the proof of the weighted compactness of the commutator of multilinear Calder′on-Zygmund operators(see[2]),we will see that the operator(k=1,2)are suitable replacement ofin our argument.

As it is well known,for a weight w ∈ A∞(Rn),there exists a positive constant θ,such that for any ball B?Rnand any measurable set E?B,

For a fixed θ∈(0,1),set

Our result concerning the operatorscan be stated as follows.

Theorem 2.1Let r1,r2 ∈ (0,∞)and δ∈ R,p1 ∈ [r1,∞)and p2 ∈ [r2,∞),Let w1,w2be weights such that∈and∈Rθfor some θ such that δ

To prove Theorem 2.1,we need the following characterization ofwhich was proved in[17].

Lemma 2.1Let w1,w2be weights,p1,p2,p∈ (0,∞)with1,2).Then the following conditions are equivalent:

(i)

(ii)

Proof of Theorem 2.1We first consider the case of pk∈(rk,∞)with k=1,2.Since the argument forandare very similar,we only consider the operatorWe will employ the ideas used in[9].Letbe the centered maximal operator de fined by

As it was pointed out in[9],it suffices to prove that for some q1,q2∈(0,1),

For each fixed k,we know by Lemma 2.1 thatand so there exists a positive constant σk>1 such that for any ball B,

For k=1,2,let

It is obvious thatγk>1,and

An application of the H¨older inequality gives that

and

On the other hand,we have by the inequalities(2.3)–(2.5)that

Note that

Combining the inequalities(2.6)–(2.8)then yields

Recall thatThus for each ball B,

Similarly,we have that

Therefore,for each fixed x∈Rnand ball B containing x,

This,along with the fact thatand the fact thatleads to that

sinceThis establishes(2.2).

For the case of pk=rkwith k=1,2,the proof is similar to the case of pk∈(rk,∞)and is more simple.In fact,for each x∈Rnand ball B?Rncontaining x,as in the proof of(2.2),we can verify that for k=1,2,

which implies that

and then shows thais bounded from

3 Proof of Theorem 1.1

Let σ ∈ L∞(R2n)and Φ ∈ S(R2n)satisfy(1.3).For κ ∈ Z,de fine

Then=σκ(2?κξ1,2?κξ2)and

where F?1f denotes the inverse Fourier transform of f.For a positive integer N,let

For an integer k with 1≤k≤m and x,y1,y2,x′∈Rn,let

Lemma 3.1Let q1,q2∈[2,∞),and s1,s2≥ 0.Then

For the proof of Lemma 3.1,see Appendix A in[8].

Lemma 3.2Let σ be a bilinear multiplier satisfying(1.5)for some s ∈ [0,∞),r1,r2∈ (1,2]and γ∈(0,s].Then for every x∈Rnand R>0,

and

Furthermore,if γ ∈ (0,s]and?γ++1>0,then

ProofBy the H¨older inequality and Lemma 3.1,we have that for each l∈ Z,

and

Therefore,

which gives(3.1)directly.We can also obtain from(3.5)(with l=0)that

Finally,(3.6)implies that

since?γ++1>0.This completes the proof of Lemma 3.2.

Remark 3.1Let σ be a bilinear multiplier satisfying(1.5)for some s ∈ [0,∞),r1,r2∈ (1,2]and γ ∈(0,s].As in the proof of(3.2),we can verify that,for each R>0 and x,y∈ R with|x?y|

Lemma 3.3Let σ be a bilinear multiplier satisfying(1.5)for some s ∈ [0,∞),r1,r2∈ (1,2]and γ ∈(0,s].For R>0 and x ∈ Rnwith|x|>4R,set

and

for positive integer l.Then for any weights w1,w2and pk∈(rk,∞)with k=1,2,

and

ProofNote that when|y1|≤R and|x|>2R,|x?y1|≥As in the proof of Lemma 3.2,we obtain by Lemma 3.1 and the H¨older inequality,

Similarly,for l≥1,we have that

This completes the proof of Lemma 3.3.

Lemma3.4Let σ be a multiplier which satis fies(1.5),Then for each R>0,x,x′∈ Rnwithnonnegative integers j1,j2with j?=max{j1,j2}≥2,

ProofWe employ some estimates in[17].Without loss of generality,we may assume that j?=j1.For l∈ Z,set

and

It was pointed out in[17]that

On the other hand,by the proof of the inequality(3.6)in[17],we know that

Therefore,

This completes the proof of Lemma 3.4.

Lemma 3.5Let σ be a multiplier which satis fies(1.5)for some s∈(n,2n],t1,t2∈[1,2)such thatLet pk∈(tk,∞)for k=1,2 and w1,w2be weights such that w ∈Then for b1,b2∈BMO(Rn),

Proof The proof here is fairly standard(see[4,17]).For each fixed positive integer N,let Tσ,Nbe the bilinear operator with kernel KNin the sense that

Let b1,b2∈ BMO(Rn),[b1,Tσ,N]1and[b2,Tσ,N]2be the commutator of Tσ,Nas in(1.8)and(1.9)respectively.As in the proof of Theorem 3.1 in[17],we can prove that if r1,r2∈(1,2]such thatthen for ?∈ (0,t)with

Now let pk∈(tk,∞),w1,w2be weights such thatWe can choose δ∈(0,1)which is close to 1,such thatandfor k=1,2.Recall that by Lemma 2.2,implies thatIt then follows that for k=1,2,

if b1,b2∈L∞(Rn).Note that for b1,b2∈L∞(Rn)and f1,f2∈S(Rn),

holds for almost everywhere x∈Rn.Thus,by the Fatou lemma,for k=1,2,b1,b2∈L∞(Rn)and f1,f2∈S(Rn),

This,via a standard argument leads to our desired conclusion.

For a positive integer N,let Tσ,Nbe the operator de fined by

Lemma3.6Let σ be a multiplier which satis fies(1.5)for some s∈ (n,2n],r1,r2∈ (1,2]such that s∈Then for any γ< τ∈(0,min{1,r})withand x∈Rn,

ProofWe employ the ideas used in[9,13].For each fixed ?>0,let

and

For functions f1and f2,let

A trivial computation shows that for y∈

We obtain from Lemma 3.5 that

On the other hand,it follows from(3.7)that for y∈

where in the last inequality,we have invoked the estimate

since?s+

Combining the estimates(3.9)–(3.12)then leads to that for y ∈

Recall that Tσ,Nis bounded from Lr1(Rn)× Lr2(Rn)to Lr,∞(Rn)(see[8,17]).Applying the argument in the proof of the Kolmogorov inequality(see also[9,13]),tells us that for τ∈(0,min{1,r}),

Therefore,for each x ∈ Rnand ?>0,

which gives us the desired conclusion.

Let φ be a non-negative function inwhich satis fies that suppφ ?{(x,y1,y2):max{|x|,|y1|,|y2|}<1},1.For β >0,let χβ= χβ(x,y1,y2)be the characteristicfunction of the setand let

whereAs it was pointed out in[2],ψβ∈ C∞(R3n),suppψβ?and ψβ(x,y1,y2)=1 if|x?yk|≥2β.For a fixed N ∈ N,letbe the bilinear operator de fined by

As usual,for b1,b2∈ BMO(Rn),letbe the commutators ofas in(1.8)–(1.9).

Lemma 3.7Let σ be a multiplier satisfying(1.5)for some s ∈ (n,2n],be the operators de fined by(3.8)and(3.13)respectively.Let r1,r2∈(1,2]such that s∈Then for any

ProofWithout loss of generality,we assume that=1.We deduce from Lemma 3.2 that

This completes the proof of Lemma 3.7.

Lemma 3.8Let r∈(1,∞),w∈Ar(Rn),K?Lr(Rn,w).Suppose that

(i)

(ii)

(iii)

Then K is precompact in Lr(Rn,w).

This lemma was given in[5].

Proof of Theorem 1.1We will employ some ideas from[2].By Lemma 3.5,it suffices to prove that when b1,b2∈(Rn),the conclusion in Theorem 1.1 is true forWe only consider[b1,Tσ]1for simplicity.Without loss of generality,we assume that+∥?b1∥L∞(Rn)=1.

Let t1,t2∈(1,2]such thatpk∈ (tk,∞)with k=1,2,w1,w2be weights such thatRecalling thatwe know that∈Rθfor some θ ∈ (0,1).Also,by Corollary 2.1 in[17],we can choose δ∈ (0,1)which is close to 1,such thatand

Letand rk=with k=1,2.We claim that for each β ∈ (0,1)and ?>0,

(a)there exists a constant A=A(?)which is independent of N,f1and f2,such that

(b)there exists a constant ρ = ρ?which is independent of N,f1and f2,such that for all u∈Rnwith 0<|u|<ρ,

If we can prove this,it then follows from the Fatou lemma that both(3.15)and(3.16)are true with f1,f2∈S(Rn)andis replaced byhereis de fined by

Since S(Rn)is dense in Lpk(Rn,wk),we then know that(3.15)and(3.16)are true whenis replaced byThis,via Lemma 3.8,tells us thatis compact from Lp1(Rn,w1)×Lp2(Rn,w2)to Lp(Rn,νw).On the other hand,(3.14)together with the Fatou lemma and a familiar density argument,leads to that

Therefore,[b1,Tσ]1is compact from Lp1(Rn,w1)×Lp2(Rn,w2)to Lp(Rn,νw).

We first prove the conclusion(a).Let R>0 be large enough such that suppb1?B(0,R).For every fixed x∈Rnwith|x|>2R,set

and

We deduce from Lemma 3.3 that for integers N>0 and l≥0,

if we choose γ4R.Recall that p>1.It then follows directly that

It is easy to verify that

On the other hand,noting thatthere exists a constantsuch that

which,in turn,implies that

if we choose

Thus,forwe have that for some constant η>0,

This leads to the conclusion(a).

We turn our attention to conclusion(b).Let

Set

and set

As in the proof of Lemma 3.7,we obtain by Lemma 3.2 that

Thus,

Note that|ψβ(x+u;y1;y2)?ψβ(x;y1,y2)|.and

Let|u|≤By Lemma 3.2 and Lemma 3.4 and the argument used in the proof of Lemma 3.7,we deduce that

withNote that

The conclusion(b)now follows from(3.17)–(3.18),Lemma 3.6 and Theorem 2.1,if we choose γ such that

[1]B′enyi,A.and Torres,R.H.,Compact bilinear operators and commutators,Proc.Amer.Math.Soc.,141,2013,3609–3621.

[2]Benyi,A.,Dami′an,W.,Moen,K.and Torres,R.,Compactness of bilinear commutator:The weighted case,Michigan Math.J.,64,2015,39–51.

[3]Bourdaud,G.,Lanze de Cristoforis,M.and Sickel,W.,Functional calculus on BMO and related spaces,J.Func.Anal.,189,2002,515–538.

[4]Bui,A.T.and Duong,X.T.,Weighted norm inequalities for multilinear operators and applications to multilinear Fourier multipliers,Bull.Sci.Math.,137,2013,63–75.

[5]Clop,A.and Cruz,V.,Weighted estimates for Beltrami equations,Ann.Acad.Sci.Fenn.Math.,38,2013,91–113.

[6]Coifman,R.R.and Meyer,Y.,Au del`a des op′erateurs pseudo-diff′erentiels,Ast′eriaque,57,1978,1–185.

[7]Coifman,R.R.and Weiss,G.,Extension of Hardy spaces and their use in analysis,Bull.Amer.Math.Soc.,83,1977,569–645.

[8]Fujita,M.and Tomita,N.,Weighted norm inequalities for multilinear Fourier multipliers,Trans.Amer.Math.Soc.,364,2012,6335–6353.

[9]Grafakos,L.,Liu,L.and Yang,D.,Multilple weighted norm inequalities for maximal singular integrals with non-smooth kernels,Proc.Royal Soc.Edinb.Ser.A,141,2011,755–775.

[10]Grafakos,L.,Miyachi,A.and Tomita,N.,On multilinear Fourier multipliers of limited smoothness,Canad.J.Math.,65,2013,299–330.

[11]Grafakos,L.and Si,Z.,The H¨ormander multiplier theorem for multilinear operators,J.Reine.Angew.Math.,668,2012,133–147.

[12]Grafakos,L.and Torres,R.H.,Multilinear Calder′on-Zygmund theory,Adv.Math.,165,2002,124–164.

[13]Grafakos,L.and Torres,R.H.,Maximal operator and weighted norm inequalities for multilinear singular integrals,Indiana Univ.Math.J.,51,2002,1261–1276.

[14]Hu,G.,The compactness of the commutator of bilinear Fourier multiplier operator,Taiwanese J.Math.,18,2014,661–675.

[15]Hu,G.and Lin,C.,Weighted norm inequalities for multilinear singular integral operators and applications,Anal.Appl.,12,2014,269–291.

[16]Hu,G.and Yi,W.,Estimates for the commutators of bilinear Fourier multiplier,Czech.Math.J.,63,2013,1113–1134.

[17]Jiao,Y.,A weighted norm inequality for the bilinear Fourier multiplier operator,Math.Inequal.Appl.,17,2014,899–912.

[18]Kenig,C.and Stein,E.M.,Multilinear estimates and fractional integral,Math.Res.Lett.,6,1999,1–15.

[19]Lerner,A.,Ombrossi,S.,P′erez,C.,et al.,New maximal functions and multiple weights for the multilinear Calder′on-Zygmund theorey,Adv.Math.,220,2009,1222–1264.

[20]Miyachi,A.and Tomita,N.,Minimal smoothness conditions for bilinear Fourier multiplier,Rev.Mat.Iberoamericana,29,2013,495–530.

[21]Tomita,N.,A H¨ormander type multiplier theorem for multilinear operator,J.Funct.Anal.,259,2010,2028–2044.

[22]Zhou,J.and Li,P.,Compactness of the commutator of multilinear Fourier multiplier operator on weighted Lebesgue space,J.Funct.Spaces,2014,Article ID 606504.http://dx.doi.org/10.1155/2014/606504