Yuexiang Heand Yueshan Wang

Department of Mathematics,Jiaozuo University,Jiaozuo 454003,China

?

Toeplitz Type Operator Associated to Singular Integral Operator with Variable Kernel on Weighted Morrey Spaces

Yuexiang He?and Yueshan Wang

Department of Mathematics,Jiaozuo University,Jiaozuo 454003,China

Abstract.Suppose Tk,1and Tk,2aresingular integrals with variablekernels and mixed homogeneity or±I(the identity operator).Denote the Toeplitz type operator by

where Mbf=bf.In this paper,the boundedness of Tbon weighted Morrey space are obtained when b belongs to the weighted Lipschitz function space and weighted BMO function space,respectively.

Toeplitz type operator,singular integral operator,variable Calder′on-Zygmund kernel,weighted BMO function,weighted Lipschitz function,weighted Morrey space.

AMS Subject Classifications:42B20,40B35

Analysis in Theory and Applications

Anal.Theory Appl.,Vol.32,No.1(2016),pp.90-102

1　Introduction

The classical Morrey spaces,introduced by Morrey[1]in 1938,have been studied intensively by various authors,and it,together with weighted Lebesgue spaces play an important role in the theory of partial differential equations,see[2,3].The boundedness of the Hardy-Littlewood maximal operator,singular integral operator,fractional integral operator and commutator of these operators in Morrey spaces have been studied by Chiarenza and Frasca in[4].Komori and Shirai[5]introduced a version of the weighted Morrey space Lp,κ(ω),which is a natural generalization of the weighted Lebesgue space Lp(ω).

As the development of singular integral operators,their commutators have been well studied[6–8].In[7],the authors proved that the commutators[b,T],which generated byCalder′on-Zygmund singular integral operator and BMO functions,are bounded on Lpfor 1＜p＜∞.The commutator generated by the Calder′on-Zygmund operator T and a locally integrable function b can be regarded as a special case of the Toeplitz operator

where Tk,1and Tk,2are the Calder′on-Zygmund operators or±I(the identity operator), Mbf=bf.When b∈BMO,the Lp-boundednessof Tbwas discussed,see[9,10].In[11,12], the authors studied the boundedness of Tbon Morrey spaces.

Let K(x,ξ):Rn×Rn{0}→R be a variable kernel with mixed homogeneity.The singular integral operator is defined by

The variable kernel K(x,ξ)depends on some parameter x and possesses good properties with respect to the second variable ξ,which was firstly introduced by Fabes and Rievi′eve in[13].They generalized the classical Calder′on-Zygmund kernel and the parabolic kernel studied by Jones in[14].By introducing a new metric ρ,Fabes and Rievi′eve studied(1.2)in Lp(Rn),where Rnwas endowed with the topology induced by ρ and defined by ellipsoids.

By using this metric ρ,Softova in[15]obtained that the integral operator(1.2)and its commutator were continuous in generalized Morrey space Lp,ω(Rn),1＜p＜∞,ω satisfying suitable conditions.Ye and Zhu in[16]discussed the continuity of(1.2)and its multilinear commutator in the weighted Morrey spaces Lp,κ(ω),1＜p＜∞,0＜κ＜1,and ω is Apweight.

Suppose Tk,1and Tk,2are singular integrals whose kernels are variable kernel with mixed homogeneity or±I(the identity operator).In this paper,we study the boundedness of Toeplitz operators Tbas(1.1)in weighted Morrey spaces when b belongs to weighted Lipschitz spaces and weighted BMO spaces,respectively.The main results are as follows.

Theorem 1.1.Suppose that Tbis a Toeplitz type operator associated to singular integral operator with variable kernel,ω∈A1,and b∈Lipβ,ω.Let 0＜κ＜p/q,1＜p＜n/β and 1/q=1/p?β/n. If T1(f)=0 for any f∈Lp,κ(ω),then there exists a constant C＞0 such that,

Theorem 1.2.Suppose that Tbis a Toeplitz type operator associated to singular integral operator with variable kernel,ω∈A1,and b∈BMO(ω).Let 1＜p＜∞,and 0＜κ＜1.If T1(f)=0 for any f∈Lp,κ(ω),then there exists a constant C＞0 such that,

2　Some preliminaries

Letα1,···,αnberealnumbers,αi≥1anddefineFollowingFabesandRivie′re[6], the functionfor any fixed x,is a decreasing one with respect to ρ＞0 and the equation F(x,ρ)=1 is uniquely solvable in ρ(x).It is easy to check that ρ(x?y) defines a distance between any two points x,y∈Rn.Thus Rnendowed with the metric ρ results a homogeneous metric space[13,15].The balls with respect to ρ(x)centered at the origin and of radius r are the ellipsoids

with Lebesgue measure|εr|=C(n)rα.It is easy to see that ε1(0)coincides with the unit sphere Sn?1with respect to the Euclidean metric.

Definition 2.1.The function K(x,ξ):Rn×Rn{0}→R is called a variable kernel with mixed homogeneity if:

(i)for every fixed x,the function K(x,·)is a constant kernel satisfying

(1)K(x,·)∈C∞(Rn{0}),

(2)for anyμ＞0,αi≥1,

Note that in the special case αi=1,1≤i≤n,Definition 2.1 gives rise to the classical Calder′on-Zygmund kernels.When αi=1,1≤i≤n?1,and αn≥1,we obtain the kernel studied by Jones in[14]and discussed in[13].

A weight ω is a nonnegative,locally integrable function on Rn.Let ε=εr(x0)denote the ellipsoid with the center x0and radius r.For a given weight function ω and a measurable set E,we also denote the Lebesguemeasure of E by|E|and set weighted measure ω(E)=REω(x)dx.For any given weight function ω on Rn,0＜p＜∞,denote by Lp(ω) the space of all function f satisfying

A weight ω is said to belong to the Muckenhoupt class Apfor 1＜p＜∞,if there exists a constant C such that

for every ellipsoid ε.The class A1is defined by replacing the above inequality with

for every ball ε.

The classical Apweight theory was first introduced by Muckenhoupt in the study of weighted Lp-boundedness of Hardy-Littlewood maximal function in[17].

Lemma 2.1.Suppose ω∈A1.Then

(i)there exists a ?＞0 such that

(ii)there exist two constant C1and C2,such that

Let us recall the definition of weighted Lipschitz function space and weighted BMO function space.

Definition 2.2.For 1≤p＜∞,0＜β＜1,and ω∈A∞.A locally integrable function b is said to be in the weighted Lipschitz function space if

The Banach space of such functions modulo constants is denoted by Lipβ,p(ω).The smallest bound C satisfying conditions above is thentakento be the normof b denotedbyPut Lipβ,ω=Lipβ,1(ω).Obviously,for the case ω=1,the Lipβ,p(ω)space is the classical Lipβspace.Let ω∈A1.Garc′ía-Cuerva in[18]proved that the spaces Lipβ,p(ω) coincide,and the normsare equivalent with respect to different values of p provided that 1≤p＜∞.Since we always discuss under the assumption ω∈A1in the following,then we denote the norm offor 1≤p＜∞.

Definition 2.3(see[6]).Let b be a locally integrable function and ω be a weight function.A locally integrable function b is said to be in the weighted BMO function space BMO(ω),if there exists a constant C such that

If ω∈A1,Garc′ín-Cuera in[19]showed that

for 1≤p＜∞.

Now we shall introduce the Hardy-Littlewood maximal operator and several variants.

For a given measurable functiondefine the Hardy-Littlewood maximal operator Mf and the sharp maximal operatoras Z

For 1≤r＜∞,the weighted maximal operator Mω,rf is defined by

For 0＜β＜n,and 1≤r＜∞,we define the fractional weighted maximal operator Mβ,ω,rf by

where the supremum is taken over all ellipsoids ε.

Definition 2.4.Let 1≤p＜∞,0≤κ＜1 and ω be a weight function.Then for two weights μand ν,the weighted Morrey space is defined by

where

and the supremum is taken over all ellipsoids ε.

If ν=μ,then we have the classical Morrey space Lp,κ(μ)with measureμ.

(i)If 1≤r＜p＜∞,and 0＜κ＜1,then

(ii)If 0＜β＜n,1≤r＜p＜n/β,1/q=1/p?β/n and 0＜κ＜p/q,then

Lemma 2.3(see[16]).Let T be a singular integral operator with variable kernel,1＜p＜∞and 1＜κ＜1.If ω∈Ap,then there exists a constant C＞0 such that

In view of Proposition 3.1 in[20],we have

Lemma2.4.Let0＜κ＜1 and 1＜p＜∞.Ifμ,ν∈A∞,then for every f∈Llocwiththere exists a constant C such that

The following lemmas play a critical role in the proof of our theorems.

Lemma 2.5.Suppose ω∈A1,and b∈Lipβ,ω(0＜β＜1).Then there exist a sufficiently large number s and a constant C＞0 such that,for every f∈Lp(ω)with p＞1 and 1＜r＜p,we have

where 1/s+1/s′=1.

Proof.Let r2=r/s′,r3=?/(s′?1)and 1/r1+1/r2+1/r3=1,where ? is the constant in Lemma 2.1.Choosing a sufficiently large number s such that 1＜s′＜r(1+?)/(r+?),then r1,r2,r3＞1.By H¨older’s inequality,we have

Since b∈Lipβ,ω,and ω∈A1,by(2.1),(2.2)we get

Similar to the proof of Lemma 2.5,we have

Lemma 2.6.Suppose ω∈A1,and b∈BMO(ω).Then there exist sufficiently large number s and constant C＞0 such that,for every f∈Lp(ω)with p＞1 and 1＜r＜p,we have

where 1/s+1/s′=1.

Finally,we need the spherical harmonics and their properties(see more detail in[13, 15]).Recall that any homogeneouspolynomial P:Rn→R of degree m that satisfies?P=0 is called an n-dimensional solid harmonic of degree m.Its restriction to the unit sphere Sn?1will be called an n-dimensional spherical harmonic of degree m Denote by Hmthe space of all n-dimensional spherical harmonics of degree m.In general it results in a finite dimensional linear space with gm=dimHmsuch that g0=1,g1=n and

for any integer l.In particular,the expansion of φ into spherical harmonics converges uniformly to φ.For the proof of the above results see[21].

Let x,y∈Rn,and

In view of the properties of the kernel K with respect to the second variable and the complete of{Ysm(x)}in L2(Sn?1),we get

Replacing the kernel with its series expansion,(1.2)can be written as

From the properties of(2.9)-(2.11),the series expansion

where the integer l is preliminarily chosen greater than(3n?2)/4.Along with the ρ(x?y)?αf(y)∈L1(Rn)for almost everywhere x∈Rn,by the Fubini dominated convergence theorem,we have

where

and Hsmsatisfies pointwise H¨ormander condition as following

for each x∈ε and y/∈2ε(see[15,Lemma 3.2]).Then

is a classical Calder′on-Zygmund operator with a constant kernel.

3　Proof of theorems

Proof of Theorem 1.1.We only give the proof of Theorem 1.1,since the proof of Theorem 1.2 is similar to Theorem 1.1.Let

Without loss generality,we may assume Tk,1(k=1,···,Q)are singular integral operators with variable kernel.By(2.12),

where

are classical Calder′on-Zygmund operator with constant kernel as(2.14).Set ε for the ellipsoid centered at x0and of radius r,and let ε?x.Since T1(g)=0 for any g∈Lp,κ(ω), then

We first prove

Taking c=U2(x0),then

Choosing a sufficiently large number s and by H¨older’s inequality,the boundedness ofin Ls′(Rn)and Lemma 2.5,we have

For any y∈ε,and z∈(2ε)c,we have ρ(y?z)～ρ(x0?z).Then by(2.13)we get,

Note that ω∈A1,and

then by(2.2),we get

By H¨older’s inequality,

Hence

Combining the estimates for M1and M2,we finish the proof of(3.1).

Since ω∈A1implies ω1?q∈Aq,by Lemma 2.4,(3.1),Lemma 2.2 and Lemma 2.3,we

have

Choosing l＞(3n?2)/4,then

This finishes the proof of Theorem 1.1.□

Acknowledgments

The authors are very grateful to the anonymous refereesfor his/herinsightful comments and suggestions.

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10.4208/ata.2016.v32.n1.8

12 June 2015;Accepted(in revised version)8 January 2016

?Corresponding author.Email addresses:heyuexiang63@163.com(Y.He),wangys1962@163.com(Y.Wang)