# Weighted Estimates for Maximal Commutators of Multilinear Singular Integrals

Authors: Dongxiang Chen; Suzhen Mao
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

This paper is concerned with the pointwise estimates for the sharp function of the maximal multilinear commutatorsTΣb* and maximal iterated commutator TΠb*, generalized by m-linear operator T and a weighted Lipschitz function b. The (Lp1(μ)×⋯×Lpm(μ),Lr(μ1-r)) boundedness and the (Lp1(μ)×⋯×Lpm(μ),Lr(μ1-mr)) boundedness are obtained for maximal multilinear commutator TΣb* and maximal iterated commutator TΠb*, respectively.


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## Body

## 1. Introduction and Notation

The theory of multilinear Calderón-Zygmund singular integral operators,originated from the work of Coifman and Meyers’, has an important role in harmonic analysis. Its study has been attracting a lot of attention in the last few decades. So far, a number of properties for multilinear operators are parallel to those of the classical linear Calderón-Zygmund operators but new interesting phenomena have also been observed. A systematic analysis of many basic properties of such multilinear operators can be found in the articles by Coifman and Meyer [1], Grafakos and Torres [2–4], and Lerner et al. [5]. So we first recall the definition and results of multilinear Calderón-Zygmund operators as well as the corresponding maximal multilinear operators.Definition 1.1.
LetT be a multilinear operator initially defined on the m-fold product of Schwartz space and taking values into the space of tempered distributions:
(1.1)T:𝒮(ℝn)×⋯×𝒮(ℝn)→𝒮′(ℝn).
Following [2], we say that T is an m-linear Calderón-Zygmund operator if for some 1≤qj<∞, it extends to a bounded multilinear operator from Lq1×⋯×Lqm to Lq, where 1/q=(1/q1)+⋯+(1/qm), and if there exists a function K, defined off the diagonal x=y1=⋯=ym in (ℝn)m+1, satisfying
(1.2)T(f1,…,fm)(x)=∫(ℝn)m‍K(x,y1,…,ym)f1(y1)⋯fm(ym)dy1⋯dym,
for all x∉⋂j=1m‍suppfj.
And(1.3)|K(y0,y1,…,ym)|≤A(∑k,l=0m|yk-yl|)mn,(1.4)|K(y0,…,yj,…,ym)-K(y0,…,yj′,…,ym)|≤A|yj-yj′|ε(∑k,l=0m|yk-yl|)mn+ε,
for some ε>0 and all 0≤j≤m, where |yj-yj′|≤(1/2)max0≤k≤m|yj-yk|.The maximal multilinear singular integral operator was defined by(1.5)T*(f→)(x)=supδ>0|Tδ(f1,…,fm)(x)|,
where Tδ is the smooth truncation of T given by
(1.6)Tδ(f1,…,fm)(x)=∫|x-y1|2+⋯+|x-ym|2>δ2‍K(x,y1,…,ym)f1(y1)⋯fm(ym)dy1⋯dym.
As pointed in [4], T*(f→) is pointwise well defined when fj∈Lqj(ℝn) with 1≤qj<∞.The study for the multilinear singular integral operator and its maximal operators attracts many authors’ attention. For maximal multilinear operatorT*, one can see [4] for details. We list some results for T* as follows.Theorem A (see [4]).
Let1≤qj<∞ and q such that 1/q=(1/q1)+⋯+(1/qm) and ω∈Aq1∩⋯∩Aqm. Let T be an m-linear Calderón-Zygmund operator. Then there exists a constant Cn,q<∞, such that for all f→=(f1,…,fm) satisfying
(1.7)∥T*(f→)∥Lq(ω)≤Cn,q(A+W)∏j=1m∥fj∥Lqj(ω),
where W is the norm of T in the mapping T: L1×⋯L1→L1/m,∞.Theorem B (see [4]).
LetT be an m -linear Calderón-Zygmund operator. Then, for all exponents p,p1,…,pm, satisfying (1/p1)+⋯+(1/pm)=1/p, one has
(1.8)T*:Lp1×⋯×Lpm→Lp,
when 1<p1,…,pm≤∞, one also has
(1.9)T*:Lp1×⋯×Lpm→Lp,∞,
when at least one pj is equal one. In either cases the norm of T* is controlled by a constant multiple of A+W.Definition 1.2 (see [5] (commutators in the jth entry)).
Given a collection of locally integrable functionb→=(b1,…,bm), we define the commutators of the m-linear Calderón-Zygmund operator T to be
(1.10)TΣb(f1,…,fm)=∑j=1mTbjj(f→),
where each term is the commutator of bj and T in the jth entry of T, that is
(1.11)Tbjj(f→)=bjT(f1,…,fj,…,fm)-T(f1,…,bjfj,…,fm).In [6], the following more general iterated commutators of multilinear Calderón-Zygmund operators and pointwise multiplication with functions in BMO were defined and studied in products of Lebesgue spaces, including strong type and weak end-point estimates with multiple Ap→ weights. That is,
(1.12)TΠb(f→)(x)=[b1,[b2,…,[bm-1,[bm,T]m]m-1]2]1=∫(ℝn)m‍K(x,y1,…,ym)∏j=1m‍(bj(x)-bj(yj))f1(y1)⋯fm(ym)dy1⋯dym.For the operator[b,T], when T is the Calderón-Zygmund singular integral operator and b∈Λ˙β(ℝn) (the homogeneous Lipschitz spaces), Paluszyński [7] established the (Lp,Lq) boundedness with 1<p<1/β and 1/q=1/p+β/n. Hu and Gu [8] extended this results to the case: b∈Lipβ,μ with μ∈A1.Now we present the definitions of two classes of maximal commutators of multilinear singular integral operators. One is(1.13)TΣb*(f→)(x)=supδ>0|∑j=1m‍∫∑i=1m|x-yi|2>δ2‍K(x,y1,…,ym)f1(y1)⋯(bj(x)-bj(yj))fj(yj)⋯fm(ym)dy→|,
the other is
(1.14)TΠb*(f→)(x)=supδ>0|[b1,[b2,…,[bm-1,[bm,Tδ]m]m-1,…]2]1(f→)(x)|=supδ>0|∫∑i=1m|x-yi|2>δ2‍K(x,y1,…,ym)∏j=1m‍(bj(x)-bj(yj))f1(y1)⋯fm(ym)dy→|,
where dy→=dy1⋯dym. It is obvious to see that
(1.15)TΣb*(f→)(x)≤∑j=1mTbj*,j(f→)(x).
The main purpose of this paper is to extend the results in [8] to the maximal commutators generated by multilinear singular integrals T and Lipβ,μ functions b→.We can formulate our result as following.Theorem 1.3.
Assume that the kernelK satisfies (1.3) and (1.4). Let 1<q1,…,qm,q<∞ be given numbers satisfying 1/q=(1/q1)+⋯+(1/qm). And assume that T maps Lq1(ℝn)×⋯×Lqm(ℝn) to Lq(ℝn). For j∈{1,…,m} and let 1/r=(1/p)-(β/n), 1<p<r<∞, 0<β<1, and 1/p= 1/p1+⋯+1/pm with 1<pi<∞, i=1,…,m. Given μ such that μ∈A1(ℝn) and bj∈Lipβ,μ(ℝn)  (j=1,…,m), then one has
(1.16)∥Tbj*,j(f→)∥Lr(μ1-r)≤C∥bj∥Lipβ,μ∏i=1m∥fi∥Lpi(μ),j=1,…,m.
From (1.15) and (1.16), one can get
(1.17)∥TΣb*(f→)∥Lr(μ1-r)≤C∑j=1m∥bj∥Lipβ,μ∏i=1m∥fi∥Lpi(μ).Ifμ=1, one can get the following.Theorem 1.4.
Assume that the kernelK satisfies (1.3) and (1.4). Let 1<q1,…,qm,q<∞ be given numbers satisfying 1/q=(1/q1)+⋯+(1/qm). And assume that T maps Lq1(ℝn)×⋯×Lqm(ℝn) to Lq(ℝn). For j∈{1,…,m} and let 1/r=(1/p)-(β/n), 1<p<r<∞, 0<β<1 and 1/p= (1/p1)+⋯+(1/pm) with 1<pi<∞, i=1,…,m. Set bj∈Lipβ(ℝn)(j=1,…,m), then one has
(1.18)∥Tbj*,j(f→)∥Lr(ℝn)≤C∥bj∥Lipβ∏i=1m∥fi∥Lpi(ℝn),j=1,…,m.
From (1.15), one can get
(1.19)∥TΣb*(f→)∥Lr(ℝn)≤C∑j=1m∥bj∥Lipβ∏i=1m∥fi∥Lpi(ℝn).The following theorem states the weighted estimates with two different weights for maximal iterated commutator of multilinear singular integrals.Theorem 1.5.
Assume that the kernelK satisfies (1.3) and (1.4). Let 1<q1,…,qm,q<∞ be given numbers satisfying 1/q=(1/q1)+⋯+(1/qm). And assume that T maps Lq1(ℝn)×⋯×Lqm(ℝn) to Lq(ℝn). Let 1/ri=(1/pi)-(βi/n), 1<pi<ri<∞, 0<βi<1, i=1,…,m with 1/p= (1/p1)+⋯+(1/pm), 1/r=(1/r1)+⋯+(1/rm), and β=β1+⋯+βm, 0<β<1. Given μ such that μ∈A1(ℝn) and bi∈Lipβi,μ(ℝn)(i=1,…,m), then one has
(1.20)∥TΠb*(f→)∥Lr(μ1-mr)≤C∏i=1m∥bi∥Lipβi,μ∥fi∥Lpi(μ).Similarly as Theorem1.4, one also obtains the unweighted estimates of maximal iterated commutators.Theorem 1.6.
Assume that the kernelK satisfies (1.3) and (1.4). Let 1<q1,…,qm,q<∞ be given numbers satisfying 1/q=(1/q1)+⋯+(1/qm). And assume that T maps Lq1(ℝn)×⋯×Lqm(ℝn) to Lq(ℝn). Let 1/ri=(1/pi)-(βi/n), 1<pi<ri<∞, 0<βi<1, i=1,…,m with 1/p=(1/p1)+⋯+(1/pm), 1/r=(1/r1)+⋯+(1/rm), and β=β1+⋯+βm, 0<β<1. Set bi∈Lipβi(ℝn)(i=1,…,m), then one has
(1.21)∥TΠb*(f→)∥Lr(ℝn)≤C∏i=1m∥bi∥Lipβi∥fi∥Lpi(ℝn).The rest of this paper is organized as follows. In Section2, we recall some standard definitions and lemmas. Section 3 is devoted to the proof of our theorems. Throughout this paper, we use the letter C to denote a positive constant that varies line to line, but it is independent of the essential variable. For any 1≤p≤∞, the p' is always used to denote the dual index such that (1/p)+(1/p')=1.


## 2. Preliminaries

A nonnegative functionμ defined on ℝn is called weight if it is locally integrable. A weight μ is said to belong to the Muckenhoupt class Ap(ℝn), 1<p<∞, if there exists a constant C such that
(2.1)supB(1|B|∫B‍μ(x)dx)(1|B|∫B‍μ(x)-1/(p-1)dx)p-1≤C<∞,
for every ball B⊂ℝn. A weight μ is said to belong to class A1(ℝn) if
(2.2)(1|B|∫B‍μ(x)dx)≤Cinfx∈Bμ(x),almostallx∈ℝn,
for every ball B∋x. The class A∞(ℝn) can be characterized as A∞=⋃1≤p<∞‍Ap.Many properties of weights can be found in the book [9], we only collect some of them in the following lemma which will be used bellow.Lemma 2.1.
(
i
)  Ap⊂Aq for 1≤p<q≤∞;(ii)
ifμ∈A1, then μθ∈A1 for 0≤θ≤1;(iii)
for1<p<∞, μ∈Ap if and only if μ1-p′∈Ap′.A locally integrable functionf belongs to the weighted Lipschitz space Lipβ,μp(ℝn) for 1≤p≤∞, 0<β<1 and μ∈A∞ if
(2.3)supB∋x1μ(B)β/n(1B∫B|f(x)-fB|pμ(x)1-p‍dx)1/p≤C<∞.
The smallest bound C satisfying (1.19) is then taken to be the norm of f denoted by ∥f∥Lipβ,μp. Put Lipβ,μ=Lipβ,μ1.Ifμ∈A1,b∈Lipβ,μ(0<β<1), from the definition of ∥f∥Lipβ,μ, it is obvious to see
(2.4)|bB-b2k+1B|≤Ckμ(x)μ(B(x,2k+1R))β/n∥b∥Lipβ,μ,
where bB=(1/|B|)∫B‍b(y)dy.The important properties of the weights are the weighted estimates for the maximal function, the sharp maximal function and their variants. One first recalls the maximal function defined by(2.5)M(f)(x)=supB∋x1|B|∫B‍|f(y)|dy,
It is well known that for 1<p<∞,  M maps Lp(μ) into itself if and only if μ∈Ap, see [10].The sharp maximal function is defined by(2.6)M#(f)(x)=supB∋x1|B|∫B‍|f(x)-fB|dy≈supB∋xinfc1|B|∫B‍|f(x)-c|dy.
One also recalls the variants Mδ(f)(x)=(M(|f|δ)(x))1/δ, and Mδ#(f)(x)=(M#(|f|δ)(x))1/δ. We denote the weighted fractional maximal operators by
(2.7)Mα,μ,s(f)(x)=supB∋x(1μ(B)1-(sα/n)∫B‍|f(y)|sμ(y)dy)1/s.
Recall that Mα∶=Mα,1,1 is the weighted fractional maximal operators, that is
(2.8)Mα(f)(x)=supB∋x(1μ(B)1-(α/n)∫B‍|f(y)|μ(y)dy).
The following lemmas are all from [11].Lemma 2.2 (Kolmogorov’s inequality).
Let(X,μ) be a probability measure space and let 0<p<q<∞ then there exists a constant C=Cp,q such that for any measurable function f(2.9)∥f∥Lp(μ)≤C∥f∥Lq,∞(μ).Lemma 2.3.
Let0<p,δ<∞ and μ∈A∞(ℝn), there exists C>0 depending on the A∞(ℝn) constant of μ such that
(2.10)∥Mδ(f)∥Lp(μ)≤C∥Mδ#(f)∥Lp(μ),
for any function f for which the left side of the above inequality is finite.Lemma 2.4.
Suppose that0<α<n, 0<s<p<α/n, 1/q=(1/p)-(α/n). If μ∈A∞(ℝn), then there exists a constant C=Cp,q such that for any measurable function f(2.11)∥Mα,μ,s(f)∥Lq(μ)≤C∥f∥Lp(μ).Lemma 2.5.
Suppose that0<α<n, 0<s<p<α/n, 1/q=(1/p)-(α/n). If μ∈A1+(q/p′)(ℝn), then there exists a constant C=Cp,q such that for any measurable function f(2.12)∥Mα(f)∥Lq(μ)≤C∥f∥Lp(μp/q).


## 3. Two Estimates for Maximal Multilinear Commutators

We will prove our theorems in this section. To begin, we prepare another two iterated operators to control the commutators.Letφ,ψ∈C∞([0,+∞)) such that |φ'(t)|≤C/t, |ψ'(t)|≤C/t and satisfying
(3.1)χ[2,∞)(t)≤φ(t)≤χ[1,∞)(t),χ[1,2](t)≤φ(t)≤χ[1/2,3](t).
We define the maximal operators
(3.2)Φ*(f→)(x)=supη>0|∫(ℝn)m‍K(x,y1,…,ym)φ(|x-y1|+⋯+|x-ym|η)∏i=1mfi(yi)dy→|,Ψ*(f→)(x)=supη>0|∫(ℝn)m‍K(x,y1,…,ym)ψ(|x-y1|+⋯+|x-ym|η)∏i=1m‍fi(yi)dy→|.
For simplicity, we denote Kφ,η(x,y1,…,ym)=K(x,y1,…,ym)φ(|x-y1|+…+|x-ym|/η), Kψ,η(x,y1,…,ym)=K(x,y1,…,ym)ψ(|x-y1|+⋯+|x-ym|/η) and
(3.3)Φη(f→)(x)=∫(ℝn)m‍Kφ,η(x,y1,…,ym)∏i=1m‍fi(yi)dy→,Ψη(f→)(x)=∫(ℝn)m‍Kψ,η(x,y1,…,ym)∏i=1m‍fi(yi)dy→.
The kernels of Φη and Ψη satisfy conditions (1.3) and (1.4) uniformly in η, respectively. And by the same argument in [4], both Φ* and Ψ* have the same weighted estimates to T* that appeared in Theorems A and B.It is easy to see thatT*(f→)≤Φ*(f→)+Ψ*(f→). Moreover,
(3.4)TΣb*(f→)≤ΦΣb*(f→)+ΨΣb*(f→),TΠb*(f→)≤ΦΠb*(f→)+ΨΠb*(f→),
where
(3.5)ΦΣb*(f→)(x)=supη>0|∑j=1m‍∫(ℝn)m‍Kφ,η(x,y1,…,ym)f1(y1)⋯(bj(x)-bj(yj))fj(yj)⋯fm(ym)dy→∑j=1m‍|≤∑j=1m‍supη>0|∫(ℝn)m‍Kφ,η(x,y1,…,ym)f1(y1)⋯(bj(x)-bj(yj))fj(yj)⋯fm(ym)dy→∫(ℝn)m‍|=∑j=1m‍Φbj*,j(f→)(x),ΨΣb*(f→)(x)=supη>0|∑j=1m‍∫(ℝn)m‍Kψ,η(x,y1,…,ym)f1(y1)⋯(bj(x)-bj(yj))fj(yj)⋯fm(ym)dy→∑j=1m‍|≤∑j=1m‍supη>0|∫(ℝn)m‍Kψ,η(x,y1,…,ym)f1(y1)⋯(bj(x)-bj(yj))fj(yj)⋯fm(ym)dy→∫(ℝn)m‍|=∑j=1m‍Ψbj*,j(f→)(x),ΦΠb*(f→)(x)=supη>0|[b1,[b2,…,[bm-1,[bm,Φη]m]m-1⋯]2]1(f→)(x)|=supη>0|∫(ℝn)m‍Kφ,η(x,y1,…,ym)∏j=1m‍(bj(x)-bj(yj))∏i=1mfi(yi)dy→‍|,ΨΠb*(f→)(x)=supη>0|[b1,[b2,…,[bm-1,[bm,Ψη]m]m-1…]2]1(f→)(x)|=supη>0|∫(ℝn)m‍Kψ,η(x,y1,…,ym)∏j=1m‍(bj(x)-bj(yj))∏i=1m‍fi(yi)dy→|.
For simplicity, we will only prove for the case m=2. The arguments for the case m>2 are similar. For the similarity to the two commutators ΦΣb*(f→) and ΨΣb*(f→), we might as well consider the former. We only consider the former. And we establish the following crucial lemma.Lemma 3.1.
Letμ∈A1(ℝn) and bj∈Lipβ,μ, with 0<β<1, j=1,2. Let 0<δ<1/2<1<s<n/β. Then one has
(3.6)Mδ#[Φbj*,j(f1,f2)](x)≤Cμ(x)∥bj∥Lipβ,μMβ,μ,s(Φ*(f1,f2))(x)+Cμ(x)∥bj∥Lipβ,μ×(Mβ,μ,s(f1)(x)M(f2)(x)+M(f1)(x)Mβ,μ,s(f2)(x)),(3.7)Mδ#[Ψbj*,j(f1,f2)](x)≤Cμ(x)∥bj∥Lipβ,μMβ,μ,s(Ψ*(f1,f2))(x)+Cμ(x)∥bj∥Lipβ,μ×(Mβ,μ,s(f1)(x)M(f2)(x)+M(f1)(x)Mβ,μ,s(f2)(x)).Proof.
Without loss of generality, we only consider the casej=1 and denote b1 by b for convenience. Fix x∈ℝn and let B=B(x,R), λ=bB* be the average of b on B*, where B*=B(x,2R). To proceed, we decompose fi=fi0+fi∞, where fi0=fiχB*, i=1,2. Let c be a constant to be fixed along the proof.
Since0<δ<1, we have
(3.8)(1|B|∫B‍||Φb*,1(f1,f2)(y)|δ-|c|δ|dy)1/δ≤(1|B|∫B|Φb*,1(f1,f2)(y)-c|δ‍dy)1/δ≤(1|B|∫B‍|(b(y)-λ)Φ*(f1,f2)(y)|δdy)1/δ+(1|B|∫B‍|Φ*((b-λ)f10,f20)(y)|δdy)1/δ+(1|B|∫B|Φ*((b-λ)f10,f2∞)(y)|δdy‍)1/δ+(1|B|∫B‍|Φ*((b-λ)f1∞,f20)(y)δ|dy)1/δ+(1|B|∫B‍supη>0|Φη((b-λ)f1∞,f2∞)(y)-c|δdy)1/δ∶=I+II+III+IV+V.
For I, since 0<δ<1, μ∈A1 and b∈Lipβ,μ, by Hölder’ inequality, we have
(3.9)I≤1|B|∫B‍|(b(y)-λ)Φ*(f1,f2)(y)|dy≤C|B*|∫B*‍|b(y)-bB*|μ(y)-1/sΦ*(f1,f2)(y)μ(y)1/sdy≤C|B*|∫B*(|b(y)-bB*|s′μ(y)1-s′dy)1/s′(∫B*‍Φ*(f1,f2)(y)sμ(y)dy)1/s‍≤C1μ(B*)β/n(1μ(B*)∫B*|b(y)-bB*|s′μ(y)1-s′dy‍)1/s′μ(B*)(β/n)+(1/s′)|B*|×(1μ(B*)1-(sβ/n)∫B*‍Φ*(f1,f2)(y)sμ(y)dy)1/sμ(B*)(1/s)-(β/n)≤C∥b∥Lipβ,μMβ,μ,s(Φ*(f1,f2))(x)μ(B*)|B*|≤Cμ(x)∥b∥Lipβ,μMβ,μ,s(Φ*(f1,f2))(x).
To estimate the second termII. Since 0<δ<1/2, using Kolmogorov’s inequality with p=δ, q=1/2, X=B, ω=dx/|B| and the (L1(ℝn)×L1(ℝn),L1/2,∞(ℝn))-boundedness of Φ*, we derive that
(3.10)II≤∥Φ*((b-λ)f10,f20)∥L1/2,∞(dy/|B|)≤C(1|B|∫B*‍|(b(y1)-bB*)f1(y1)|dy1)(1|B|∫B*‍|f2(y2)|dy2)≤C(1|B*|∫B*‍|(b(y1)-bB*)f1(y1)|dy1)(1|B*|∫B*‍|f2(y2)|dy2)≤Cμ(x)∥b∥Lipβ,μMβ,μ,s(f1)(x)M(f2)(x),
where we have used the analogous technique in I to get the last inequality.
For the termIII, using the fact |y-y2|~|y2-x| for any y2∈(B*)c, y∈B, and note that Kφ,η satisfies (1.3) uniformly in η, we obtain
(3.11)III≤1|B|∫B‍|Φ*((b-λ)f10,f2∞)(y)|dy≤1|B|∫B‍∫B*×(ℝn∖B*)‍A|b(y1)-λ||f1(y1)||f2(y2)|(|y-y1|+|y-y2|)2ndy1dy2dy≤∫B*‍|b(y1)-bB*∣f1(y1)|dy1∫ℝn∖B*‍|f2(y2)||y2-x|2ndy2≤C(∫B*‍|b(y1)-bB*||f1(y1)|dy1)(∑k=1∞∫2kB*∖2k-1B*‍|f2(y2)||y2-x|2ndy2)≤C1|B*|(∫B*‍|b(y1)-bB*||f1(y1)|dy1)(∑k=1∞2-kn1|2kB*|∫2kB*‍|f2(y2)|dy2)≤Cμ(x)∥b∥Lipβ,μMβ,μ,s(f1)(x)M(f2)(x).
For the term IV, using the fact |y-y1|~|y1-x| for any y1∈(B*)c, y∈B, and note that Kφ,η satisfies (1.3) uniformly in η, and using (2.4), we obtain
(3.12)IV≤1|B|∫B‍|Φ*((b-λ)f1∞,f20)(y)|dy≤1|B|∫B‍∫(ℝn∖B*)×B*‍A|b(y1)-λ||f1(y1)||f2(y2)|(|y-y1|+|y-y2|)2ndy1dy2dy≤C∫ℝn∖B*‍|b(y1)-bB*||f1(y1)||x-y1|2ndy1∫B*‍|f2(y2)|dy2≤C(∑k=0∞‍∫2k+1B*∖2kB*‍|b(y1)-bB*||f1(y1)||x-y1|2ndy1)(∫B*‍|f2(y2)|dy2)≤C(∑k=0∞‍1|2kB*|2∫2k+1B*‍|b(y1)-bB*||f1(y1)|dy1)(∫B*‍|f2(y2)|dy2)≤C(∑k=0∞‍2-kn1|2k+1B*|∫2k+1B*‍|b(y1)-bB*||f1(y1)|dy1)(1|B*|∫B*‍|f2(y2)|dy2)≤CM(f2)(x)∑k=0∞‍2-kn1|2k+1B*|(∫2k+1B*‍|b(y1)-b2k+1B*||f1(y1)|dy1+|bB*-b2k+1B*|∫2k+1B*‍|f1(y1)|dy1)≤CM(f2)(x)∑k=0∞‍2-kn(μ(B(x,2k+1R))β/n|2k+1B*|μ(x)∥b∥Lipβ,μMβ,μ,s(f1)(x)+kμ(x)∥b∥Lipβ,μμ(B(x,2k+1R))β/n|2k+1B*|∫2k+1B*‍|f1(y1)|dy1)≤Cμ(x)∥b∥Lipβ,μM(f2)(x)Mβ,μ,s(f1)(x).
ForV, fix the value of c by taking c=Φ*((b-λ)f1∞,f2∞)(x), recall that Kφ,η satisfies (1.4) uniformly in η, then we can obtain
(3.13)V≤1|B|∫B‍|Φ*((b-λ)f1∞,f2∞)(y)-Φ*((b-λ)f1∞,f2∞)(x)|dy≤1|B|∫B‍supη>0|Φη((b-λ)f1∞,f2∞)(y)-Φη((b-λ)f1∞,f2∞)(x)|dy≤1|B|∫B‍∫(ℝn∖B*)2‍supη>0|Kφ,η(y,y1,y2)-Kφ,η(x,y1,y2)||b(y1-λ)||f1(y1)|×|f2(y2)|dy1dy2dy≤C|B|∫B‍∫(ℝn∖B*)2‍|x-y|ε(|y-y1|+|y-y2|)2n+ε|(b(y1)-bB*)f1(y1)f2(y2)|dy1dy2dy≤C|B|∑k=0∞‍∫2k+1B*∖2kB*‍|x-y|ε|y1-x|2n+ε|(b(y1)-bB*)f1(y1)||f2(y2)|dy1dy2dy≤C(∑k=0∞‍|B*|ε/n|2kB*|2+ε/n∫2k+1B*‍|(b(y1)-bB*)f1(y1)|dy1)(∫2k+1B*‍|f2(y2)|dy2)≤C(∑k=0∞‍2-kε1|2k+1B*|∫2k+1B*‍|(b(y1)-bB*)f1(y1)|dy1)(1|2k+1B*|∫2k+1B*‍|f2(y2)|dy2)≤Cμ(x)∥b∥Lipβ,μMβ,μ,s(f1)(x)M(f2)(x),
where in the last inequality, we use the same computation in the IV term.
Consequently, combining the estimates ofI,II,III, and V, we conclude the proof of Lemma 3.1.Now we are ready to return to prove Theorem1.3.Proof.
First, by Lemma2.1, we have that μ∈A1⊂Ar', and hence μ1-r∈Ar⊂A∞. Then by Lemma 2.3, we obtain
(3.14)∥Φbj*,j(f1,f2)(x)∥Lr(μ1-r)≤∥Mδ(Φbj*,j(f1,f2))(x)∥Lr(μ1-r)≤∥Mδ#(Φbj*,j(f1,f2))(x)∥Lr(μ1-r).
For j=1,2, by Lemma 3.1, we reduce to bound the ∥·∥Lr(μ1-r) norm of the right-hand side of (3.6). For the first term, since 1/r=(1/p)-(β/n) and taking s such that 1<s<p<n/β, by Lemma 2.4, and Theorem B(ii), we have
(3.15)∥μMβ,μ,s(Φ*(f1,f2))∥Lr(μ1-r)=∥Mβ,μ,s(Φ*(f1,f2))∥Lr(μ)≤C∥Φ*(f1,f2)∥Lp(μ)≤C∥f1∥Lp1(μ)∥f2∥Lp2(μ).
For the second term, we let1/r=1/p2+1/l, and 1/l=1/p1-β/n. Then by Lemma 2.4 again, together with Hölder’s inequality, we obtain
(3.16)∥μMβ,μ,s(f1)M(f2)∥Lr(μ1-r)=∥Mβ,μ,s(f1)M(f2)∥Lr(μ)≤∥Mβ,μ,s(f1)∥Ll(μ)∥M(f2)∥Lp2(μ)≤C∥f1∥Lp1(μ)∥f2∥Lp2(μ).
We can obtain that
(3.17)∥Φb*,1(f1,f2)∥Lr(μ1-r)≤C∥b∥Lipβ,μ∥f1∥Lp1(μ)∥f2∥Lp2(μ).
Similarly, we have
(3.18)∥Ψb*,1(f1,f2)∥Lr(μ1-r)≤C∥b∥Lipβ,μ∥f1∥Lp1(μ)∥f2∥Lp2(μ).
Consequently, by the above arguments, we conclude the proof of Theorem 1.3.Similarly as the proof of Lemma3.1 and that of Theorem 1.3, we only consider the case m=2 and establish the following sharp maximal function for ΦΠb*.Lemma 3.2.
Letμ∈A1(ℝn) and bi∈Lipβ1,μ,i=1,2; β=β1+β2, and 0<β<1. And let 0<δ<1/3<1<si<n/βi, i=1,2. Then one has
(3.19)Mδ#(ΦΠb*(f1,f2))(x)≤μ(x)2∏j=12‍∥bj∥Lipβj,μ×(Mβ,μ,s(Φ*(f1,f2))(x)+Mβ,μ,s(f1)(x)Mβ,μ,s(f2)(x)),(3.20)Mδ#(ΨΠb*(f1,f2))(x)≤μ(x)2∏j=12‍∥bj∥Lipβj,μ×(Mβ,μ,s(Φ*(f1,f2))(x)+Mβ,μ,s(f1)(x)Mβ,μ,s(f2)(x)).Proof.
Fixx∈ℝn and let B=B(x,R) with n>0. Taking λi=(bi)B*, the average of bi on B*, i=1,2, where B*=B(x,2R). Let c be a constant to be fixed along the proof. We split ΦΠb*(f1,f2)(y) in the following way:
(3.21)ΦΠb*(f1,f2)(y)=supη>0|(b1(y)-λ1)(b2(y)-λ2)Φη(f1,f2)(y)-(b1(y)-λ1)Φη(f1,(b2-λ2)f2)(y)-(b2(y)-λ2)Φη×((b1-λ1)f1,f2)(y)+Φη((b1-λ1)f1,(b2-λ2)f2)(y)|.
Since 0<δ<1/3, then we have
(3.22)(1|B|∫B‍||ΦΠb*(f1,f2)(y)|δ-|c|δ|dy)1/δ≤(1|B|∫B|ΦΠb*(f1,f2)(y)-c|δ‍dy)1/δ≤(1|B|∫B|(b1(y)-λ1)(b2(y)-λ2)Φ*(f1,f2)(y)|δ‍dy)1/δ+(1|B|∫B‍(supη>0|(b1(y)-λ1)Φη(f1,(b2-λ2)f2)(y)|)δdy)1/δ+(1|B|∫B‍(supη>0|(b2(y)-λ2)Φη((b1-λ1)f1,f2)(y)|)δdy)1/δ+(1|B|∫B‍supη>0|Φη((b1-λ1)f1,(b2-λ2)f2)-c|δdy)1/δ∶=U1+U2+U3+U4.
For the term U1, since 0<δ<1/3, and β=β1+β2, then by Hölder’s inequality, we have
(3.23)U1≤(1|B|∫B|b1(y)-λ1|3δ‍dy)1/3δ(1|B|∫B‍|b2(y)-λ2|3δdy)1/3δ×(1|B|∫B‍|Φ*(f1,f2)(y)|3δdy)1/3δ≤C(1|B*|∫B*|b1(y)-(b1)B*|dy‍)(1|B*|∫B*‍|b2(y)-(b2)B*|dy)×(1|B*|∫B*‍|Φ*(f1,f2)(y)|dy)≤C(1|B*|∫B*|b1(y)-(b1)B*|s′‍μ(y)1-s′dy)1/s′(1|B*|∫B*‍μ(y)dy)1/s×(1|B*|∫B*‍|b2(y)-(b2)B*|s′μ(y)1-s′dy)1/s′(1|B*|∫B*‍μ(y)dy)1/s×(1|B*|∫B*‍|Φ*(f1,f2)(y)|sμ(y)dy)1/s(1|B*|∫B*‍μ(y)-s′/sdy)1/s′≤C1μ(B*)β1/n(1μ(B*)∫B*‍|b1(y)-(b1)B*|s′μ(y)1-s′dy)1/s′μ(B*)(β1/n)+1|B*|×1μ(B*)β2/n(1μ(B*)∫B*‍|b2(y)-(b2)B*|s′μ(y)1-s′dy)1/s′μ(B*)(β2/n)+1|B*|×(1μ(B*)1-(sβ/n)∫B*‍Φ*(f1,f2)(y)sμ(y)dy)1/sμ(B*)(1/s)-(β/n)μ(B*)(1/s′)-1|B*|≤Cμ(x)2∥b1∥Lipβ1,μ∥b2∥Lipβ2,μMβ,μ,s(Φ*(f1,f2))(x).
For the termU2, noting that 0<δ<1/3, we use the facts 1=δ+(1-δ) and 0<δ/(1-δ)<1/2, then by Hölder’s inequality and Komolgorov’s inequality (Lemma 2.2) and Theorem B, we have
(3.24)U2≤C(1|B*|∫B*‍|b1(y)-(b1)B*|dy)(1|B*|∫B*‍supη>0|Φη(f1,(b2-λ2)f2)(y)|δ/(1-δ)dy)(1-δ)/δ≤(1|B*|∫B*‍|b1(y)-(b1)B*|dy)(1|B*|∫B*|Φ*(f1,(b2-λ2)f2)(y)|δ/(1-δ)‍dy)1-δ/δ≤(1|B*|∫B*‍|b1(y)-(b1)B*|dy)∥Φ*(f1,(b2-λ2)f2)∥L1/2,∞(dy/|B*|)≤(1|B*|∫B*‍|b1(y)-(b1)B*|dy)(1|B*|∫B*‍|f1(y1)|dy1)×(1|B*|∫B*‍|(b2(y2)-(b2)B*)f2(y2)|dy2)≤(1|B*|∫B*‍|b1(y)-(b1)B*|s′μ(y)1-s′dy)1/s′(∫B*‍μ(y)dy)1/s×(1|B*|∫B*‍|f1(y1)|sμ(y1)dy1)1/s(1|B*|∫B*‍μ(y1)-s′/sdy1)1/s'×1|B*|(∫B*‍|b2(y2)-(b2)B*|s′μ(y2)1-s′dy2)1/s′(∫B*‍|f2(y2)|sμ(y2)dy2)1/s≤1μ(B*)β1/n(1μ(B*)∫B*‍|b(y)-bB*|s′μ(y)1-s′dy)1/s'μ(B*)(β1/n)+1|B*|×(1μ(B*)1-(sβ/n)∫B*|f1(y1)|s‍μ(y1)dy1)1/sμ(B*)(1/s)-(β/n)μ(B*)-1/s|B*|×1μ(B*)β2/n(1μ(B*)∫B*|b2(y)-(b2)B*|s′‍μ(y)1-s′dy2)1/s'μ(B*)(β2/n)+(1/s')|B*|×(1μ(B*)1-(sβ/n)∫B*|f2(y2)|s‍μ(y2)dy2)1/sμ(B*)(1/s)-(β/n)≤Cμ(x)2∥b1∥Lipβ1,μ∥b2∥Lipβ2,μMβ,μ,s(f1)(x)Mβ,μ,s(f2)(x).
Similarly, for the term U3, we have
(3.25)U3≤Cμ(x)2∥b1∥Lipβ,μ∥b2∥Lipβ,μMβ,μ,s(f1)(x)Mβ,μ,s(f2)(x).
Now we turn to estimate the last termU4. To proceed, we denote that fi=fi0+fi∞, where fi0=fiχB*, i=1,2. Let c=c1+c2+c3, where
(3.26)c1=Φη((b1-λ1)f10,(b2-λ2)f2∞)(x),c2=Φη((b1-λ1)f1∞,(b2-λ2)f20)(x),c3=Φη((b1-λ1)f1∞,(b2-λ2)f2∞)(x).
We split IV in the following way:
(3.27)U4≤U41+U42+U43+U44,
where
(3.28)U41=(1|B|∫B‍supη>0|Φη((b1-λ1)f10,(b2-λ2)f20)(y)|δdy)1/δ,U42=(1|B|∫B‍supη>0|Φη((b1-λ1)f10,(b2-λ2)f2∞)(y)-Φη((b1-λ1)f10,(b2-λ2)f2∞)(x)|δdy1|B|)1/δ,U43=(1|B|∫B‍supη>0|Φη((b1-λ1)f1∞,(b2-λ2)f20)(y)-Φη((b1-λ1)f1∞,(b2-λ2)f20)(x)|δdy1|B|)1/δ,U44=(1|B|∫B‍supη>0|Φη((b1-λ1)f1∞,(b2-λ2)f2∞)(y)-Φη((b1-λ1)f1∞,(b2-λ2)f2∞)(x)|δdy1|B|)1/δ.
For the termU41, we choose 1<p0<1/2δ and use Kolmogorov’s inequality and Theorem B, then we use the same computation as U2 to deduce that
(3.29)U41≤(1|B|∫B|Φ*((b1-λ1)f10,(b2-λ2)f20)(y)|p0δdy)1/p0δ≤∥Φ*((b1-λ1)f10,(b2-λ2)f20)∥L1/2,∞(dy/|B|)≤(1|B|∫B‍|(b1(y1)-λ1)f10(y1)|dy1)(1|B|∫B‍|(b2(y2)-λ2)f20(y2)|dy2)≤C(1|B*|∫B*|(b1(y1)-(b1)B*)f1(y1)|dy1‍)(1|B*|∫B*|(b2(y2)-(b2)B*)f2(y2)|‍dy2)≤Cμ(x)2∥b1∥Lipβ1,μ∥b2∥Lipβ2,μMβ,μ,s(f1)(x)Mβ,μ,s(f2)(x).
ForU42, by the fact |y-y2|~|y2-x|, for any y2∈(B*)c, y∈B, and note that Kφ,η satisfies (1.4) uniformly in η, then we get that
(3.30)U42≤(1|B|∫B‍supη>0|Φη((b1-λ1)f10,(b2-λ2)f2∞)(y)-Φη((b1-λ1)f10,(b2-λ2)f2∞)(x)|dy1|B|)≤C|B|∫B‍(∫B*‍|(b1(y1)-λ1)f1(y1)|dy1∫(B*)c‍|y-x|ε|b2(y2)-λ2||f2(y2)|(|y-y1|+|y-y2|)2n+εdy2)dy≤(∫B*|(b1(y1)-(b1)B*)f1(y1)|dy1‍)×(∑k=0∞‍∫2k+1B*∖2kB*‍|y-x|ε|b2(y2)-(b2)B*||f2(y2)||x-y2|2n+εdy2)≤(∫B*‍|(b1(y1)-(b1)B*)f1(y1)|dy1)×(∑k=0∞‍|B*|ε/n|2kB*|2+ε/n∫2k+1B*|b2(y2)-(b2)B*|‍|f2(y2)|dy2)≤(1|B*|∫B*|(b1(y1)-(b1)B*)f1(y1)|dy1‍)×(∑k=0∞‍2-kε1|2k+1B*|∫2k+1B*|b2(y2)-(b2)B*|‍|f2(y2)|dy2)≤Cμ(x)∥b1∥Lipβ1,μMβ,μ,s(f1)(x)∑k=1∞‍2-kε1|2k+1B*|(∫2k+1B*‍|b2(y2)-(b2)2k+1B*||f2(y2)|dy2+|(b2)B*-(b2)2k+1B*|∫2k+1B*‍|f2(y2)|dy2)≤Cμ(x)2∥b1∥Lipβ1,μ∥b2∥Lipβ2,μMβ,μ,s(f1)(x)Mβ,μ,s(f2)(x),
where we have used the same computation of IV to gain the last inequality.
Similarly asU42, we can get the estimates for U43,
(3.31)U43≤Cμ(x)2∥b1∥Lipβ,μ∥b2∥Lipβ,μMβ,μ,s(f1)(x)Mβ,μ,s(f2)(x).
Now we turn toU44, by the fact |y-y1|~|y1-x|, |y-y2|~|y2-x| for any y1,y2∈(B*)c, y∈B, and recalling that Kφ,η satisfies (1.4) uniformly in η, then we can obtain
(3.32)|Φη((b1-λ1)f1∞,(b2-λ2)f2∞)(y)-Φη((b1-λ1)f1∞,(b2-λ2)f2∞)(x)|≤C∑k=0∞‍∫(2k+1B*∖2kB*)2‍|x-y|ε(|y-y1|+|y-y2|)2n+ε|(b1(y1)-λ1)f1(y1)(b2(y-2)-λ2)f2(y2)|dy1dy2≤C∑k=0∞‍∫(2k+1B*∖2kB*)2‍|x-y|ε|y-y1|2n+ε|(b1(y1)-λ1)f1(y1)(b2(y-2)-λ2)f2(y2)|dy1dy2≤C∑k=0∞‍|B*|ε/n|2kB*|2+ε/n∫2k+1B*‍|(b1(y1)-λ1)f1(y1)|dy1∫2k+1B*‍|(b2(y2)-λ2)f2(y2)|dy2≤C∑k=0∞‍2-kε(1|2k+1B*|∫2k+1B*‍|(b1(y1)-λ1)f1(y1)|dy1)×(1|2k+1B*|∫2k+1B*‍|(b2(y2)-λ2)f2(y2)|dy2)≤Cμ(x)2∥b1∥Lipβ1,μ∥b2∥Lipβ2,μMβ,μ,s(f1)(x)Mβ,μ,s(f2)(x).
Therefore,
(3.33)U44≤1|B|∫B‍supη>0|Φη((b1-λ1)f1∞,(b2-λ2)f2∞)(y)-Φη((b1-λ1)f1∞,(b2-λ2)f2∞)(x)|dy≤Cμ(x)2∥b1∥Lipβ1,μ∥b2∥Lipβ2,μMβ,μ,s(f1)(x)Mβ,μ,s(f2)(x).
Consequently, the estimate for U1, together with those of U2,U3, and U4, can conclude the proof of Lemma 3.2.Now we return to prove Theorem1.5.Proof.
Similarly as the proof of Theorem1.3,
(3.34)∥ΦΠb*(f1,f2)∥Lr(μ1-2r)≤∥Mδ(ΦΠb*(f1,f2))∥Lr(μ1-2r)≤∥Mδ#(ΦΠb*(f1,f2))∥Lr(μ1-2r).
We reduce to bound the ∥·∥Lr(μ1-r) norm of the right-hand side of (3.19). We estimate each term as follows. For the first term, since 1/r=(1/p)-(β/n) and choosing s such that 1<s<p<n/β, by Lemma 2.4 and Theorem A and observe that μ∈A1, we obtain
(3.35)∥μ2Mβ,μ,s(Φ*(f1,f2))∥Lr(μ1-2r)=∥Mβ,μ,s(Φ*(f1,f2))∥Lr(μ)≤C∥Φ*(f1,f2)∥Lp(μ)≤C∥f1∥Lp1(μ)∥f2∥Lp2(μ).
For the second term, since 1/r=(1/r1)+(1/r2), by Hölder’s inequality and Lemma 2.4, we get
(3.36)∥μ2Mβ,μ,s(f1)Mβ,μ,s(f2)∥Lr(μ1-2r)=∥Mβ,μ,s(f1)Mβ,μ,s(f2)∥Lr(μ)≤∥Mβ,μ,s(f1)∥Lr1(μ)∥Mβ,μ,s(f2)∥Lr2(μ)≤C∥f1∥Lp1(μ)∥f2∥Lp2(μ).
Similarly, we also have
(3.37)∥ΨΠb*(f1,f2)∥Lr(μ1-2r)≤C∥f1∥Lp1(μ)∥f2∥Lp2(μ).
This estimate together with that for ΦΠb*(f1,f2) finishes the proof of Theorem 1.5.



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*Source: 128520-2012-10-16.xml*