# New Weighted Norm Inequalities for Pseudodifferential Operators and Their Commutators

Authors: The Anh Bui
Journal: International Journal of Analysis
 (2013)

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## Abstract

This paper is dedicated to study weightedLp inequalities for pseudodifferential operators
with amplitudes and their commutators by using the new class of weights Ap∞ and the new BMO function space BMO∞ which are larger than the Muckenhoupt class of weights Ap and
classical BMO space BMO, respectively. The obtained results therefore improve substantially
some well-known results.


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## Body

## 1. Introduction and the Main Results

Forf∈C0∞(ℝn) a pseudodifferential operator given formally by
(1)Taf(x)=1(2π)n∬ℝna(x,y,ξ)ei〈x-y,ξ〉f(y)dydξ,
where the amplitude a satisfies certain growth conditions. The boundedness of pseudodifferential operators has been studied extensively by many mathematicians; see, for example, [1–7] and the references therein. One of the most interesting problems is studying the weighted norm inequalities for pseudodifferential operators and their commutators with BMO function; see, for example, [5–9].In this paper we consider the following classes of symbols and amplitudesa (in what follows we set 〈x〉=(1+|x|2)1/2).Definition 1.
Leta:ℝn×ℝn×ℝn→ℝn and m∈ℝ, ρ∈[0,1] and δ∈[0,1].(a)
We saya∈Aρ,δm when for each triple of multi-indices α, β, and γ there exists a constant C such that
(2)|∂ξα∂xβ∂yγa(x,y,ξ)|≤C〈ξ〉m-ρ|α|+δ|β+γ|.(b)
We saya∈L∞Aρ,δm when for each triple of multi-indices α, β, and γ there exists a constant C such that
(3)∥∂ξα∂yβa(·,y,ξ)∥L∞≤C〈ξ〉m-ρ|α|+δ|β|.Definition 2.
Leta:ℝn×ℝn→ℝn and m∈ℝ, ρ∈[0,1] and δ∈[0,1].(a)
We saya∈Sρ,δm when for each pair of multi-indices α and β there exists a constant C such that
(4)|∂ξα∂xβa(x,ξ)|≤C〈ξ〉m-ρ|α|+δ|β|.(b)
We saya∈L∞Sρm when for each multi-indices α there exists a constant C such that
(5)∥∂ξαa(·,ξ)∥L∞≤C〈ξ〉m-ρ|α|.It is easy to see thatSρ,δm⊂Aρ,δm, L∞Sρm⊂L∞Aρ,δm, Sρ,δm⊂L∞Sρm, and Aρ,δm⊂L∞Aρ,δm. The classes Aρ,δm and Sρ,δm were studied in [3, 8]. For further information about these two classes, we refer the reader to, for example, [3, 10]. The class L∞Sρm was introduced by [11], and it is the natural generalization of the class Sρ,δm. This class is much rougher than that considered in [6, 7]. The amplitude class L∞Aρ,δm in Definition 1 is rough in the x variable, but smooth in the y variable. This is smaller than the class L∞Aρm introduced in [5] but includes the class Aρ,δm.The aim of this paper is to study the weighted norm inequalities for pseudodifferential operatorsTa and their commutators by using the new BMO functions and the new class of weights. Firstly, we would like to give brief definitions on the new class of weights and the new BMO function space (we refer to Section 2 for details).The new classes of weightsAp∞=∪θ>0Apθ for p≥1, where Apθ, θ≥0, is the set of those weights satisfying
(6)(∫Bw)1/p(∫Bw-1/(p-1))1/p′≤C|B|(1+rB)θ
for all ball B=B(xB,rB). We denote that A∞∞=∪p≥1Ap∞. It is easy to see that the new class Ap∞ is strictly larger than the Muckenhoupt class Ap. Indeed, for example, the weight w(x)=1+|x|γ with γ>n(p-1) belongs to the class Ap∞, but it is not in Ap, for p>1, see, for example, [12].The new BMO spaceBMOθ with θ≥0 is defined as a set of all locally integrable functions b satisfying
(7)1|B|∫B|b(y)-bB|dy≤C(1+rB)θ,
where B=B(xB,rB) and bB=(1/|B|)∫Bb. A norm for b∈BMOθ, denoted by ∥b∥θ, is given by the infimum of the constants satisfying (12). Clearly BMOθ1⊂BMOθ2 for θ1≤θ2 and BMO0=BMO. We define BMO∞=∪θ>0BMOθ.Our main result is the following theorem.Theorem 3.
Leta∈L∞Aρ,δm with m<n(ρ-1) or a∈L∞A1,δ0,δ∈[0,1]. If Ta is bounded on Lp for all 1<p<∞, then(a)
Ta is bounded on Lp(w) for 1<p<∞ and w∈Ap∞;(b)
for anyb∈BMO∞, the commutator [b,Ta] bounded on Lp(w) for 1<p<∞ and w∈Ap∞.
In particular, the obtained results in (a) and (b) still hold forw(x)=1+|x|γ with γ>n(p-1).We would like to specify some applications of Theorem3.In [8], the author studied the weighted Lp inequalities of Ta when the symbol a belongs to the class S1,δ0⊂L∞A1,δ0 with δ∈(0,1). It was proved that Ta is bounded on Lp(w) for 1<p<∞, w∈Ap. Recently, the author in [9] showed that Ta and its commutator with a BMO function [b,Ta] are bounded on Lp(w) for 1<p<∞ and w∈Ap∞ by the different approach. Here, by using Theorem 3, we not only reobtain the boundedness of Ta on Lp(w) for 1<p<∞ and w∈Ap∞ but also obtain the new result on the boundedness of its commutator with BMO∞ functions.Corollary 4.
Leta∈S1,δ0⊂L∞A1,δ0,0<δ<1. Then we have the following:(i)
Ta is bounded on Lp(w) for 1<p<∞ and w∈Ap∞;(ii)
for eachb∈BMO∞, the commutator [b,Ta] is bounded on Lp(w) for 1<p<∞ and w∈Ap∞.
In particular, the obtained results in (i) and (ii) still hold forw(x)=1+|x|γ with γ>n(p-1).Now we consider the classL∞Sρm. If a∈L∞Sρm with ρ∈[0,1] and m<n(ρ-1), then the authors in [5] proved that the pseudodifferential operator Ta and its commutators with BMO functions [b,Ta] are bounded on Lp(w) for 1<p<∞ and w∈Ap; see [5, Theorems 3.3 and 4.5]. So, Theorem 3 leads us to the following result.Corollary 5.
Leta∈L∞Sρm with ρ∈[0,1] and m<n(ρ-1). Then we have the following:(i)
Ta is bounded on Lp(w) for 1<p<∞ and w∈Ap∞;(ii)
for eachb∈BMO∞, the commutator [b,Ta] is bounded on Lp(w) for 1<p<∞ and w∈Ap∞.
In particular, the obtained results in (i) and (ii) still hold forw(x)=1+|x|γ with γ>n(p-1).It was proved in [5, Theorem 3.7] that if a∈L∞Aρm with 0≤ρ≤1 and m<n(ρ-1), then Ta and [b,Ta] are bounded on Lp(w) for 1<p<∞ and w∈Ap with b∈BMO. Therefore, in the light of Theorem 3, we have the following:Corollary 6.
Leta∈L∞Aρ,δm with 0≤ρ≤1 and m<n(ρ-1). Then we have the following:(i)
Ta is bounded on Lp(w) for 1<p<∞ and w∈Ap∞;(ii)
for eachb∈BMO∞, the commutator [b,Ta] is bounded on Lp(w) for 1<p<∞ and w∈Ap∞.
In particular, the obtained results in (i) and (ii) still hold forw(x)=1+|x|γ with γ>n(p-1).For smooth amplitudes, we have the following result.Corollary 7.
Leta∈Aρ,δn(ρ-1) with 0<ρ≤1,0≤δ<1. Then we have the following:(i)
Ta is bounded on Lp(w) for 1<p<∞ and w∈Ap∞;(ii)
for eachb∈BMO∞, the commutator [b,Ta] is bounded on Lp(w) for 1<p<∞ and w∈Ap∞.
In particular, the obtained results in (i) and (ii) still hold forw(x)=1+|x|γ with γ>n(p-1).Proof.
The remark in [1, page 11] tells us that Ta is bounded on Lp for 1<p<∞. Thanks to Theorem 3, we conclude that Ta and [b,Ta],b∈BMO∞ are bounded on Lp(w) for 1<p<∞ and w∈Ap∞.The outline of the paper is as follows. In Section2, we first recall some definitions of the new class of weights Ap∞ and the new BMO function spaces BMO∞. Then we also review some basic properties concerning Ap∞ and BMO∞. Section 3 represents some kernel estimates for the pseudodifferential operator Ta. The proof of the main result will be given in Section 4.


## 2. Preliminaries

To simplify notation, we will often just useB for B(xB,rB) and |E| for the measure of E for any measurable subset E⊂ℝn. Also given λ>0, we will write λB for the λ-dilated ball, which is the ball with the same center as B and with radius rλB=λrB. For each ball B⊂ℝn we set that
(8)S0(B)=B,Sj(B)=2jB∖2j-1Bforj∈ℕ.


### 2.1. The New Class of Weights and NewBMO Function Spaces

Recently, in [12], a new class of weights associated to Schrödinger operators L:=-Δ+V, where the potential V∈RHn/2, the reverse Hölder class has been introduced. According to [12], the authors defined the new classes of weights ApL=∪θ≥0ApL,θ for p≥1, where ApL,θ, θ≥0, is the set of those weights satisfying
(9)(∫Bw)1/p(∫Bw-1/(p-1))1/p′≤C|B|(1+rρ(x))θ
for all ball B=B(x,r). We denote that A∞L=∪p≥1ApL, where the critical radius function ρ(·) is defined by
(10)ρ(x)=sup{r>0:1rn-2∫B(x,r)V≤1},x∈ℝn.In this paper, we consider the particular case whenρ(·)≡1. In this situation the new classes of weights are defined by Ap∞=∪θ≥0Apθ for p≥1, where Apθ,θ≥0 is the set of those weights satisfying
(11)(∫Bw)1/p(∫Bw-1/(p-1))1/p′≤C|B|(1+rB)θ
for all ball B=B(xB,rB). We denote that A∞∞=∪p≥1Ap∞.It is easy to see that the new classAp∞ is larger than the Muckenhoupt class Ap. The following properties hold for the new classes Ap∞; see [12, Proposition 5].Proposition 8.
The following statements hold:(i)
Ap∞⊂Aq∞ for 1≤p≤q<∞,(ii)
ifw∈Ap∞ with p>1, then there exists ϵ>0 such that w∈Ap-ϵ∞. Consequently, Ap∞=∪q<pAq∞.Similarly, by adapting the ideas to [13], the new BMO space BMOθ with θ≥0 is defined as a set of all locally integrable functions b satisfying
(12)1|B|∫B|b(y)-bB|dy≤C(1+rB)θ,
where B=B(xB,rB) and bB=(1/|B|)∫Bb. A norm for b∈BMOθ, denoted by ∥b∥θ, is given by the infimum of the constants satisfying (12). Clearly BMOθ1⊂BMOθ2 for θ1≤θ2 and BMO0=BMO. We define BMO∞=∪θ>0BMOθ.The following result can be considered to be a variant of John-Nirenberg inequality for the spacesBMOθ.Proposition 9.
Letθ>0,s≥1. If b∈BMOθ, then for all balls B
(i)(13)(1|B|∫B|b(y)-bB|sdx)1/s≲∥b∥θ(1+rB)θ;
(ii)(14)(1|2kB|∫2kB|b(y)-bB|sdx)1/s≲∥b∥θk(1+2krB)θ
for all k∈ℕ.The proof is similar (even easier) to [13, Lemma 1 and Proposition 3] and hence we omit details.


### 2.2. Weighted Estimates for Some Localized Operators

A ball of the formB(xB,rB) is called a critical ball if rB=1. We have the following result.Proposition 10.
There exists a sequence of pointsxj,j≥1 in ℝn so that the family of critical balls {Qj}j where Qj:=B(xj,1), j≥1 satisfies the following:(i)
∪jQj=ℝn,(ii)
there exists a constantC such that for any σ>1, ∑jχσQj≤Cσn.Note that the more general version of Proposition10 is obtained by [14]. However, in our particular situation, for convenience, we would like to give a simple proof of this proposition.Proof.
Let us consider the family of balls{B(x,1/5):x∈ℝn}. Using Vitali covering lemma, we can pick the subfamily of balls {Bj:=B(xj,1/5):j≥1} so that {Qj}j is pairwise disjoint and ℝn⊂∪jQj where Qj=5Bj=B(xj,1). This gives (i).
To prove (ii), pick anyx∈ℝn. Let ℑ be the set of all indices j so that x∈σQj. Note that if x∈σQj, then σQj⊂B(x,2σ). Therefore, B(xj,1/5)⊂B(x,2σ) for all j∈ℑ. Since {B(xj,1/5)}j∈ℑ is pairwise disjoint, ∑j∈ℑ|B(xj,1/5)|≤|B(x,2σ)|. This is equivalent to that |ℑ|/5n≤Cσn. Hence, |ℑ|≤Cσn. This completes our proof.We consider the following maximal functions forg∈Lloc1(ℝn) and x∈ℝn:
(15)Mloc,αg(x)=supx∈B∈ℬα1|B|∫B|g|,Mloc,α♯g(x)=supx∈B∈ℬα1|B|∫B|g-gB|,
where ℬα={B(y,r):y∈ℝnandr≤α}.Also, given a ballQ, we define the following maximal functions for g∈Lloc1(ℝn) and x∈Q:
(16)MQg(x)=supx∈B∈ℱ(Q)1|B∩Q|∫B∩Q|g|,MQ♯g(x)=supx∈B∈ℱ(Q)1|B∩Q|∫B∩Q|g-gB∩Q|,
where ℱ(Q)={B(y,r):y∈Q,r>0}.We have the following lemma.Lemma 11.
For1<p<∞, let {Qk}k be a sequence of balls as in Proposition 10. Then
(17)∫ℝn|Mloc,1/2g(x)|pw(x)dx≲∫ℝn|Mloc,4♯g(x)|pw(x)dx+∑kw(Qk)(1|2Qk|∫2Qk|g|‍)p
for all g∈Lloc1(ℝn) and w∈A∞∞.Proof.
We adapt the argument in [13, Lemma 2] to our present situation.
By Proposition10, we have
(18)∫ℝn|Mloc,1/2g(x)|pw(x)dx≤C∑k∫Qk|Mloc,1/2g(x)|pw(x)dx.
It can be verified that, for x∈Qk, Mloc,1/2g(x)≤M2Qk(gχ2Qk). Note that since gχ2Qk is supported in 2Qk, operators M2Qk and M2Qk♯ are Hardy-Littlewood and sharp maximal functions defined in 2Qk viewed as a space of homogeneous type with the Euclidean metric and the Lebesgues measure restricted to 2Qk. Moreover, by definition of A∞∞, if w∈A∞∞, then w∈A∞(2Qk), where A∞(2Qk)=∪p≥1Ap(2Qk), and Ap(2Qk) is the class of Muckenhoupt weights on the spaces of homogeneous type 2Qk. Moreover, due to [12, Lemma 5], [w]A∞(2Qk)≤C for all k≥1. Therefore, using Proposition 3.4 in [15] gives
(19)∫ℝn|Mloc,1/2g(x)|pw(x)dx≤C∑k∫Qk|Mloc,1/2g(x)|pw(x)dx≤C∑k∫Qk|M2Qk(gχ2Qk)(x)|pw(x)dx≤C∑k∫2Qk|M2Qk♯(gχ2Qk)(x)|pw(x)dx+C∑kw(2Qk)(1|2Qk|∫2Qk|g(x)|dx)p.
To complete the proof, we need only to check that M2Qk♯(gχ2Qk)(x)≤CMloc,4♯(g)(x) for x∈2Qk. We have
(20)Mloc,4♯(g)(x)=supB∈ℱ(2Qk):B∋x1|B∩2Qk|∫B∩2Qk|f-fB∩2Qk|.
If rB≥4, due to r2Qk=2, 2Qk⊂B. Hence, in this situation, we have
(21)1|B∩2Qk|∫B∩2Qk|f-fB∩2Qk|=1|Qk|∫2Qk|f-f2Qk|≤Mloc,4♯(g)(x).
Otherwise, if rB<4, it is obvious that |B∩2Qk|≈|B|. So we have
(22)1|B∩2Qk|∫B∩2Qk|f-fB∩2Qk|≤21|B∩2Qk|∫B∩2Qk|f-fB|≤C1|B|∫B|f-fB|≤CMloc,4♯(g)(x).
This completes our proof.LetN>0. For κ≥1 and p≥1, we define the following functions for g∈Lloc1(ℝn) and x∈ℝn:
(23)Gκ,pNf(x)=supQ∋x;Qiscritical∑k=0∞2-Nk(1|2kQ^|∫2kQ^|f(z)|pdz)1/p,
where Q^=κQ.Whenκ=1, we write GpN instead of G1,pN. The following result gives the weighted estimates for Gκ,p.Proposition 12.
Letp>s>1 and w∈Ap/sθ, θ≥0. Then we have
(24)∥Gκ,sNf∥Lp(w)≲∥f∥Lp(w)
provided that N>θ/s+n/p.Without loss of generality, we assume thatκ=1. Assume that Q=B(x0,1). For x∈Q, Q⊂2B(x,1). This implies that
(25)GsNf(x)≤C∑k=0∞2-Nk(1|2kB(x,1)|∫Bk(x,1)|f(z)|sdz)1/s,
where Bk(x,1)=B(x,2k+1).Let{Qj} be the family of critical balls given by Proposition 10. Note that if x∈Qj, Bk(x,1)⊂Qjk where Qjk=2k+2Qj. These estimates and Hölder’s inequalities give
(26)∥GpNf∥Lp(w)≤C∑k=0∞2-Nk(∑j∫Qj(1|2kB(x,1)|∫Bk(x,1)|f(z)|sdz)p/sqqqqqqqqqqqq×w(x)dx∑j∫Qj(1|2kB(x,1)|∫Bk(x,1)|f(z)|sdz)p/s)1/p≤C∑k=0∞‍2-Nk×(∑j∫Qj(1|2kQj|∫Qjk|f(z)|sdz)p/sw(x)dx)1/p≤C∑k=0∞2-Nk(∑jw(Qj)|2kQj|p/s(∫Qjk|f(z)|sdz)p/s)1/p≤C∑k=0∞2-Nk(∑jw(Qjk)|2kQj|p/s(∫Qjkw-(p/s)′/(p/s))(p/s)/(p/s)′aaaaaaaaaaaa×(∫Qjk|f(z)|pw(z)dz)∑jw(Qjk)|2kQj|p/s(∫Qjkw-(p/s)′/(p/s))(p/s)/(p/s)′)1/p.
Since w∈Ap/sθ, by definition of the classes Apθ, we have
(27)w(Qjk)(∫Qjkw-(p/s)′/(p/s))(p/s)/(p/s)′≤C|Qjk|p/s2kθ×(p/s).This together with (26) gives
(28)∥GsNf∥Lp(w)≤C∑k‍2-k(N-θ/s)(∑j∫Qjk|f(z)|pw(z)dz)1/p≤C∑k2-k(N-θ/s-n/p)∥f∥Lp(w)≤C∥f∥Lp(w).
This completes our proof.For a family of balls{Qk}k given by Proposition 10, we define the operator M~s,s≥1, as
(29)M~sf=∑kχQkMs(fχQ~k),
where Q~j=8Qj and Msf=M(|f|s)1/s with M being the Hardy-Littlewood maximal function. We have the following result.Proposition 13.
Ifp>s>1 and w∈Ap/sθ, θ>0, then M~s is bounded on Lp(w).Proof.
We have(30)∫ℝn|M~sf(x)|pw(x)dx=∑j∫Qj|Ms(fχQ~k)|pw(x)dx.
For each k, if we consider Q~k as a space of homogeneous type with the Euclidean metric and the Lebesgues measure restricted to Q~k, then w∈Ap/s(Q~k). Moreover, it can be verified that
(31)∥Ms(fχQ~k)∥Lp(w,Q~k)≤C∥f∥Lp(w,Q~k),
and the constant C is independent of k.
Therefore, by (ii) of Proposition10,
(32)∫ℝn|M~sf(x)|pw(x)dx≤C∑j∫Q~k|f(x)|pw(x)dx≤C∥f∥Lp(w)p.
This completes our proof.


## 2.1. The New Class of Weights and NewBMO Function Spaces

Recently, in [12], a new class of weights associated to Schrödinger operators L:=-Δ+V, where the potential V∈RHn/2, the reverse Hölder class has been introduced. According to [12], the authors defined the new classes of weights ApL=∪θ≥0ApL,θ for p≥1, where ApL,θ, θ≥0, is the set of those weights satisfying
(9)(∫Bw)1/p(∫Bw-1/(p-1))1/p′≤C|B|(1+rρ(x))θ
for all ball B=B(x,r). We denote that A∞L=∪p≥1ApL, where the critical radius function ρ(·) is defined by
(10)ρ(x)=sup{r>0:1rn-2∫B(x,r)V≤1},x∈ℝn.In this paper, we consider the particular case whenρ(·)≡1. In this situation the new classes of weights are defined by Ap∞=∪θ≥0Apθ for p≥1, where Apθ,θ≥0 is the set of those weights satisfying
(11)(∫Bw)1/p(∫Bw-1/(p-1))1/p′≤C|B|(1+rB)θ
for all ball B=B(xB,rB). We denote that A∞∞=∪p≥1Ap∞.It is easy to see that the new classAp∞ is larger than the Muckenhoupt class Ap. The following properties hold for the new classes Ap∞; see [12, Proposition 5].Proposition 8.
The following statements hold:(i)
Ap∞⊂Aq∞ for 1≤p≤q<∞,(ii)
ifw∈Ap∞ with p>1, then there exists ϵ>0 such that w∈Ap-ϵ∞. Consequently, Ap∞=∪q<pAq∞.Similarly, by adapting the ideas to [13], the new BMO space BMOθ with θ≥0 is defined as a set of all locally integrable functions b satisfying
(12)1|B|∫B|b(y)-bB|dy≤C(1+rB)θ,
where B=B(xB,rB) and bB=(1/|B|)∫Bb. A norm for b∈BMOθ, denoted by ∥b∥θ, is given by the infimum of the constants satisfying (12). Clearly BMOθ1⊂BMOθ2 for θ1≤θ2 and BMO0=BMO. We define BMO∞=∪θ>0BMOθ.The following result can be considered to be a variant of John-Nirenberg inequality for the spacesBMOθ.Proposition 9.
Letθ>0,s≥1. If b∈BMOθ, then for all balls B
(i)(13)(1|B|∫B|b(y)-bB|sdx)1/s≲∥b∥θ(1+rB)θ;
(ii)(14)(1|2kB|∫2kB|b(y)-bB|sdx)1/s≲∥b∥θk(1+2krB)θ
for all k∈ℕ.The proof is similar (even easier) to [13, Lemma 1 and Proposition 3] and hence we omit details.


## 2.2. Weighted Estimates for Some Localized Operators

A ball of the formB(xB,rB) is called a critical ball if rB=1. We have the following result.Proposition 10.
There exists a sequence of pointsxj,j≥1 in ℝn so that the family of critical balls {Qj}j where Qj:=B(xj,1), j≥1 satisfies the following:(i)
∪jQj=ℝn,(ii)
there exists a constantC such that for any σ>1, ∑jχσQj≤Cσn.Note that the more general version of Proposition10 is obtained by [14]. However, in our particular situation, for convenience, we would like to give a simple proof of this proposition.Proof.
Let us consider the family of balls{B(x,1/5):x∈ℝn}. Using Vitali covering lemma, we can pick the subfamily of balls {Bj:=B(xj,1/5):j≥1} so that {Qj}j is pairwise disjoint and ℝn⊂∪jQj where Qj=5Bj=B(xj,1). This gives (i).
To prove (ii), pick anyx∈ℝn. Let ℑ be the set of all indices j so that x∈σQj. Note that if x∈σQj, then σQj⊂B(x,2σ). Therefore, B(xj,1/5)⊂B(x,2σ) for all j∈ℑ. Since {B(xj,1/5)}j∈ℑ is pairwise disjoint, ∑j∈ℑ|B(xj,1/5)|≤|B(x,2σ)|. This is equivalent to that |ℑ|/5n≤Cσn. Hence, |ℑ|≤Cσn. This completes our proof.We consider the following maximal functions forg∈Lloc1(ℝn) and x∈ℝn:
(15)Mloc,αg(x)=supx∈B∈ℬα1|B|∫B|g|,Mloc,α♯g(x)=supx∈B∈ℬα1|B|∫B|g-gB|,
where ℬα={B(y,r):y∈ℝnandr≤α}.Also, given a ballQ, we define the following maximal functions for g∈Lloc1(ℝn) and x∈Q:
(16)MQg(x)=supx∈B∈ℱ(Q)1|B∩Q|∫B∩Q|g|,MQ♯g(x)=supx∈B∈ℱ(Q)1|B∩Q|∫B∩Q|g-gB∩Q|,
where ℱ(Q)={B(y,r):y∈Q,r>0}.We have the following lemma.Lemma 11.
For1<p<∞, let {Qk}k be a sequence of balls as in Proposition 10. Then
(17)∫ℝn|Mloc,1/2g(x)|pw(x)dx≲∫ℝn|Mloc,4♯g(x)|pw(x)dx+∑kw(Qk)(1|2Qk|∫2Qk|g|‍)p
for all g∈Lloc1(ℝn) and w∈A∞∞.Proof.
We adapt the argument in [13, Lemma 2] to our present situation.
By Proposition10, we have
(18)∫ℝn|Mloc,1/2g(x)|pw(x)dx≤C∑k∫Qk|Mloc,1/2g(x)|pw(x)dx.
It can be verified that, for x∈Qk, Mloc,1/2g(x)≤M2Qk(gχ2Qk). Note that since gχ2Qk is supported in 2Qk, operators M2Qk and M2Qk♯ are Hardy-Littlewood and sharp maximal functions defined in 2Qk viewed as a space of homogeneous type with the Euclidean metric and the Lebesgues measure restricted to 2Qk. Moreover, by definition of A∞∞, if w∈A∞∞, then w∈A∞(2Qk), where A∞(2Qk)=∪p≥1Ap(2Qk), and Ap(2Qk) is the class of Muckenhoupt weights on the spaces of homogeneous type 2Qk. Moreover, due to [12, Lemma 5], [w]A∞(2Qk)≤C for all k≥1. Therefore, using Proposition 3.4 in [15] gives
(19)∫ℝn|Mloc,1/2g(x)|pw(x)dx≤C∑k∫Qk|Mloc,1/2g(x)|pw(x)dx≤C∑k∫Qk|M2Qk(gχ2Qk)(x)|pw(x)dx≤C∑k∫2Qk|M2Qk♯(gχ2Qk)(x)|pw(x)dx+C∑kw(2Qk)(1|2Qk|∫2Qk|g(x)|dx)p.
To complete the proof, we need only to check that M2Qk♯(gχ2Qk)(x)≤CMloc,4♯(g)(x) for x∈2Qk. We have
(20)Mloc,4♯(g)(x)=supB∈ℱ(2Qk):B∋x1|B∩2Qk|∫B∩2Qk|f-fB∩2Qk|.
If rB≥4, due to r2Qk=2, 2Qk⊂B. Hence, in this situation, we have
(21)1|B∩2Qk|∫B∩2Qk|f-fB∩2Qk|=1|Qk|∫2Qk|f-f2Qk|≤Mloc,4♯(g)(x).
Otherwise, if rB<4, it is obvious that |B∩2Qk|≈|B|. So we have
(22)1|B∩2Qk|∫B∩2Qk|f-fB∩2Qk|≤21|B∩2Qk|∫B∩2Qk|f-fB|≤C1|B|∫B|f-fB|≤CMloc,4♯(g)(x).
This completes our proof.LetN>0. For κ≥1 and p≥1, we define the following functions for g∈Lloc1(ℝn) and x∈ℝn:
(23)Gκ,pNf(x)=supQ∋x;Qiscritical∑k=0∞2-Nk(1|2kQ^|∫2kQ^|f(z)|pdz)1/p,
where Q^=κQ.Whenκ=1, we write GpN instead of G1,pN. The following result gives the weighted estimates for Gκ,p.Proposition 12.
Letp>s>1 and w∈Ap/sθ, θ≥0. Then we have
(24)∥Gκ,sNf∥Lp(w)≲∥f∥Lp(w)
provided that N>θ/s+n/p.Without loss of generality, we assume thatκ=1. Assume that Q=B(x0,1). For x∈Q, Q⊂2B(x,1). This implies that
(25)GsNf(x)≤C∑k=0∞2-Nk(1|2kB(x,1)|∫Bk(x,1)|f(z)|sdz)1/s,
where Bk(x,1)=B(x,2k+1).Let{Qj} be the family of critical balls given by Proposition 10. Note that if x∈Qj, Bk(x,1)⊂Qjk where Qjk=2k+2Qj. These estimates and Hölder’s inequalities give
(26)∥GpNf∥Lp(w)≤C∑k=0∞2-Nk(∑j∫Qj(1|2kB(x,1)|∫Bk(x,1)|f(z)|sdz)p/sqqqqqqqqqqqq×w(x)dx∑j∫Qj(1|2kB(x,1)|∫Bk(x,1)|f(z)|sdz)p/s)1/p≤C∑k=0∞‍2-Nk×(∑j∫Qj(1|2kQj|∫Qjk|f(z)|sdz)p/sw(x)dx)1/p≤C∑k=0∞2-Nk(∑jw(Qj)|2kQj|p/s(∫Qjk|f(z)|sdz)p/s)1/p≤C∑k=0∞2-Nk(∑jw(Qjk)|2kQj|p/s(∫Qjkw-(p/s)′/(p/s))(p/s)/(p/s)′aaaaaaaaaaaa×(∫Qjk|f(z)|pw(z)dz)∑jw(Qjk)|2kQj|p/s(∫Qjkw-(p/s)′/(p/s))(p/s)/(p/s)′)1/p.
Since w∈Ap/sθ, by definition of the classes Apθ, we have
(27)w(Qjk)(∫Qjkw-(p/s)′/(p/s))(p/s)/(p/s)′≤C|Qjk|p/s2kθ×(p/s).This together with (26) gives
(28)∥GsNf∥Lp(w)≤C∑k‍2-k(N-θ/s)(∑j∫Qjk|f(z)|pw(z)dz)1/p≤C∑k2-k(N-θ/s-n/p)∥f∥Lp(w)≤C∥f∥Lp(w).
This completes our proof.For a family of balls{Qk}k given by Proposition 10, we define the operator M~s,s≥1, as
(29)M~sf=∑kχQkMs(fχQ~k),
where Q~j=8Qj and Msf=M(|f|s)1/s with M being the Hardy-Littlewood maximal function. We have the following result.Proposition 13.
Ifp>s>1 and w∈Ap/sθ, θ>0, then M~s is bounded on Lp(w).Proof.
We have(30)∫ℝn|M~sf(x)|pw(x)dx=∑j∫Qj|Ms(fχQ~k)|pw(x)dx.
For each k, if we consider Q~k as a space of homogeneous type with the Euclidean metric and the Lebesgues measure restricted to Q~k, then w∈Ap/s(Q~k). Moreover, it can be verified that
(31)∥Ms(fχQ~k)∥Lp(w,Q~k)≤C∥f∥Lp(w,Q~k),
and the constant C is independent of k.
Therefore, by (ii) of Proposition10,
(32)∫ℝn|M~sf(x)|pw(x)dx≤C∑j∫Q~k|f(x)|pw(x)dx≤C∥f∥Lp(w)p.
This completes our proof.


## 3. Some Kernel Estimates

Letφ0:ℝn→ℝ be a smooth radial function which is equal to 1 on the unit ball centered at origin and supported on its concentric double. Set φ(ξ)=φ0(ξ)-φ0(2ξ) and φk(ξ)=φ(2-kξ). Then, we have
(33)∑k=0∞φk(ξ)=1∀ξ∈ℝn
and supp φk⊂{ξ:2k-1≤|ξ|≤2k+1} for all k≥1. Moreover, for any multi-index α and N≥0, we have
(34)|∂ξαφk(ξ)|≤cα2-k|α|.Lemma 14.
Leta∈L∞Aρ,δm with m∈ℝ, ρ∈[0,1] and δ∈[0,1]. Let ak(x,y,ξ)=a(x,y,ξ)φk(ξ) for k≥0.(a)
For eachℓ≥0,
(35)|z|ℓ|∫ak(x,y,ξ)ei〈z,ξ〉dξ|≤C2k(n+m-ρℓ).(b)
Ifa∈L∞Aρ,δm with m<n(ρ-1) and ρ,δ∈[0,1], then, for each N>0, there exist ϵ,ϵ′>0 so that for any ball B⊂ℝn, y,y¯∈B, and x∈Sj(B), j≥2 so that
(36)|∫ak(x,y,ξ)ei〈x-y,ξ〉-ak(x,y¯,ξ)ei〈x-y¯,ξ〉dξ|≤C2-jϵ(2jrB)-nmin{1,(2jrB)-N}2-kϵ′.(c)
Ifa∈L∞A1,δ0, δ∈[0,1], then there exist ϵ,ϵ′>0 so that for any ball B⊂ℝn, y,y¯∈B, and x∈Sj(B), j≥2 so that
(37)|∫ak(x,y,ξ)ei〈x-y,ξ〉-ak(x,y¯,ξ)ei〈x-y¯,ξ〉dξ|≤C2-jϵ(2jrB)-nmin{1,(2jrB)-N}(2krB)ϵ′
as long as 2krB≤1; and
(38)|∫ak(x,y,ξ)ei〈x-y,ξ〉-ak(x,y¯,ξ)ei〈x-y¯,ξ〉dξ|≤C2-jϵ(2jrB)-nmin{1,(2jrB)-N}(2krB)-ϵ′
as long as 2krB>1.Proof.
We refer to Lemma 3.1 in [5] for the proof of (a).
(b) We first note that sincea∈L∞Aρ,δm, we have
(39)|∂ξαak(x,y,ξ)|≤cα2k(m-ρ|α|)∀k=1,2,….
Sincex∈Sj(B), j≥2 and y,y¯∈B, we have x-y≈x-y¯. If |y-y¯|>2-k, using (a) with ℓ=n+ϵ so that m-n(ρ-1)-ρϵ+ϵ<0 gives
(40)LHS≔|∫ak(x,y,ξ)ei〈x-y,ξ〉-ak(x,y¯,ξ)ei〈x-y¯,ξ〉dξ|≤|∫ak(x,y,ξ)ei〈x-y,ξ〉dξ|+|∫ak(x,y¯,ξ)ei〈x-y¯,ξ〉dξ|≤C|x-y|-n-ϵ2k(n+m-ρn-ρϵ)≤C(2jrB)-n-ϵ2k(m-n(ρ-1)-ρϵ).
This together with the fact that |y-y¯|>2-k gives
(41)LHS≤C(2jrB)-n-ϵ2k(m-n(ρ-1)-ρϵ)≤C(2jrB)-n+12k((m-n(ρ-1))-ρϵ+ϵ)|y-y¯|ϵ≤C2-jϵ(2jrB)-n2-kϵ′,
where ϵ′=-[(m-n(ρ-1))-ρϵ+ϵ]>0.
If|y-y¯|≤2-k, we have
(42)LHS≤|∫ak(x,y,ξ)(1-ei〈y-y¯,ξ〉)ei〈x-y,ξ〉dξ|+|∫(ak(x,y,ξ)-ak(x,y¯,ξ))ei〈x-y¯,ξ〉dξ|:=E1+E2.
We will claim that, for allℓ≥0, we have
(43)E1≤C(2jrB)ℓ2k(m+n-ρℓ+1)|y-y¯|.
Indeed, we have for all integersℓ≥0,
(44)E1≤|x-y|-ℓ|x-y|ℓ×|∫ak(x,y,ξ)(1-ei〈y-y¯,ξ〉)ei〈x-y,ξ〉dξ|≤(2jrB)-ℓ×|∑|α|=ℓ∫(x-y)αak(x,y,ξ)(1-ei〈y-y¯,ξ〉)ei〈x-y,ξ〉dξ|≤(2jrB)-ℓ×|∑|α|=ℓ∫ak(x,y,ξ)(1-ei〈y-y¯,ξ〉)∂ξαei〈x-y,ξ〉dξ|.
Using integration by parts, we get that
(45)E1≤(2jrB)-ℓ×|∑|α|=ℓ∫‍∂ξα[ak(x,y,ξ)(1-ei〈y-y¯,ξ〉)]ei〈x-y,ξ〉dξ|.
We write
(46)∑|α|=ℓ∂ξα[ak(x,y,ξ)(1-ei〈y-y¯,ξ〉)]=∑|α|+|β|=ℓ∂ξαak(x,y,ξ)∂ξβ(1-ei〈y-y¯,ξ〉).
If |β|=0, |1-ei〈y-y¯,ξ〉|≤C|y-y¯||ξ|≤C2k|y-y¯|. Therefore, in this situation,
(47)|∑|α|=ℓ∫‍∂ξα[ak(x,y,ξ)](1-ei〈y-y¯,ξ〉)ei〈x-y,ξ〉dξ|≤C2k(n+m+1-ρ|α|)|y-y¯|=C2k(n+m+1-ρℓ)|y-y¯|.
Otherwise, |∂ξβ(1-ei〈y-y¯,ξ〉)|≤C|y-y¯||β|. This together with (39) gives
(48)|∫‍∂ξαak(x,y,ξ)∂ξβ(1-ei〈y-y¯,ξ〉)ei〈x-y,ξ〉dξ|≤C2k(n+m-ρ|α|)|y-y¯||β|≤C2k(n+m+1-ρ|α|-|β|)|y-y¯|≤C2k(n+m+1-ρℓ)|y-y¯|.
Therefore,
(49)E1≤C(2jrB)-ℓ2k(m+n-ρℓ+1)|y-y¯|.
The general statement for noninteger values of ℓ follows by interpolation of the inequality for i and i+1, where i<ℓ<i+1. Therefore, (43) holds for all ℓ>0. Now taking ℓ=n+ϵ so that ϵ′=-(m+n-ρn-ρϵ+ϵ)>0, we have
(50)E1≤C(2jrB)-n-ϵ2k(m+n-ρn-ρϵ+ϵ)|y-y¯|ϵ(2k|y-y¯|)1-ϵ≤C(2jrB)-n-ϵ2-kϵ′|y-y¯|ϵ≤C2-jϵ(2jrB)-n2-kϵ′.
It remains to take care of the termE2. Repeating the previous arguments we also obtain
(51)E2≤(2jrB)-ℓ×|∑|α|=ℓ∫‍∂ξα[ak(x,y,ξ)-ak(x,y¯,ξ)]ei〈y-y¯,ξ〉dξ|.
At this stage, using the mean value theorem (applied for each component of ak) and then using the definition of the class L∞Aρ,δm give
(52)E2≤C(2jrB)-ℓ|y-y¯|2k(n+m-ρℓ+δ)≤C(2jrB)-ℓ|y-y¯|2k(n+m-ρℓ+1)
for all integer ℓ≥0. Hence, by interpolation again,
(53)E2≤C(2jrB)-ℓ|y-y¯|2k(n+m-ρℓ+1)
for all ℓ≥0. Repeating the arguments used to estimate E1, we conclude that
(54)E2≤C2-jϵ(2jrB)-n2-kϵ′.
Therefore, LHS≤C2-jϵ(2jrB)-n2-kϵ′. It remains to show that
(55)LHS≤C2-jϵ(2jrB)-n-N2-kϵ′.
To do this, we repeat the arguments above with ℓ=N+n+ϵ. Since the proof of this part is analogous to (55), and hence we omit details here. This completes our proof.
(c) If2-k≤rB, using the argument as in (b), we have
(56)LHS≔|∫‍ak(x,y,ξ)ei〈x-y,ξ〉-ak(x,y¯,ξ)ei〈x-y¯,ξ〉dξ|≤|∫‍ak(x,y,ξ)ei〈x-y,ξ〉dξ|+|∫‍ak(x,y¯,ξ)ei〈x-y¯,ξ〉dξ|≤C|x-y|-n-ϵ2-kϵ≤C(2jrB)-n-ϵrBϵ2-kϵrBϵ=C2-jϵ(2jrB)n(1rB2k)ϵ.
IfrB<2-k, we have
(57)LHS≤|∫‍ak(x,y,ξ)(1-ei〈y-y¯,ξ〉)ei〈x-y,ξ〉dξ|+|∫‍(ak(x,y,ξ)-ak(x,y¯,ξ))ei〈x-y¯,ξ〉dξ|:=E1+E2.
The previous arguments in (b) show that
(58)E1+E2≤C(2jrB)-n-ϵ2k(-ϵ+1)|y-y¯|≤C(2jrB)-n-ϵ2k(-ϵ+1)rB=C(2jrB)-n-ϵrBϵ(rB2k)(-ϵ+1)≤C2-jϵ(2jrB)-n(rB2k)(1-ϵ).
Hence,
(59)|∫‍ak(x,y,ξ)ei〈x-y,ξ〉-ak(x,y¯,ξ)ei〈x-y¯,ξ〉dξ|≤C2-jϵ(2jrB)-n(2krB)ϵ′if2krB≤1,|∫‍ak(x,y,ξ)ei〈x-y,ξ〉-ak(x,y¯,ξ)ei〈x-y¯,ξ〉dξ|≤C2-jϵ(2jrB)-n(2krB)-ϵ′if2krB>1.
By taking ℓ=n+N+ϵ and repeating the previous arguments, we obtain that
(60)|∫‍ak(x,y,ξ)ei〈x-y,ξ〉-ak(x,y¯,ξ)ei〈x-y¯,ξ〉dξ|≤C2-jϵ(2jrB)-n-N(2krB)ϵ′if2krB≤1,|∫‍ak(x,y,ξ)ei〈x-y,ξ〉-ak(x,y¯,ξ)ei〈x-y¯,ξ〉dξ|≤C2-jϵ(2jrB)-n-N(2krB)-ϵ′if2krB>1.
This completes the proof of (c).Since the associated kernelK(x,y) of the operator Ta is given by
(61)K(x,y)=1(2π)n∫‍a(x,y,ξ)ei〈x-y,ξ〉dξ=∑k≥01(2π)n∫‍ak(x,y,ξ)ei〈x-y,ξ〉dξ
with ak(x,ξ) as in Lemma 14, from Lemma 14 we deduce directly the following result.Lemma 15.
Leta∈L∞Aρ,δm with m<n(ρ-1) or a∈L∞A1,δ0, δ∈[0,1], and let K*(x,y) be the associated kernel of the operator Ta*, the conjugate of Ta.(a)
For anyN>0, we have
(62)|K*(x,y)|≤C|x-y|-N,x≠y.(b)
For anyN>0, there exists ϵ>0 so that any ball B⊂ℝn, y,y¯∈B, x∈Sj(B), j≥2, we have
(63)|K*(y,x)-K*(y¯,x)|≤C2-jϵ(2jrB)-nmin{1,(2jrB)-N}.


## 4. Proof of Theorem3

Note that, by duality argument, the linear operatorT is bounded on Lp(w), 1<p<∞ if and only if its conjugate T* is bounded on Lp′(w1-p′). Moreover, by Hölder’s inequality, it can be verified that w∈Ap∞ if and only if w1-p′∈Ap′∞. Therefore, it suffices to prove (a) and (b) for Ta* and Ta*,b=[b,Ta*] with b∈BMO∞. Before coming to the proof of Theorem 3, we need the following results.Lemma 16.
Leta∈L∞Aρ,δm with m<n(ρ-1) or a∈L∞A1,δ0, δ∈[0,1], and b∈BMOθ, θ≥0. If Ta is bounded on Lp for all 1<p<∞, then for any p>1 and N>0 there exists C>0 such that for all balls Q=Q(x0,1),
(a)(64)1|Q|∫Q‍|Ta*f(x)|dx≤Cinfy∈QGpN(y);
(b)(65)1|Q|∫Q‍|Ta*,bf(x)dx|≤Cinfy∈QGpN-n-θf(y)∥b∥θ;Proof.
(a) We splitf=f1+f2 where f1=fχ4Q. For each j≥0, we have
(66)1|Q|∫Q‍|Ta*f(x)|dx≤1|Q|∫Q‍|Ta*f1|+1|Q|∫Q‍|Ta*f2|:=I1+I2.
Using Hölder’s inequality and the fact that Ta* is bounded on Lp, 1<p<∞, we write
(67)I1≤C(1|Q|∫Q‍|Ta*f1|p)1/p≤(1|4Q|∫4Q‍|f|p)1/p≤Cinfy∈QGpNf(y).
For the term I2 we have, for x∈Q,
(68)Ta*f2(x)=∫Rn∖4Q‍K*(x,y)f(y)dy=∫Rn∖4Q‍K*(y,x)f(y)dy=∑k≥3∫Sk(Q)‍K*(y,x)f(y)dy.
Applying (a) of Lemma 15, we have
(69)Ta*f2(x)=∑k≥3∫Sk(Q)K*(x,y)f(y)dy≤∑k≥3∫Sk(Q)‍f(y)|x-y|n+Ndy≤Cinfy∈QGNf(y)≤Cinfy∈QGpNf(y).
This completes the proof of (a).
(b) Taking1<r<p, we write
(70)Ta*,bf=(b-bQ)Ta*f-Ta*((b-bQ)f).
So, we have
(71)1|Q|∫Q‍|Ta*,bf(x)|dx≤1|Q|∫Q‍|(b-bQ)Ta*f|dx+1|Q|∫Q‍|Ta*((b-bQ)f)(x)|dx:=I1+I2.
We now take care of I1. By Hölder’s inequality, we can write
(72)I1≤C∥b∥θ(1|Q|∫Q‍|Ta*f|p)1/p≤C∥b∥θ((1|Q|∫Q‍|Ta*f1|p)1/p+(1|Q|∫Q‍|Ta*f2|p)1/p):=I11+I12,
where f=f1+f2 with f1=fχ4Q.
Due toLp boundedness of Ta*, one has
(73)I11≤C(1|4Q|∫4Q‍|f|p)1/p≤Cinfy∈QGpNf(y).
To estimateI12, using (69) gives I12≤Cinfy∈QGpNf(y).
The estimate forI2 can be proceeded in the same method. Indeed, we write
(74)1|Q|∫Q‍|Ta*((b-bQ)f)(x)|dx≤1|Q|∫Q‍|Ta*((b-bQ)f1)(x)|dx+1|Q|∫Q‍|Ta*((b-bQ)f2)(x)|dx:=I21+I22,
where f=f1+f2 and f1=fχ4Q.
To estimateI21, using Hölder’s inequality, we have
(75)1|Q|∫Q|Ta*((b-bQ)f1)(x)|dx‍≤(1|Q|∫Q‍|Ta*((b-bQ)f1)(x)|rdx)1/r≤(1|Q|∫Q‍|((b-bQ)f1)(x)|rdx)1/r≤(1|4Q|∫4Q‍|f(x)|pdx)1/p×(1|4Q|∫4Q‍|b(x)-bQ|νdx)1/ν(ν=prp-r)≤C∥b∥θinfy∈QGpNf(y).
For the termI22, due to (a) of Lemma 15, we can write
(76)Ta*((b-bQ)f2)(x)=∑k≥3∫Sk(Q)‍K*(x,y)((b-bQ)f)(y)dy≤C∑k≥32-kN∫Sk(Q)‍|(b(y)-bQ)f(y)|dy.
By Hölder’s inequality and Proposition9, we give
(77)∫Sk(Q)|(b(y)-bQ)f(y)|dy‍≤|2kQ|(1|2kQ|∫2kQ‍|f|p)1/p×(1|2kQ|∫2kQ‍|b-bQ|p′)1/p′≤k2kθ|2kQ|∥b∥θ(1|2kQ|∫2kQ‍|f|p)1/p≤k2k(θ+n)∥b∥θ(1|2kQ|∫2kQ‍|f|p)1/p.
From (77) and (76) we obtain that
(78)Ta*((b-bQ)f2)(x)≤C∥b∥θinfy∈QGpN-n-θf(y).
This completes our proof.Remark 17.
The result in Lemma16 still holds if we replace the critical ball Q by 2Q.Lemma 18.
Leta∈L∞Aρ,δm with m<n(ρ-1) or a∈L∞A1,δ0, δ∈[0,1] and b∈BMOθ, θ≥0. If Ta is bounded on Lp for all 1<p<∞, then for any p>1 and N>0 there exists C>0 so that, for all f and x,y∈B=B(xB,rB) with rB<4, we have
(a)(79)∫ℝn∖2B‍|(K*(x,z)-K*(y,z))f(z)|dz≤C(infu∈BG4,pNf(u)+infu∈BM~pf(u));
(b)(80)∫ℝn∖2B‍|(K*(x,z)-K*(y,z))((b-bB)f)(z)|dz≤C∥b∥θinfu∈B(infu∈BG4,pN-n-θf(u)+infu∈BM~pf(u)).Proof.
(a) Using (b) of Lemma15, we write
(81)∫ℝn∖2B|(K*(x,z)-K*(y,z))f(z)|dz‍≤C∑k≥2∫Sk(B)‍|(K*(x,z)-K*(y,z))f(z)|dz≤C∑k≥22-kϵ(2krB)-nmin{1,(2jrB)-N}∫Sk(B)‍|f(z)|dz≤C∑k≥22-kϵmin{1,(2jrB)-N}1|2kB|∫Sk(B)‍|f(z)|dz=∑k=2k0⋯+∑k>k0⋯:=I1+I2,
where k0 is the smallest integer so that 2k0+1rB>4.
To estimateI1, let {Ql} and {Q~l} be families of balls as in (29). If x∈Ql∩B, then 2kB⊂Q~l for all k=1,2,…,k0. This implies that
(82)1|2kB|∫2kB‍|f(z)|dz≤infu∈BM~pf(u)
for all k=1,2,…,k0.
Hence(83)I1≤∑k=2k02-kϵinfu∈BM~pf(u)≤Cinfu∈BM~pf(u).
For the termI2, since 2k0rB≥4 we have
(84)I2≤∑k≥k02-kϵ(2krB)-N1|2kB|∫Sk(B)‍|f(z)|dz≤∑k≥k02-kϵ(2k-k02k0rB)-N1|2k-k02k0B|∫2k-k02k0B‍|f(z)|dz≤∑k≥k02-kϵ(2k-k0)-N1|2k-k02k0B|∫2k-k02k0B‍|f(z)|dz≤∑k≥02-kϵ2-kN1|2k2k0B|∫2k2k0B‍|f(z)|dz.
Note that 2k0B⊂Q^=4Q here Q=B(x0,1) and |Q|≈|2k0B|. So, we have
(85)I2≤∑k≥0‍2-kϵ2-kN(1|2kQ^|∫2kQ^‍|f(z)|dz)≤Cinfu∈BG4,pNf(u).
Hence, we get (a).
(b) Using Hölder’s inequality and (b) of Lemma15, we obtain that
(86)∫ℝn∖2B|(K*(x,z)-K*(y,z))((b-bB)f)(z)|dz‍=∑k≥2∫Sk(B)‍|(K*(x,z)-K*(y,z))((b-bB)f)(z)|dz≤C∑k≥22-kϵmin{1,(2jrB)-N}1|2kB|×∫Sk(B)‍|((b-bB)f)(z)|dz≤C∑k≥2‍2-kϵmin{1,(2jrB)-N}(1|2kB|∫2kB‍|f(z)|pdz)1/p×(1|2kB|∫2kB‍|b(z)-bB|p′dz)1/p′.
Now using Proposition9, we get that
(87)∫ℝn∖2B‍|(K*(x,z)-K*(y,z))((b-bB)f)(z)|dz≤C∑k≥2‍k2-kϵ(2jrB)θmin{1,(2jrB)-N}∥b∥θ×(1|2kB|∫2kB‍|f(z)|pdz)1/p.
At this stage, repeating the same argument as in (a), we complete the proof of (b).We are now in position to prove Theorem3.Proof of Theorem3.
(a) Using the standard argument, see, for example, [13], fix 1<p<∞ and w∈Ap∞. Let N>0 which will be fixed later. So, by Proposition 8, we can pick r>1 and ν≥0 so that w∈Ap/rν. By Lemma 11 we have
(88)∥Ta*f∥Lp(w)p≤∥Mloc,βTa*f∥Lp(w)p≤C∥Mloc,4♯Ta*f∥Lp(w)p+C∑kw(Qk)(12Qk∫2Qk‍|Ta*f|)p:=I1+I2.
Let us estimate I1 first. By Lemma 16 and Remark 17, we have
(89)12Qk∫2Qk‍|Ta*f|≤Cinfy∈QkGrNf(y).
Invoking Proposition12, we conclude that
(90)∑kw(Qk)(12Qk∫2Qk‍|Ta*f|)p≤∑k∫Qk‍|GrNf(x)|pw(x)dx≤C∫ℝn‍|GrNf(x)|pw(x)dx≤C∥f∥Lp(w)p
as long as N>n/p+ν/r. We now take care of I2. For any ball B(x0,rB) with rB≤4 and x∈B, we write
(91)1|B|∫B|Ta*f(x)-(Ta*f)B|dx‍≤2|B|∫B‍|Ta*f1(x)|dx+1|B|∫B‍|Ta*f2(x)-(Ta*f2)B|dx:=E1+E2,
where f=f1+f2 with f1=fχ2B.
ForE1, since Ta* is bounded on Lr, we have
(92)1|B|∫B‍|Ta*f1(x)|dx≤(1|B|∫B‍|Ta*f1(x)|rdx)r≤C(1|2B|∫2B‍|f|rdx)r≤Cinfu∈BM~rf(u).
Due to Lemma 18, we can write
(93)E2≤1|B|2×∬B(∫ℝn∖2B‍|(K*(u,z)-K*(y,z))f(z)|dz)dydu≤C(infu∈BG4,rNf(u)+infu∈BM~rf(u)).
These two estimates of E1 and E2 tell us that
(94)Mloc,4♯Ta*f(x)≤C(G4,rNf(x)+M~rf(x)).
Applying Proposition 12 and the weighted estimates of M~r, we get that
(95)∥Mloc,4♯Ta*f∥Lp(w)≤C∥f∥Lp(w)
provided that M>n/p+ν/r.
From (90) and (95), we obtain that
(96)∥Ta*f∥Lp(w)≤C∥f∥Lp(w).
This completes our proof.
(b) Fixed1<p<∞, b∈BMOθ, θ≥0 and w∈Ap∞. So, we can pick r>1 and ν≥0 so that w∈Ap/rν. Then we have by Lemma 11(97)∥Ta*,bf∥Lp(w)p≤∫ℝn‍|Mloc,β(Ta*,bf)(x)|pw(x)dx≤C∫ℝn‍|Mloc,4♯(Ta*,bf)(x)|pw(x)dx+∑k‍w(Qk)(1|2Qk|∫2Qk‍|Ta*,bf|)p,
where {Qk} is a family of critical balls given in Lemma 11.
The analogous argument to that in (a) gives(98)∑kw(Qk)(1|2Qk|∫2Qk‍|Ta*,bf|)p‍≤C∥b∥θp∥f∥Lp(w)p.
It remains to estimate∫ℝn‍|Mloc,4♯(Ta*,bf)(x)|pw(x)dx. For any ball B(x0,rB) with rB≤4 and x∈B, we write
(99)1|B|∫B|Ta*,bf(x)-(Ta*,bf)B|dx‍≤2|B|∫B‍|(b-bB)Ta*f(x)|dx+2|B|∫B‍|Ta*((b-bB)f1)(x)|dx+1|B|∫B‍|(Ta*((b-bB)f2))Ta*((b-bB)f2)(x)dddddddd-(Ta*((b-bB)f2))B|dx:=E1+E2+E3,
where f=f1+f2 with f1=fχ2B.
Hölder’s inequality and Proposition9 show that
(100)E1≤C(1|B|∫B‍|b-bB|r′)1/r′(1|B|∫B‍|Ta*|r)1/r≤C∥b∥θ(1|B|∫B‍|Ta*f|r)1/r.
For any critical ballQj such that x∈Qj∩B. It can be verified that B⊂Q~j:=8Qj. This yields that
(101)E1≤C∥b∥θ×infy∈BM~r(Ta*f)(y).
Using Hölder’s inequality and Proposition 9 again, we have, for 1<s<r,
(102)E2≤C(1|B|∫B‍|Ta*((b-bB)f1)|s)1/s≤C(1|B|∫2B‍|(b-bB)f1|s)1/s≲(1|B|∫2B‍|(b-bB)|γ)1/γ(1|B|∫2B‍|f|r)1/rforsomeγ>s≲∥b∥θ×infy∈BM~r(f)(y).
To estimateE3, using Lemma 18, we conclude that
(103)E3≤C1|B|2∬B(∫Rn∖2B|K*(u,z)-K*(y,z)|aaaaaaaaaaaa×|b(z)-bB||f(z)|dz∫Rn∖2B|K*(u,z)-K*(y,z)|)dydu≤C∥b∥θ(G4,rN-θ-nf(x)+M~r(f)(x)).
These three estimates of E1, E2, and E3 give
(104)Mloc,4♯(Ta*,bf)(x)≤C∥b∥θ(G4,rN-θ-nM~r(Ta*f)(x)dddddddi+G4,rN-θ-n(x)+M~r(f)(x)).
This implies that
(105)∥Mloc,4♯(Ta*,bf)∥Lp(w)≤C∥b∥θ(∥M~p0(Ta*f)∥Lp(w)+∥G4,rN-θ-nf∥Lp(w)+∥M~r(f)∥Lp(w)).
Since M~r, G4,rN-θ-n, and Ta* are bounded on Lp(w) as long as N>n+θ+n/p+ν/s, we obtain the desired results.
This completes our proof.



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*Source: 798528-2013-01-27.xml*