# WeightedBMO Estimates for Toeplitz Operators on Weighted Lebesgue Spaces

Authors: Yan Lin; Mengmeng Zhang
Journal: Journal of Function Spaces
 (2015)

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## Abstract

The authors establish the weightedBMO estimates for a class of Toeplitz operators related to strongly singular Calderón-Zygmund operators on weighted Lebesgue spaces. Moreover, the corresponding result for the Toeplitz operators related to classical Calderón-Zygmund operators can be deduced.


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## Body

## 1. Introduction

The introduction of the strongly singular Calderón-Zygmund operator is motivated by a class of multiplier operators with symbols given byei|ξ|α/|ξ|β away from the origin, where 0<α<1 and β>0. Fefferman and Stein [1] enlarged the multiplier operators onto a class of convolution operators. Coifman [2] also considered a related class of operators for n=1. The strongly singular nonconvolution operator was introduced by Alvarez and Milman [3], whose properties are similar to those of the classical Calderón-Zygmund operator, but the kernel is more singular near the diagonal than that of the standard case.Definition 1.
LetT:S→S′ be a bounded linear operator. T is called a strongly singular Calderón-Zygmund operator if the following conditions are satisfied.(1)
T can be extended into a continuous operator from L2(Rn) into itself.(2)
There exists a functionK(x,y) continuous away from the diagonal {(x,y):x=y} such that (1)Kx,y-Kx,z+Ky,x-Kz,x≤Cy-zδx-zn+δ/α,if2|y-z|α≤|x-z| for some 0<δ≤1 and 0<α<1. 〈Tf,g〉=∬Kx,yfygxdydx, for f,g∈S with disjoint support.(3)
For somen(1-α)/2≤β<n/2, both T and its conjugate operator T∗ can be extended into continuous operators from Lq to L2, where 1/q=1/2+β/n.Alvarez and Milman [3, 4] discussed the properties of the strongly singular Calderón-Zygmund operator on Lebesgue spaces. Lin [5] proved the boundedness of the strongly singular Calderón-Zygmund operator and its commutators on Morrey type spaces. The authors in [6] established the weighted BMO estimates for the commutator of the strongly singular Calderón-Zygmund operator on weighted Morrey spaces.The commutator generated by the Calderón-Zygmund operator and a locally integrable functionb can be regarded as a special case of the Toeplitz operator Tb=∑j=1mTj,1MbTj,2, where Tj,1 and Tj,2 are the Calderón-Zygmund operators or ±I (I is the identity operator), Mbf(x)=b(x)f(x). When b∈BMO, Krantz and Li [7] discussed the Lp boundedness of Tb on the homogeneous spaces. The commutator generated by the fractional integral operator and a locally integrable function b can be regarded as a special case of another class of Toeplitz operators Θα0b=∑j=1m(Tj,1MbIα0Tj,2+Tj,3Iα0MbTj,4), where Tj,1 are the Calderón-Zygmund operators or ±I, Tj,2 and Tj,4 are the bounded linear operators on Lp, Tj,3=±I, and Iα0f(x)=∫Rnf(y)/|x-y|n-α0dy is the fractional integral operator. When b∈BMO, Qiu [8] obtained the boundedness of Θα0b on the homogeneous spaces from Lp(X) to Lq(X), where 1/q=1/p-α0. Moreover, Lin and Lu [9] discussed the boundedness of the two kinds of Toeplitz operators Tb and Θα0b related to strongly singular Calderón-Zygmund operators and BMO functions on Lebesgue spaces.In this paper, we are interested in the Toeplitz operatorsΘα0b related to strongly singular Calderón-Zygmund operators and weighted BMO functions. Actually, in this situation, Tj,1 are the strongly singular Calderón-Zygmund operators or ±I, Tj,2 and Tj,4 are the bounded linear operators on weighted Lebesgue spaces, Tj,3=±I, and b is a weighted BMO function. And we will focus on the boundedness of this kind of Toeplitz operators on weighted Lebesgue spaces.Before stating our main results, first we need to recall some definitions and notations as follows.Definition 2.
A nonnegative measurable functionω is said to be in the Muckenhoupt class Ap with 1<p<∞ if, for every cube Q in Rn, there exists a positive constant C independent of Q such that (2)1Q∫Qωxdx1Q∫Qωx1-p′dxp-1≤C,where Q denotes a cube in Rn with the side parallel to the coordinate axes and 1/p+1/p′=1. When p=1, a nonnegative measurable function ω is said to belong to A1, if there exists a constant C>0 such that, for any cube Q, (3)1Q∫Qωydy≤Cωx,a.e.x∈Q.
DenoteA∞=⋃1≤p<∞Ap. It is well known that if ω∈Ap with 1<p<∞, then ω∈Ar for all r>p and ω∈Aq for some 1<q<p.Definition 3.
A weighted functionω belongs to the reverse Hölder class RHr if there exist two constants r>1 and C>0 such that the following reverse Hölder inequality (4)1Q∫Qωxrdx1/r≤C1Q∫Qωxdxholds for every cube Q in Rn.It is well known that ifω∈Ap with 1≤p<∞, then there exists a r>1 such that ω∈RHr. It follows directly from Hölder’s inequality that ω∈RHr implies ω∈RHs for all 1<s<r.Definition 4.
Suppose thatω is a nonnegative locally integrable function on Rn. Define ω∈Ap,q(1<p,q<∞), if there exists a constant C>0, such that, for any cube Q in Rn, (5)supQ1Q∫Qωxqdx1/q1Q∫Qωx-p′dx1/p′≤C<∞.Definition 5.
Let1≤p<∞ and let ω be a weighted function. A locally integrable function b is said to be in the weighted BMO space BMOp(ω) if (6)bBMOpω=supQ1ωQ∫Qbx-bQpωx1-pdx1/p<∞,where bQ=1/|Q|∫Qb(y)dy and the supremum is taken over all cubes Q⊂Rn. Moreover, we denote BMOp(ω) simply by BMO(ω) when p=1.Definition 6.
The Hardy-Littlewood maximal operatorM is defined by (7)Mfx=supQ∋x1Q∫Qfydy.We set Ms(f)=M(|f|s)1/s, where 0<s<∞.
The sharp maximal operatorM♯ is defined by (8)M♯fx=supQ∋x1Q∫Qfy-fQdy~supQ∋xinfa∈C1Q∫Qfy-ady,where fQ=1/|Q|∫Qf(x)dx.
Letω be a weight. The weighted maximal operator Mω is defined by (9)Mωfx=supQ∋x1ωQ∫Qfyωydy.We also set Ms,ω(f)=Mω(|f|s)1/s, where 0<s<∞. The fractional maximal operator Mα0 and the weighted case Mα0,s,ω are defined by (10)Mα0fx=supQ∋x1Q1-α0/n∫Qfydy,(11)Mα0,s,ωfx=supQ∋x1ωQ1-α0s/n∫Qfysωydy1/s,0<α0<n,respectively.


## 2. Main Results

Now we state our main results as follows.Theorem 7.
LetTj,1 be strongly singular Calderón-Zygmund operators or ±I, let α,β,δ be the same as in Definition 1, and n(1-α)/2<β<n/2. Suppose 0<α0<2nβ/n(1-α)+2β, n2(1-α)/α0[n(1-α)+2β]+1<p<n/α0, 1/q=1/p-α0/n, ωq∈A1, and  ω∈RHr with r>q/p, b∈BMO(ω), Tj,3=±I; Tj,2 and Tj,4 are bounded operators on Lp(ω). When f∈Cc∞(Rn), Θα01(f)=0. Then Θα0b is bounded from Lp(ω) to Lq(ωq/p-q).If we consider the extreme casesα→1 and β→0 in Definition 1, then the strongly singular Calderón-Zygmund operator comes back to the classical Calderón-Zygmund operator. Thus, we can get the boundedness of the Toeplitz operators related to classical Calderón-Zygmund operators on weighted Lebesgue spaces as a corollary of Theorem 7.Corollary 8.
LetTj,1 be classical Calderón-Zygmund operators or ±I. Suppose 0<α0<n, 1<p<n/α0, 1/q=1/p-α0/n, ωq∈A1, and ω∈RHr with r>q/p, b∈BMO(ω), Tj,3=±I; Tj,2 and Tj,4 are bounded operators on Lp(ω). When f∈Cc∞(Rn), Θα01(f)=0. Then Θα0b is bounded from Lp(ω) to Lq(ωq/p-q).


## 3. Lemmas

Before giving the proof of our main results, we need some necessary lemmas.Lemma 9.
Ifωq∈A1, then ω∈A1, where 1<q<∞.Lemma 10.
Ifω∈A1, then ω1/q∈Ap,q, where 1<p,q<∞.The results of Lemmas9 and 10 can be deduced directly by Definitions 2 and 4 and Hölder’s inequality. We omit the details.Lemma 11.
Ifωq∈A1, then ω∈Ap,q and ω1/p∈Ap,q, where 1<p,q<∞.Proof.
Sinceωq∈A1, then ω=(ωq)1/q∈Ap,q by Lemma 10. It follows from (ωq/p)p=ωq∈A1, p>1, and Lemma 9 that ωq/p∈A1. Then applying Lemma 10, we have ω1/p=(ωq/p)1/q∈Ap,q.Lemma 12 (see [6]).
Letω∈A1 and let f be a function in BMO(ω). Suppose 1≤p<∞, x∈Rn, and r1,r2>0. Then(12)1Bx,r1∫Bx,r1fy-fBx,r2pωy1-pdy1/p≤CfBMOωωx1+ln⁡r2r1·ωBx,r1Bx,r1-1/p′.Lemma 13 (see [6]).
Givenɛ>0, there is lnx≤1/ɛxɛ, for all x≥1.Lemma 14 (see [10]).
The fractional integral operatorIα0 is bounded from Lp(Rn) to Lq(Rn), where 1/q=1/p-α0/n, 1<p<n/α0, and 0<α0<n.Lemma 15 (see [11, 12]).
Letω∈A1. Then, for any 1≤p<∞, there exists an absolute constant C>0 such that bBMOp(ω)≤CbBMO(ω).Lemma 16 (see [3, 4]).
IfT is a strongly singular Calderón-Zygmund operator, then T can be defined to be a continuous operator from L∞ to BMO and T is of weak (L1,L1) type.By Lemma16, Definition 1, and the interpolation theory, we can get that T is bounded on Lp, 1<p<∞. Besides the (Lp,Lp)-boundedness, the strongly singular Calderón-Zygmund operator T still has another kind of boundedness properties on Lebesgue spaces. If we interpolate between (L2,Lq′) and (L∞,BMO), where q is given as in Definition 1 and 1/q+1/q′=1, then T is bounded from Lu to Lv with 2≤u<∞ and v=uq′/2. It is easy to see that 0<u/v≤α in this situation. Then we interpolate between (L2,Lq′) and weak (L1,L1) to obtain the boundedness of T from Lu to Lv, where 1<u≤2 and v=uq′/2q′-uq′+2u-2. In this situation, 0<u/v≤α if and only if n(1-α)+2β/2β≤u≤2. In a word, the boundedness properties of the strongly singular Calderón-Zygmund operator on Lebesgue spaces can be summarized as follows.Remark 17.
The strongly singular Calderón-Zygmund operatorT is bounded on Lp for 1<p<∞. And T is bounded from Lu to Lv, n(1-α)+2β/2β≤u<∞, and 0<u/v≤α. In particular, if we restrict n(1-α)/2<β<n/2 in (3) of Definition 1, then T is bounded from Lu to Lv, where n(1-α)+2β/2β<u<∞ and 0<u/v<α.Lemma 18 (see [13]).
Let1<p<∞ and ω∈A∞. Then MωfLp(ω)≤CfLp(ω).Lemma 19.
If1<p<∞ and ω∈A∞, then, for any 1<s<p, we have (13)Ms,ωfLpω≤CfLpω.Proof.
By Lemma18, we have (14)Ms,ωfLpω=MωfsLp/sω1/s≤CfsLp/sω1/s=CfLpω.Lemma 20.
If0<s<∞, r>1, and ω∈RHr, then Ms,ω(f)(x)≤CMsr,ωr(f)(x).Proof.
For any ballB⊂Rn which contains x, we have (15)1ωB∫Bfysωydy1/s≤1ωB1/s∫Bfysrωrydy1/srB1/s-1/sr≤Msr,ωrfxωrB1/srωB1/sB1/s-1/sr.
Since(16)ωrB=∫Bωrydy≤C1B∫BωydyrB=CωBrB1-r,then (17)1ωB∫Bfysωydy1/s≤CMsr,ωrfxωB1/sB1/sr-1/sωB1/sB1/s-1/sr=CMsr,ωrfx.Thus we can get the desired conclusion by taking over all balls B which contain x.Lemma 21 (see [13]).
Suppose that1≤s<p<n/α0, 1/q=1/p-α0/n, 0<α0<n, and ω∈Ap,q. Then Mα0,s,ωfLq(ω)≤CfLp(ω).Lemma 22 (see [14]).
Suppose that0<α0<n, 1<p<n/α0, 1/q=1/p-α0/n, and ω∈Ap,q. Then (18)∫RnIα0fxωxqdx1/q≤C∫Rnfxωxpdx1/p.Lemma 23 (see [14]).
Suppose that0<α0<n, 1<p<n/α0, and 1/q=1/p-α0/n. If ω∈Ap,q, then (19)∫RnMα0fxωxqdx1/q≤C∫Rnfxωxpdx1/p,where C is independent of f.Lemma 24 (see [14]).
Letω∈Ap, 1≤p0≤p<∞. If f is such that Mf∈Lp0(ω), then (20)∫RnMfxpωxdx≤C∫RnM♯fxpωxdx.Lemma 25.
LetTj,1 be strongly singular Calderón-Zygmund operators or ±I, let α,β,δ be the same as in Definition 1, and n(1-α)/2<β<n/2. Suppose 0<α0<2nβ/n(1-α)+2β, n(1-α)+2β/2β<s<n/α0, and ω∈A1∩RHr with r>[n(1-α)+2β](s-1)/2βs-[n(1-α)+2β], b∈BMO(ω), and Tj,3=±I. Suppose also Θα01(f)=0, when f∈Cc∞(Rn). Then for a.e. x∈Rn we have (21)M♯Θα0bfx≤C∑j=1mTj,1bBMOωωx·Ms,ωIα0Tj,2fx+C∑j=1mbBMOωωx·Ms,ωIα0Tj,2fx+C∑j=1mIα0bBMOω·ωx1-α0/nMα0,s,ωTj,4fx+C∑j=1mbBMOω·ωx1-α0/nMα0,s,ωTj,4fx+C∑j=1mbBMOω·ωxMα0Tj,4fx.Proof.
DenoteΘα0b=Aα0b+Bα0b, where (22)Aα0b=∑j=1mTj,1MbIα0Tj,2,Bα0b=∑j=1mTj,3Iα0MbTj,4.For x∈Rn, let B be a ball centered at x with radius rB. Then (23)Θα0bf=Θα0b-b2Bf=Aα0b-b2Bf+Bα0b-b2Bf.
We considerBα0b-b2B(f) first. Write (24)Bα0b-b2Bf=∑j=1mTj,3Iα0Mb-b2Bχ2BTj,4f+∑j=1mTj,3Iα0Mb-b2Bχ2BcTj,4f≕f11+f12.Then (25)1B∫BBα0b-b2Bfy-f12xdy≤1B∫Bf11ydy+1B∫Bf12y-f12xdy≕I1+I2.
Sincen(1-α)+2β/2β<s<n/α0 and r>[n(1-α)+2β](s-1)/2βs-[n(1-α)+2β], then n(1-α)+2β/2β<rs/r+s-1. There is l1 such that n(1-α)+2β/2β<l1<rs/r+s-1. Denote 1/l2=1/l1-α0/n. By Lemma 14 we can get that Iα0 is bounded from Ll1 to Ll2. Then (26)I1≤∑j=1m1B∫BIα0Mb-b2Bχ2BTj,4fyl2dy1/l2≤∑j=1mIα0l1,l2B-1/l2∫2Bby-b2Bl1·ωyl1/s-l1/sTj,4fyl1dy1/l1.
Sincen(1-α)+2β/2β<l1<rs/r+s-1<s, there is an l(1<l<∞) such that 1/l1=1/s+1/l. Let p0=(r-1)(s-l1)/(r-1)(s-l1)-s(l1-1). Since l1<rs/r+s-1, we have 1<p0<∞,  p0′=(r-1)(s-l1)/s(l1-1),  1<p0′<∞, and r=lp0′/s′-p0′/p0. By Hölder’s inequality, we get(27)I1≤∑j=1mIα0l1,l2B-1/l2·∫2Bby-b2Blωy-l/sdy1/l·∫2BTj,4fysωydy1/s≤∑j=1mIα0l1,l2·B-1/l2∫2Bby-b2Blωy-l/sdy1/l·ω2B1/s-α0/n1ω2B1-α0s/n·∫2BTj,4fysωydy1/s≤∑j=1mIα0l1,l2·B-1/l2ω2B1/s-α0/nMα0,s,ωTj,4fx·∫2Bby-b2Blωy1/p0-l+l/s′-1/p0dy1/l≤∑j=1mIα0l1,l2B-1/l2ω2B1/s-α0/nMα0,s,ωTj,4f·x∫2Bby-b2Blp0ωy1-lp0dy1/lp0·∫2Bωylp0′/s′-p0′/p0dy1/lp0′≤∑j=1mIα0l1,l2·B-1/l2ω2B1/s-α0/nMα0,s,ωTj,4fxbBMOω·ω2B1/lp02B1/lp0′12B∫2Bωyrdy1/lp0′≤C∑j=1mIα0l1,l2Mα0,s,ωTj,4fxbBMOω·ω2B2B1-α0/n≤C∑j=1mIα0l1,l2bBMOω·ωx1-α0/nMα0,s,ωTj,4fx.
Note thatω∈A1; a direct calculation shows that (28)b2k+1B-b2B≤CbBMOωkωx.Applying Hölder’s inequality to I2, we can get (29)I2≤∑j=1m1B∫B∫2Bcbz-b2BTj,4fz1y-zn-α0-1x-zn-α0dzdy≤C∑j=1m∑k=1∞∫2k+1B∖2kBbz-b2BTj,4fz·rBx-zn-α0+1dz≤C∑j=1m∑k=1∞rB2krBn-α0+1·∫2k+1Bbz-b2k+1BTj,4fzdz+b2k+1B-b2B∫2k+1BTj,4fzdz≤C∑j=1m∑k=1∞rB2krBn-α0+1·∫2k+1Bbz-b2k+1Bs′ωz1-s′dz1/s′·∫2k+1BTj,4fzsωzdz1/s+C∑j=1m∑k=1∞rB2krBn-α0+1kbBMOωωx·∫2k+1BTj,4fzdz≤C∑j=1m∑k=1∞rB2krBn-α0+11ω2k+1B1-α0s/n·∫2k+1BTj,4fzsωzdz1/s·ω2k+1B1/s-α0/n+1/s′1ω2k+1B∫2k+1Bbz-b2k+1Bs′ωz1-s′dz1/s′+C∑j=1m∑k=1∞rB2krBn-α0+1kbBMOωωx·2k+1B1-α0/n12k+1B1-α0/n∫2k+1BTj,4fzdz≤C∑j=1m∑k=1∞rB2krBn-α0+1ω2k+1B1-α0/nbBMOω·Mα0,s,ωTj,4fx+C∑j=1m∑k=1∞rB2krBn-α0+1kbBMOωωx·2k+1rBn-α0Mα0Tj,4fx≤C∑j=1m∑k=1∞2-kbBMOωωx1-α0/nMα0,s,ωTj,4f·x+C∑j=1m∑k=1∞2-kkbBMOωωxMα0Tj,4f·x≤C∑j=1mbBMOωωx1-α0/nMα0,s,ωTj,4f·x+C∑j=1mbBMOωωxMα0Tj,4fx.Thus (30)1B∫BBα0b-b2Bfy-f12xdy≤C∑j=1mIα0bBMOωωx1-α0/nMα0,s,ωTj,4fx+C∑j=1mbBMOωωx1-α0/nMα0,s,ωTj,4fx+C∑j=1mbBMOωωxMα0Tj,4fx.
Secondly, we considerAα0b-b2B(f) in the following two cases.Case 1 (0<rB≤1).
Sincen(1-α)+2β/2β<s<n/α0 and r>[n(1-α)+2β](s-1)/2βs-[n(1-α)+2β], then n(1-α)+2β/2β<rs/r+s-1 and there is u satisfying n(1-α)+2β/2β<u<rs/r+s-1. By Remark 17, we can find v such that the strongly singular Calderón-Zygmund operator T is bounded from Lu to Lv and 0<u/v<α. So there is θ such that 0<u/v<θ<α. Let B~ be a ball centered at x whose radius is rBθ. Consider(31)Aα0b-b2Bf=∑j=1mTj,1Mb-b2Bχ2B~Iα0Tj,2f+∑j=1mTj,1Mb-b2Bχ2B~cIα0Tj,2f≕f21+f22.Then (32)1B∫BAα0b-b2Bfy-f22xdy≤1B∫Bf21ydy+1B∫Bf22y-f22xdy≕II1+II2.
Write(33)II1≤∑j=1m1B∫BTj,1Mb-b2Bχ2B~Iα0Tj,2fydy≕∑j=1mII1,j.
IfTj,1 is a strongly singular Calderón-Zygmund operator, then (34)II1,j≤1B∫BTj,1Mb-b2Bχ2B~Iα0Tj,2fyvdy1/v≤Tj,1u,vB-1/v·∫2B~by-b2BuIα0Tj,2fyudy1/u.
Sincen(1-α)+2β/2β<u<rs/r+s-1<s, there is s1(1<s1<∞) such that 1/u=1/s+1/s1. Let p1=(r-1)(s-u)/(r-1)(s-u)-s(u-1); then r=s1p1′/s′-p1′/p1. Applying Lemma 12, we have (35)II1,j≤Tj,1u,vB-1/v·∫2B~by-b2Bs1ωy-s1/sdy1/s1·∫2B~Iα0Tj,2fysωydy1/s≤Tj,1u,v·B-1/vω2B~1/sMs,ωIα0Tj,2fx·∫2B~by-b2Bs1ωy1/p1-s1+s1/s′-1/p1dy1/s1≤Tj,1u,vB-1/vω2B~1/sMs,ωIα0Tj,2fx·∫2B~ωyrdy1/s1p1′·∫2B~by-b2Bs1p1ωy1-s1p1dy1/s1p1≤Tj,1u,vB-1/vω2B~1/sMs,ωIα0Tj,2fx·2B~1/s1p1bBMOωωxω2B~2B~-1/s1p1′·1+ln⁡2rB2rBθ2B~1/s1p1′·12B~∫2B~ωyrdy1/rr/s1p1′≤CTj,1u,v·rBnθ/u-1/vMs,ωIα0Tj,2fxbBMOωωx·1+1-θln⁡1rB.
The factu/v<θ implies that ε1≔n(θ/u-1/v)>0. According to Lemma 13, we get (36)II1,j≤CTj,1u,vMs,ωIα0Tj,2fxbBMOωωx·rBnθ/μ-1/ν1+1ε1rB-ε1≤CTj,1u,v·Ms,ωIα0Tj,2fxbBMOωωxrBnθ/μ-1/ν-ε1=CTj,1u,vbBMOωωxMs,ωIα0Tj,2fx.
IfTj,1=±I, then (37)II1,j=1B∫Bby-b2BIα0Tj,2fydy≤1B∫Bby-b2Bs′ωy1-s′dy1/s′·∫BIα0Tj,2fysωydy1/s≤B1/s′-1·bBMOωωx1+ln⁡2rBrBωBB-1/s·ωB1/sMs,ωIα0Tj,2fx≤CbBMOωωx·Ms,ωIα0Tj,2fx.
By (36) and (37), we have (38)II1≤C∑j=1mTj,1bBMOωωxMs,ωIα0Tj,2fx.
Write(39)II2≤∑j=1m1B∫BTj,1Mb-b2Bχ2B~cIα0Tj,2fy-Tj,1Mb-b2Bχ2B~cIα0Tj,2fxdy≕∑j=1mII2,j.
IfTj,1 is a strongly singular Calderón-Zygmund operator, then(40)II2,j≤1B∫B∫2B~cKy,z-Kx,zbz-b2B·Iα0Tj,2fzdzdy≤C1B·∫B∑k=1∞∫2k+1B~∖2kB~y-xδx-zn+δ/αbz-b2BIα0Tj,2fzdzdy≤C1B∫B∑k=1∞rBδ2krBθn+δ/α∫2k+1B~bz-b2B·Iα0Tj,2fzdzdy≤C∑k=1∞rBδ2krBθn+δ/α·∫2k+1B~bz-b2Bs′ωz1-s′dz1/s′·∫2k+1B~Iα0Tj,2fzsωzdz1/s≤C∑k=1∞rBδ2krBθn+δ/α2k+1B~1/s′bBMOωωx1+ln⁡2rB2k+1rBθω2k+1B~2k+1B~-1/sω2k+1B~1/s·Ms,ωIα0Tj,2fx≤C∑k=1∞rBδ2krBθδ/αbBMOωωx·k1+1-θln⁡1rBMs,ωIα0Tj,2fx.
The factθ<α implies that ε2≔δ-θδ/α=δ(1-θ/α)>0. According to Lemma 13, we get (41)II2,j≤CrBδ-θδ/αbBMOωωxMs,ωIα0Tj,2fx·1+1ε2rB-ε2≤CrBδ-θδ/α-ε2bBMOωωx·Ms,ωIα0Tj,2fx=CbBMOωωx·Ms,ωIα0Tj,2fx.
Note thatII2,j=0; if Tj,1=±I, then (42)II2≤C∑j=1mbBMOωωxMs,ωIα0Tj,2fx.
Thus,(43)1B∫BAα0b-b2Bfy-f22xdy≤C∑j=1mTj,1bBMOωωxMs,ωIα0Tj,2fx+C∑j=1mbBMOωωxMs,ωIα0Tj,2fx.So (44)1B∫BΘα0bfy-f12x+f22xdy≤1B·∫BAα0b-b2Bfy-f22xdy+1B·∫BBα0b-b2Bfy-f12xdy≤C∑j=1mTj,1·bBMOωωxMs,ωIα0Tj,2fx+C∑j=1mbBMOωωxMs,ωIα0Tj,2fx+C∑j=1mIα0bBMOωωx1-α0/nMα0,s,ωTj,4f·x+C∑j=1mbBMOωωx1-α0/nMα0,s,ωTj,4f·x+C∑j=1mbBMOωωxMα0Tj,4fx.Case 2 (rB>1).
Consider(45)Aα0b-b2Bf=∑j=1mTj,1Mb-b2Bχ2BIα0Tj,2f+∑j=1mTj,1Mb-b2Bχ2BcIα0Tj,2f≕f31+f32.Then (46)1B∫BAα0b-b2Bfy-f32xdy≤1B∫Bf31ydy+1B∫Bf32y-f32xdy≕III1+III2.
Sincen(1-α)+2β/2β<rs/r+s-1, there is u1 satisfying n(1-α)+2β/2β<u1<rs/r+s-1<s. Then the strongly singular Calderón-Zygmund operator T is bounded from Lu1 to Lu1. And there is s2(1<s2<∞) such that 1/u1=1/s+1/s2. Let p2=(r-1)(s-u1)/(r-1)(s-u1)-s(u1-1). We have 1<p2<∞, p2′=(r-1)(s-u1)/s(u1-1), 1<p2′<∞, and r=s2p2′/s′-p2′/p2. Then (47)III1≤∑j=1m1B·∫BTj,1Mb-b2Bχ2BIα0Tj,2fyu1dy1/u1≤∑j=1mTj,1u1,u1B-1/u1·∫2Bby-b2Bu1Iα0Tj,2fyu1dy1/u1≤∑j=1mTj,1u1,u1B-1/u1·∫2Bby-b2Bs2ωy-s2/sdy1/s2·∫2BIα0Tj,2fysωydy1/s≤∑j=1mTj,1u1,u1·B-1/u1ω2B1/sMs,ωIα0Tj,2fx·∫2Bby-b2Bs2p2ωy1-s2p2dy1/s2p2·∫2Bωyrdy1/s2p2′≤C∑j=1mTj,1u1,u1B-1/u1·ω2B1/sMs,ωIα0Tj,2fxω2B1/s2p2·bBMOω2B1/s2p2′ω2B2Br/s2p2′≤C∑j=1mTj,1u1,u1bBMOωωxMs,ωIα0Tj,2f·x.
Write(48)III2≤∑j=1m1B∫BTj,1Mb-b2Bχ2BcIα0Tj,2fy-Tj,1Mb-b2Bχ2BcIα0Tj,2fxdy≕∑j=1mIII2,j.
IfTj,1 is a strongly singular Calderón-Zygmund operator, then (49)III2,j≤1B∫B∫2BcKy,z-Kx,zbz-b2B·Iα0Tj,2fzdzdy≤C1B·∫B∑k=1∞∫2k+1B∖2kBy-xδx-zn+δ/αbz-b2BIα0Tj,2fzdzdy≤C∑k=1∞rBδ2krBn+δ/α∫2k+1Bbz-b2Bs′ωz1-s′dz1/s′·∫2k+1BIα0Tj,2fzsωzdz1/s≤C∑k=1∞rBδ2krBn+δ/α·2k+1B1/s′bBMOωωx1+ln⁡2rB2k+1rB·ω2k+1B2k+1B-1/sω2k+1B1/sMs,ωIα0Tj,2fx≤C∑k=1∞k2-kδ/αrBδ-δ/αbBMOωωxMs,ωIα0Tj,2fx≤CbBMOωωxMs,ωIα0Tj,2fx.
SinceIII2,j=0 when Tj,1=±I, then (50)III2≤C∑j=1mbBMOωωxMs,ωIα0Tj,2fx.
Thus,(51)1B∫BAα0b-b2Bfy-f32xdy≤C∑j=1mTj,1bBMOωωxMs,ωIα0Tj,2fx+C∑j=1mbBMOωωxMs,ωIα0Tj,2fx.So (52)1B∫BΘα0bfy-f12x+f32xdy≤1B·∫BAα0b-b2Bfy-f32xdy+1B·∫BBα0b-b2Bfy-f12xdy≤C∑j=1mTj,1·bBMOωωxMs,ωIα0Tj,2fx+C∑j=1mbBMOωωxMs,ωIα0Tj,2fx+C∑j=1mIα0bBMOωωx1-α0/nMα0,s,ωTj,4f·x+C∑j=1mbBMOωωx1-α0/nMα0,s,ωTj,4f·x+C∑j=1mbBMOωωxMα0Tj,4fx.In conclusion, combining the two cases, we can get the desired estimate for M♯(Θα0b(f))(x).


## 4. Proof of Main Result

Now we are able to prove our main result.Proof of Theorem7.
Sincen2(1-α)/α0[n(1-α)+2β]+1<p, we have q/p=n/n-pα0>[n(1-α)+2β](p-1)/2βp-[n(1-α)+2β]. So r>q/p>[n(1-α)+2β](p-1)/2βp-[n(1-α)+2β]; that is, p>[n(1-α)+2β](r-1)/2βr-[n(1-α)+2β]. There exists s such that p>s>[n(1-α)+2β](r-1)/2βr-[n(1-α)+2β]>n(1-α)+2β/2β. Since s>[n(1-α)+2β](r-1)/2βr-[n(1-α)+2β], then r>[n(1-α)+2β](s-1)/2βs-[n(1-α)+2β]. Applying Lemmas 24 and 25, we have (53)Θα0bfLqωq/p-q≤MΘα0bfLqωq/p-q≤CM♯Θα0bfLqωq/p-q≤C∑j=1mTj,1·bBMOωωMs,ωIα0Tj,2fLqωq/p-q+C∑j=1mbBMOωωMs,ωIα0Tj,2fLqωq/p-q+C∑j=1mIα0bBMOω·ω1-α0/nMα0,s,ωTj,4fLqωq/p-q+C∑j=1mbBMOωω1-α0/nMα0,s,ωTj,4fLqωq/p-q+C∑j=1mbBMOωωMα0Tj,4fLqωq/p-q.
Sinceω∈RHr and r>q/p, then ω∈RHq/p. It follows from p>s and ωq∈A1 that 1<sq/p<q and ωq/p∈A1. Then ωq/p∈A∞ and ω1/p∈Ap,q by Lemma 11. Applying Lemmas 19, 20, and 22, we have (54)I1≕ωMs,ωIα0Tj,2fLqωq/p-q=Ms,ωIα0Tj,2fLqωq/p≤CMsq/p,ωq/pIα0Tj,2fLqωq/p≤CIα0Tj,2fLqωq/p≤CTj,2fLpω≤CTj,2fLpω.
Sinceωq∈A1, ω∈Ap,q by Lemma 11. Applying Lemma 21, we have (55)I2≕ω1-α0/nMα0,s,ωTj,4fLqωq/p-q=Mα0,s,ωTj,4fLqω≤CTj,4fLpω≤CTj,4fLpω.
It follows fromω1/p∈Ap,q and Lemma 23 that (56)I3≕ωMα0Tj,4fLqωq/p-q=∫RnMα0Tj,4fxωx1/pqdx1/q≤CTj,4fLpω≤CTj,4fLpω.
Therefore,(57)Θα0bfLqωq/p-q≤C∑j=1mTj,1bBMOωTj,2·fLpω+C∑j=1mbBMOωTj,2fLpω+C∑j=1mIα0bBMOωTj,4fLpω+C∑j=1mbBMOωTj,4fLpω+C∑j=1mbBMOω·Tj,4fLpω≤CbBMOω·∑j=1mTj,1Tj,2+Tj,2+Iα0Tj,4+Tj,4·fLpω.This completes the proof of Theorem 7.



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*Source: 349535-2015-06-16.xml*