# Toeplitz Type Operators Associated with Generalized Calderón-Zygmund Operator on Weighted Morrey Spaces

Authors: Bijun Ren; Enbin Zhang
Journal: Journal of Function Spaces
 (2016)

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## Abstract

LetT1 be a generalized Calderón-Zygmund operator or ±I (the identity operator), let T2 and T4 be the linear operators, and let T3=±I. Denote the Toeplitz type operator by Tb=T1MbIαT2+T3IαMbT4, where Mbf=bf and Iα is the fractional integral operator. In this paper, we investigate the boundedness of the operator Tb on weighted Morrey space when b belongs to the weighted BMO spaces.


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## Body

## 1. Introduction and Results

The classical Morrey spaces, introduced by Morrey [1] in 1938, have been studied intensively by various authors and together with weighted Lebesgue spaces play an important role in the theory of partial differential equations (see [2, 3]). Komori and Shirai [4] introduced a version of the weighted Morrey space Lp,κ(ω), which is a natural generalization of the weighted Lebesgue space Lp(ω).Definition 1.
Suppose thatT:S→S′ is a linear operator with kernel K(·,·) defined initially by (1)Tfx=∫RnKx,yfydy,f∈Cc∞Rn,x∉supp⁡f.The operator T is called a generalized Calderón-Zygmund operator provided that the following three conditions are satisfied: (1)
Tcan be extended into a continuous operator on L2(Rn).(2)
K is smooth away from the diagonal {(x,y):x=y} with (2)∫x-y>2z-yKx,y-Kx,z+Ky,x-Kz,xdx≤C,where C>0 is a constant independent of y and z.(3)
There is a sequence of positive constant numbers{Cj} such that, for each j∈N,(3)∫2jz-y≤x-y<2j+1z-yKx,y-Kx,zqdx1/q≤Cj2jz-y-n/q′,∫2jz-y≤x-y<2j+1z-yKy,x-Kz,xqdx1/q≤Cj2jz-y-n/q′,where (q,q′) is a fixed pair of positive numbers with 1/q+1/q′=1 and 1<q′<2.If we compare the generalized Calderón-Zygmund operator with the classical Calderón-Zygmund operator, whose kernelK is smooth away from the diagonal {(x,y):x=y} with (4)Kx,y≤Cx-y-n,Kx,y-Kx,z+Ky,x-Kz,x≤Cx-y-nz-yx-yδ,where |x-y|>2|z-y| for some δ>0, we can find out that the classical Calderón-Zygmund operator is a generalized Calderón-Zygmund operator defined above with Cj=2-jδ, j∈N, and any 1<q<∞.Letb be a locally integrable function on Rn. The Toeplitz type operator associated with generalized Calderón-Zygmund operator and fractional integral operator Iα is defined by (5)Tb=T1MbIαT2+T3IαMbT4,where T1 is the generalized Calderón-Zygmund operator or ±I (the identity operator), T2 and T4 are the linear operators, T3=±I, and Mbf=bf.Note that the commutators[b,Iα](f)=bIα(f)-Iα(bf) are the particular cases of the Toeplitz type operators Tb. The Toeplitz type operators Tb are the nontrivial generalization of these commutators.The boundedness of the singular integral commutators generated by BMO function was obtained in [5–8]. Motivated by these, in this paper, we investigate the boundedness of Tb on the weighted Morrey space when b belongs to weighted BMO space and we have the following result.Theorem 2.
Suppose thatTb is a Toeplitz type operator associated with generalized Calderón-Zygmund operator and fractional integral operator Iα, and b∈BMO(ω). Let 0<α<n, q′<p<n/α, 1/s=1/p-α/n, 0<κ<p/s, {jCj}∈l1, ωs/p∈A1 and the critical index of ω for the reverse Hölder condition rω>max⁡{s-1q′/(s-q′),(1-κ)/(p/s-κ)}. If T1(f)=0 for any f∈Lp,κ(ω), T2 and T4 are the bounded operators on Lp,κ(ω), and then there exists a constant C>0 such that (6)TbfLs,κs/pω1-1-α/ns,ω≤CbBMOωfLp,κω.The following results are immediately obtained from Theorem2.Corollary 3 (see [5]).
Let0<α<n, 1<p<n/α, 1/q=1/p-α/n, 0<κ<p/q, and ωq/p∈A1. Suppose that b∈BMO(ω) and the critical index of ω for the reverse Hölder condition rω>(1-κ)/(p/q-κ); then, [b,Iα] is bounded from Lp,κ(ω) to Lq,κq/p(ω1-(1-α/n)q,ω).Corollary 4.
Suppose thatTb is a Toeplitz type operator associated with generalized Calderón-Zygmund operator and fractional integral operator Iα and b∈BMO(Rn). Let 0<α<n, 1<p<n/α, 1/s=1/p-α/n, 0<κ<p/s, and {jCj}∈l1. If T1(f)=0 for any f∈Lp,κ(Rn), Tk,2 and Tk,4 are the bounded operators on Lp,κ(Rn); then, there exists a constant C>0 such that (7)TbfLs,κs/pRn≤CbBMOfLp,κRn.The paper is organized as follows. In Section2, we will introduce some notation and definitions and recall some preliminary results. In Section 3, we give the sharp estimates for Tb. In Section 4, we will give the proof of Theorem 2.


## 2. Some Preliminaries

First, let us recall some notation and the definition of weight classes.A weightω is a nonnegative, locally integrable function on Rn. Let B=Br(x0) denote the ball with the center x0 and radius r, and let λB=Bλr(x0) for any λ>0. For a given weight function ω and a measurable set E, we also denote the Lebesgue measure of E by |E| and set weighted measure ω(E)=∫Eω(x)dx. For any given weight function ω on Rn, 0<p<∞, denote by Lp(ω) the space of all function f satisfying (8)fLpω=∫Rnfxpωxdx1/p<∞.Definition 5 (see [9]).
A weightω is said to belong to the Muckenhoupt class Ap for 1<p<∞, if there exists a constant C such that (9)1B∫Bωxdx1B∫Bωx-1/p-1dxp-1≤Cfor every ball B. The class A1 is defined by replacing the above inequality with (10)1B∫Bωydy≤C·essinfx∈B⁡wx.When p=∞, we defined A∞=∪1≤p<∞Ap.Definition 6 (see [10]).
A weight functionω belongs to Ap,q for 1<p<q<∞, if for every ball B in Rn, there exists a positive constant C which is independent of B such that (11)1B∫Bωy-p′dy1/p′1B∫Bωyqdy1/q≤C,where p′ denotes the conjugate exponent of p>1; that is, 1/p+1/p′=1.From the definition ofAp,q, we can get that (12)ω∈Ap,q,iffωq∈A1+q/p′.Definition 7 (see [11]).
A weight functionω belongs to the reverse Hölder class RHu if there exist two constants u>1 and C>0 such that the following reverse Hölder inequality (13)1B∫Bωxudx1/u≤C1B∫Bωxdxholds for every ball B⊂Rn.It is well known that ifω∈Ap with 1<p<∞, then ω∈Au for all u>p and ω∈Aq for some 1<q<p. If ω∈Ap with 1≤p<∞, then there exists u>1 such that ω∈RHu. It follows directly from Hölder’s inequality that ω∈RHu implies ω∈RHs for all 1<s<u. Moreover, if ω∈RHu, u>1, then we have ω∈RHu+ε for some ε>0. We write rω=sup⁡{u>1:ω∈RHu} to denote the critical index of ω for the reverse Hölder condition.Lemma 8 (see [11]).
Supposeω∈A1. Then, there exist two constants C1 and C2, such that (14)C1ωB≤Binfx∈B⁡ωx≤C2ωB.Lemma 9 (see [11]).
Letω∈Ap, p≥1. Then, for any ball B and any λ>1, there exists an absolute constant C>0 such that (15)ωλB≤CλpnωB,where C does not depend on B or λ.Next, we will recall the definition of the Hardy-Littlewood maximal operator and several variants, the fractional integral operator, and some function spaces.Definition 10.
The Hardy-Littlewood maximal operatorMf is defined by (16)Mfx=supx∈B⁡1B∫Bfydy.For 0<δ<1, the sharp maximal operator Mδ♯f is defined by (17)Mδ♯fx=supx∈B⁡infc∈C⁡1B∫Bfyδ-cδdy1/δ.For 0≤α<n,t≥1, we define the fractional maximal operator Mα,tf by (18)Mα,tfx=supx∈B⁡1B1-αt/n∫Bfytdy1/tand define the fractional weighted maximal operator Mα,r,ωf by (19)Mα,t,ωfx=supx∈B⁡1ωB1-αt/n∫Bfytωydy1/t,where the above supremum is taken over all balls B containing x. In order to simplify the notation, we set Mα=Mα,1 and Mt,ω=M0,t,ω.Definition 11.
For0<α<n, the fractional integral operator Iα is defined by (20)Iαfx=∫Rnfyx-yn-αdy.Lemma 12.
LetIα be fractional integral operator, and let E be a measurable set in Rn. Then, for any f∈L1(Rn), there exists a constant C such that (21)∫EIαfxdx≤CfL1Eα/n.Proof.
Since(22)x∈E:Iαfx>λ≤x∈Rn:Iαfx>λ≤CfL1λn/n-α,we have(23)∫EIαfxdx=∫0∞x∈E:Iαfx>λdλ≤∫0∞min⁡CfL1λn/n-α,Edλ≤∫0CfL1Eα/n-1Edλ+∫CfL1Eα/n-1∞fL1λn/n-αdλ≤CfL1Eα/n.Definition 13.
Let1≤p<∞ and ω be a weighted function. A locally integrable function b is said to be in BMOp(ω) if (24)bBMOpω=supB⁡1ωB∫Bbx-bBpωx1-pdx1/p<∞,where bB=(1/B)∫Bb(y)dy and the supremum is taken over all balls B⊂Rn.Lemma 14 (see [12]).
Letω∈A1. Then, for any 1≤p<∞, there exists an absolute constant C>0 such that bBMOp(ω)≤CbBMO(ω).Definition 15 (see [4]).
Let1≤p<∞, 0<κ<1, and let ω be a weight function. Then, the weighted Morrey space is defined by (25)Lp,κω=f∈Llocpω:fLp,κω<∞,where (26)fLp,κω=supB⁡1ωBκ∫Bfxpωxdx1/p,and the supremum is taken over all balls B⊂Rn.In order to deal with the fractional order case, we need to consider the weighted Morrey space with two weights.Definition 16 (see [4]).
Let1≤p<∞ and 0<κ<1. Then, for two weights μ and ν, the weighted Morrey space is defined by (27)Lp,κμ,ν=f∈Llocpμ:fLp,κμ,ν<∞,where (28)fLp,κμ,ν=supB⁡1νBκ∫Bfxpμxdx1/p,and the supremum is taken over all balls B⊂Rn.We list a series of lemmas which will be used in the proof of our theorem.Lemma 17 (see [5]).
Let0<α<n, 1<p<n/α, 1/q=1/p-α/n, 0<κ<p/q, and ω∈A∞. Then, for any 1≤r<p, we have (29)Mα,r,ωfLq,κq/pω≤CfLp,κω.Lemma 18 (see [5]).
Let0<α<n, 1<p<n/α, 1/q=1/p-α/n, ωp/q∈A1, and rω>(1-κ)/(p/q-κ). Then, for every 0<κ<p/q and 1≤r<p, we have (30)Mr,ωfLq,κq/pωq/p,ω≤CfLq,κq/pωq/p,ω.Lemma 19 (see [5]).
Let0<α<n, 1<p<n/α, 1/q=1/p-α/n, and ωp/q∈A1. Then, if0<κ<p/q and rω>(1-κ)/(p/q-κ), we have (31)MαfLq,κq/pωq/p,ω≤CfLp,κω.Lemma 20 (see [4]).
Let0<α<n, 1<p<n/α, 1/q=1/p-α/n, and ωp/q∈A1. Then, if0<κ<p/q, we have (32)IαfLq,κq/pωq/p,ω≤CfLp,κω.


## 3. The Sharp Estimates forTb

In this section, we will prove the sharp estimates forTb as follows.Theorem 21.
Suppose thatTb is a Toeplitz type operator associated with generalized Calderón-Zygmund operator and fractional integral operator Iα, and b∈BMO(ω). Let 0<δ<1, 0<α<n, q′<p<n/α, {jCj}∈l1, ω∈A1, rω>q′, and t>rω-1q′/(rω-q′). If T1(f)=0 for any f∈Lp,κ(ω), then there exists a constant C>0 such that (33)Mδ♯Tbfx≤CbBMOωωxMt,ωIαT2fx+MαT4fx+CbBMOωωx1-α/nMα,t,ωT4fx.Proof.
For any ballB=B(x0,rB) which contains x, without loss generality, we may assume that T1 is a generalized Calderón-Zygmund operator. We write, by T1(f)=0, (34)Tbfy=T1MbIαT2fy+T3IαMbT4fy=Uby+Vby=Ub-b2By+Vb-b2By,where (35)Ub-b2By=T1Mb-b2Bχ2BIαT2fy+T1Mb-b2Bχ2BcIαT2fy=U1y+U2y,Vb-b2By=T3IαMb-b2Bχ2BT4fy+T3IαMb-b2Bχ2BcT4fy=V1y+V2y.Since 0<δ<1, then (36)1B∫BTbfyδ-U2x0+V2x0δdy1/δ≤1B∫BTbfy-U2x0-V2x0δdy1/δ≤1B∫BU1yδdy1/δ+1B∫BV1yδdy1/δ+1B∫BU2y-U2x0δdy1/δ+1B∫BV2y-V2x0δdy1/δ=M1+M2+M3+M4.
We are going to estimate each term, respectively. SinceT1 is bounded from L1 to WL1 and ω∈A1, then, by Kolmogorov’s inequality and Lemmas 8 and 9, we get (37)M1=1B∫BU1yδdy1/δ≤1B∫2Bby-b2BIαT2fydy≤CB∫2Bby-b2Bt′ωy1-t′dy1/t′∫2BIαT2fytωydy1/t≤CbBMOωω2BBMt,ωIαT2fx≤CbBMOωωxMt,ωIαT2fx.
SinceT3=±I, by Hölder’s inequality and Lemma 12, we have (38)M2=1B∫BT3IαMb-b2Bχ2BT4fyδdy1/δ≤CB∫BT3IαMb-b2Bχ2BT4fydy=CB∫BIαMb-b2Bχ2BT4fydy≤C1B1-α/n∫RnMb-b2Bχ2BT4fydy≤C1B1-α/n∫2Bby-b2Bt′ωy1-t′dy1/t′∫2BT4fytωydy1/t≤CbBMOωωBB1-α/n1ωB1-αt/n∫2BT4fytωydy1/t≤CbBMOωωx1-α/nMα,t,ωT4fx.
By the definition of generalized Calderón-Zygmund operator, we have(39)U2y-U2x0=T1Mb-b2Bχ2BcIαT2fy-T1Mb-b2Bχ2BcIαT2fx0≤C∫2Bcbz-b2BKy,z-Kx0,zIαT2fzdz.Then, by Hölder’s inequality, we get (40)M3=1B∫BU2y-U2x0δdy1/δ≤1B∫BU2y-U2x0dy≤1B∫BT1Mb-b2Bχ2BcIαT2fy-T1Mb-b2Bχ2BcIαT2fx0dy≤CB∫B∫2Bcbz-b2BKy,z-Kx0,zIαT2fzdzdy≤CB∑j=1∞∫B∫2jy-x0≤z-x0<2j+1y-x0bz-b2j+1BKy,z-Kx0,zIαT2fzdzdy+CB∑j=1∞b2j+1B-b2B∫B∫2jy-x0≤z-x0<2j+1y-x0Ky,z-Kx0,zIαT2fzdzdy=M31+M32.Since rω>q′>1 and t>rω-1q′/(rω-q′)>q′, then there exists 1<l<∞ such that 1/q+1/t+1/l=1. Applying Hölder’s inequality for q, l, and t and by (3) of Definition 1, we get (41)M31≤CB∑j=1∞∫B∫2jy-x0≤z-x0<2j+1y-x0Ky,z-Kx0,zqdz1/q∫2jy-x0≤z-x0<2j+1y-x0bz-b2j+1Blωz1/t′-1ldz1/l∫2jy-x0≤z-x0<2j+1y-x0IαT2fztωzdz1/tdy≤CB∑j=1∞Cj∫B2jy-x0-n/q′dy∫2j+1Bbz-b2j+1Blωz1/t′-1ldz1/l∫2j+1BIαT2fztωzdz1/t.Note that (42)∫B2jy-x0-n/q′dy≤C2jB-1/q′rn;then, (43)M31≤CMt,ωIαT2fx∑j=1∞Cj2jB-1/q′ω2jB1/t∫2j+1Bbz-b2j+1Blωz1/t′-1ldz1/l.Since t>rω-1q′/(rω-q′), we have rω>t-1q′/(t-q′). By rω=sup{u>1:ω∈RHu}, there is u such that u>t-1q′/(t-q′)>q′>1. Let p0=(u-1)/(t-1q′/t-q′-1); then, 1<p0<∞. By 1/q+1/l+1/t=1, we have l/t′=t-1q′/(t-q′). Then, p0=(u-1)/(l/t′-1); that is, u=lp0/t′-p0/p0′. Then, we have(44)1-up0l-1q′=-1,1t+1p0′l+up0l.Applying Hölder’s inequality for p0 and p0′, the definition of weighted BMO, and ω∈RHu, we have (45)∫2j+1Bbz-b2j+1Blωz1/t′-1ldz1/l≤C∫2j+1Bbz-b2j+1Blp0′ωz1-lp0′dz1/p0′l∫2j+1Bωzudz1/p0′l≤CbBMOω2jB1-u/p0lω2jB1/p0′l+u/p0l.Hence,(46)M31≤CbBMOωMt,ωIαT2fx∑j=1∞Cj2jB1-u/p0l-1/q′ω2jB1/t+1/p0′l+u/p0l≤CbBMOωMt,ωIαT2fx∑j=1∞Cjω2jB2jB≤CωxbBMOωMt,ωIαT2fx.
Note that(47)b2j+1B-b2B≤∑k=1j12kB∫2k+1Bbz-b2k+1Bdz≤CjωxbBMOω.Thus, by Hölder’s inequality, we get(48)M32≤CB∑j=1∞∫B∫2jy-x0≤z-x0<2j+1y-x0Ky,z-Kx0,zqdz1/qb2j+1B-b2B2j+1B1/l∫2j+1BIαT2fztdz1/tdy≤CBωxbBMOω∑j=1∞jCj∫B2jy-x0-n/q′dy1ω2j+1B∫2j+1BIαT2fztωzdz1/t2j+1B1/l+1/t≤CωxbBMOωMt,ωIαT2fx∑j=1∞jCj2j+1B1/l+1/t-1/q′≤CωxbBMOωMt,ωIαT2fx∑j=1∞jCj≤CωxbBMOωMt,ωIαT2fx.Then,(49)M3≤CωxbBMOωMt,ωIαT2fx.
For anyy∈B and z∈(2B)c, we have |y-z|~|x0-z|. Then, (50)M4=1B∫BV2y-V2x0δdy1/δ≤1B∫BV2y-V2x0dy≤1B∫BT3IαMb-b2Bχ2BcT4fy-T3IαMb-b2Bχ2BcT4fx0dy≤C1B∫B∫2Bcbz-b2B1y-zn-α-1x0-zn-αT4fzdzdy≤C1B∫B∫2Bcbz-b2Bx0-yx0-zn-α+1T4fzdzdy≤C∑j=1∞rB2jrBn-α+1∫2j+1Bbz-b2BT4fzdz≤C∑j=1∞2-jb2j+1B-b2B12j+1B1-α/n∫2j+1BT4fzdz+C∑j=1∞2-j12j+1B1-α/n∫2j+1Bbz-b2j+1BT4fzdz=M41+N42.Note that(51)b2j+1B-b2B≤CjbBMOω;then,(52)M41=C∑j=1∞2-jb2j+1B-b2B12j+1B1-α/n∫2j+1BT4fzdz≤CbBMOωωxMαT4fx∑j=1∞j2-j≤CbBMOωωxMαT4fx.By Hölder’s inequality, (53)M42=C∑j=1∞2-j12j+1B1-α/n∫2j+1Bbz-b2j+1BT4fzdz≤C∑j=1∞2-j12j+1B1-α/n∫2j+1Bbz-b2j+1Bt′ωz1-t′dz1/t′∫2j+1BT4fztωzdz1/t≤CbBMOω∑j=1∞2-jω2j+1B2j+1B1-α/n1ω2j+1B1-αt/n∫2j+1BT4fztωzdz1/t≤CbBMOωωx1-α/nMα,t,ωT4fx.Then,(54)M4≤CbBMOωωxMαT4fx+ωx1-α/nMα,t,ωT4fx.
Combining the estimates forM1, M2, M3, and M4, the proof of Theorem 21 is completed.


## 4. Proof of Theorem2

To prove Theorem2, we need the following analogy of the classical Fefferman-Stein inequality for the sharp maximal function Mδ♯f; its proof can be found in [13].Lemma 22.
Let0<δ<1, 0<κ<1, and 1<p<∞. If μ,ν∈A∞, then we have (55)MδfLp,κμ,ν≤CMδ♯fLp,κμ,νfor all functions f such that the left hand side is finite.Proof.
It follows fromrω>s-1q′/(s-q′) that s>rω-1q′/(rω-q′); then, there exists t such that s>t>rω-1q′/(rω-q′). Note that (56)s+1-1-αns=sp;then, by Lemma 22 and Theorem 21, we have (57)TbfLs,κs/pω1-1-α/ns,ω≤Mδ♯TbfLs,κs/pω1-1-α/ns,ω≤CbBMOωMt,ωIαT2fLs,κs/pωs/p,ω+MαT4fLs,κs/pωs/p,ω+CbBMOωMα,t,ωT4fLs,κs/pω.Since rω>(1-κ)/(p/s-κ), by Lemmas 17–20, we get (58)TbfLs,κs/pω1-1-α/ns,ω≤CbBMOωIαT2fLs,κs/pωs/p,ω+T4fLp,κω≤CbBMOωfLp,κω.
This finishes the proof of Theorem2.



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