# Multiple Weighted Estimates for Vector-Valued Multilinear Singular Integrals with Non-Smooth Kernels and Its Commutators

Authors: Dongxiang Chen; Dan Zou; Suzhen Mao
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

This note concerns multiple weighted inequalities for vector-valued multilinear singular integral operator with nonsmooth kernel and its corresponding commutators containing multilinear
commutator and iterated commutator generated by the vector-valued multilinear operator and BMO functions. By the weighted estimates for a class of new variant maximal and sharp maximal functions, the multiple weighted norm inequalities for such operators are obtained.


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## Body

## 1. Introduction

It is well known that multiple weighted norm inequalities for multilinear operators and their related commutators on various spaces of function is a center topic of harmonic analysis, which recently attracts a lot of attention, see [1–3] et al.In this paper, we will focus on the multiple weighted estimates for vector valued multilinear singular integral with nonsmooth kernel and its commutators. Now we give some information on multilinear Calderón-Zygmund operators.The multilinear operatorT we study is initially defined on the m-fold product of Schwartz space 𝒮(ℝn)×⋯×𝒮(ℝn) and taking values into the space of tempered distributions 𝒮′(ℝn); that is,
(1)T:𝒮(ℝn)×⋯×𝒮(ℝn)⟶𝒮′(ℝn).A locally integrable functionK(x,y1,…,ym) defined away from the diagonal x=y1=⋯=ym in (ℝn)m+1 is called an associated kernel of T if
(2)T(f1,…,fm)(x)=∫(ℝn)m‍K(x,y1,…,ym)f1(y1)⋯fm(ym)dy1⋯dym,
where f1,…,fm are C∞ functions with compact support and for all x∉⋂j=1m suppfj.Moreover, we assume the associated kernelK satisfies the following standard estimates:
(3)|K(y0,y1,…,ym)|≤A(∑k,l=0m|yk-yl|)mn,
for some A>0 and all (y0,…,ym) with y0≠yj for some j∈{1,…,m}, and
(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|. These facts can be founded in [4].We now turn to present the definitions ofm-linear commutator and iterated commutator of multilinear singular integral.For the multilinear operatorT and b→=(b1,…,bm) in BMOm, we define the m-linear commutator TΣb→ as the following form:
(5)TΣb→(f→)(x)=∑j=1m‍Tbjj(f→)(x):=∑j=1m‍[bj,T]j(f1,…,fm)(x),
where
(6)Tbjj(f→)(x)=bj(x)T(f1,…,fm)(x)-T(f1,…,bjfj,…,fm)(x).TΠb→(f→)(x)≔[b1,[b2,…[bm-1,[bm,T]m]m-1,…]2]1×(f1,…,fm)(x).IfT is associated with a distribution kernel, which coincides with the function K defined away from the diagonal x=y1=⋯=ym in (ℝn)m+1, then
(7)Tbjj(f→)(x)=∫(ℝn)m‍K(x,y1,…,ym)f1(y1)⋯(bj(x)-bj(yj))fj(yj)⋯fm(ym)dy1⋯dym.
And we also present the iterated commutator TΠb→,
(8)TΠb→(f→)(x)=∫(ℝn)m‍K(x,y1,…,ym)×∏j=1m‍(bj(x)-bj(yj))fj(yj)dy1⋯dym.Here the notations of commutators are taken from [5, 6].The following class of weights were introduced in [1]. Let 1≤p1,…,pm<∞, and 1/m≤p<∞, with 1/p=1/p1+⋯+1/pm, and P→=(p1,…,pm); given ω→=(ω1,…,ωm), we say that ω→ satisfies the Ap→ condition if
(9)supQ(1|Q|∫Q‍νω→)1/p∏j=1m‍(1|Q|∫Q‍ωj1-pj′)1/pj′<∞,
where νω→=∏j=1m‍ωjp/pj, when pj=1,    ((1/|Q|)∫Q‍ωj1-pj′)1/pj′ is understood as (infQωj)-1.Observe thatA(1,…,1) is contained in AP→ for each P→. However, the class AP→ is not increasing with the natural partial order, see [1] for detail. Lerner et al. [1] established multiple weighted estimates for multilinear C-Z operators and that for m-linear commutator of multilinear C-Z commutator. In 2012, Chen and Wu [2] extend their results to m-linear commutator and iterated commutator of multilinear C-Z operator with nonsmooth kernel satisfying Assumptions (H1) and (H2).Next we define the vector-valued multilinear operatorTq(q>0) associated with the operator T by
(10)Tq(f1,…,fm)(x)=|T(f1,…,fm)|q(x)=(∑k=1∞‍|T(f1k,…,fmk)(x)|q)1/q,
where fi={fik}k=1∞, for i=1,…,m.This operator was first studied by Grafakos and Martell in [7]. Later Cruz-Uribe et al. gained the weak boundedness of this one in [8]. We list them as follows.Theorem A (see [7]).
LetT be a multilinear Calderón-Zygmund operator as before, and let 1/m<p<∞,  1/p=1/p1+⋯+1/pm with 1<p1,…,pm<∞,1/m<q<∞, and 1/q=1/q1+⋯+1/qm with 1<q1,…,qm<∞. There exists a constant C>0 such that
(11)∥Tq(f→)∥Lp(ℝn)≤C∏j=1m‍∥|fj|qj∥Lpj(ℝn).For the sequence{f→k}k=1∞=(f1k,…,fmk)k=1∞, the vector-valued version of the commutators TΣb→,q and TΠb→,q are given by
(12)TΣb→,q(f→)(x)=|TΣb→(f→)|q(x)=(∑k=1∞‍|TΣb→(f1k,…,fmk)(x)|q)1/q,(13)TΠb→,q(f→)(x)=|TΠb→(f→)|q(x)=(∑k=1∞‍|TΠb→(f1k,…,fmk)(x)|q)1/q.In 2008, Tang established weighed norm inequalities for the commutators of vector-valued multilinear operator in [6], but their results are not the multiple weighted estimates that are obtained by Lerner et al. in [1].Now we restore to give some information on the kernelK which satisfy Assumptions (H1) and (H2). Let {At}t>0 be a class of integral operators which play the role of the approximation to the identity. We always assume that the operators At are associated with kernels at(x,y) in the sense that for all f∈⋃p∈[1,∞]‍Lp and x∈ℝn(14)Atf(x)=∫ℝn‍at(x,y)f(y)dy,
and that the kernels at(x,y) satisfy the following conditions
(15)|at(x,y)|≤ht(x,y):=t-n/sh(|x-y|t1/s),
where s is a positive fixed constant and h is a positive, bounded, decreasing function satisfying that for some η>0,
(16)limr→∞rn+ηh(rs)=0.Recall that thejth transpose T*,j of the m-linear operator T is defined via
(17)〈T*,j(f1,…,fm),g〉=〈T(f1,…,fj-1,g,fj+1,…,fm),fj〉,
for all f1,…,fm,g in 𝒮(ℝn). Notice that the kernel K*,j of T*,j is related to the kernel K of T via the identity
(18)K*,j(x,j1,…,yj,…,ym)=K(yj,y1,…,x,…,ym).If anm-linear operator T maps a product of Banach Spaces Xm×⋯×Xm to another Banach Space X; then transpose T*,j maps the product of Banach Spaces X1×⋯×Xj-1×Xj*×Xj+1×⋯×Xm into Xj*. Moreover, the norms of T and T*,j are equal. To maintain uniform notation, we may occasionally denote T by T*,0 and K by K*,0.Assumption (H1).Assume that for each i=1,…,m, there exist operators {Ati}t>0 with kernel ati(x,y) satisfying conditions s and η and that for every j=0,1,…,m, there exist kernels Kt*,j(i)(x,y1,…,ym) such that
(19)〈T*,j(f1,…,At(i)fj,…,fm),g〉=∫ℝn‍∫(ℝn)m‍Kt*,j(i)(x,y1,…,ym)f1(y1)⋯fm(ym)g(x)dy→dx,
for all f1,…,fm in 𝒮(ℝn) with ⋂k=1msuppfk∩suppg=∅. Also assume that for every j=0,1,…,m and every i=1,…,m, we have
(20)|K*,j(x,y→)-Kt*,j(i)(x,y→)|≤A(∑k=1m|x-yk|)mn×∑k=1,k≠im‍Φ(|yi-yk|t1/s)+Atϵ/s(∑k=1m|x-yk|)mn+ϵ,
where t1/s≤|x-yi|/2,y→=(y1,…,ym).IfT satisfies Assumption (H1), we will say that T is an m-linear operator with generalized Caderón-Zygmund kernel K. The collection of function K satisfying (19) and (20) with parameters m,A,s,η and ϵ will be denoted by m-GCZK(A,s,η,ϵ), we say that T is of class m-GCZK(A,s,η,ϵ) if T has an associated kernel K in m-GCZK(A,s,η,ϵ).Assumption (H2). Assume that there exist operators {Bt}t>0 with kernel bt(x,y) satisfying conditions (15) and (16) with constants s and η. Let
(21)Kt(0)(x,y1,…,ym)=∫ℝn‍K(z,y1,…,ym)bt(x,z)dz,
whenever 2t1/s≤min1≤j≤m|x-yj| and
(22)|K(x,y→)-Kt(0)(x′,y→)|≤A(∑k=1m|x-yk|)mn×∑k=1,k≠im‍Φ(|yi-yk|t1/s)+Atϵ/s(∑k=1m|x-yk|)mn+ϵ,
whenever 2|x-x′|≤t1/s and 2t1/s≤max1≤j≤m|x-yj|.WhenT is of m-GCZK(A,s,η,ϵ) and its kernel also satisfies Assumption (H2), Duong et al. in [5] proved that multilinear singular integral T is bounded from Lp1(ℝn)×⋯×Lpm(ℝn) to Lp(ℝn)(Lp,∞(ℝn)) for 0<p<∞, 1≤p1,…,pm<∞ with 1/p=1/p1+⋯+1/pm. And they also remarked that the above kernel which they studied has weaker regularity. It is natural to ask whether the vector-valued multilinear operator Tq with kernels satisfying the same conditions as in [5] and its commutators TΣb→,q and TΠb→,q have multiple weighted estimates or not. These problems will be addressed by our next theorems.Now we can formulate our results as follows.Theorem 1.
Assume thatTq is a vector-valued multilinear operator defined as (10) associated with T being an m-GCZK(A,s,η,ϵ) whose kernel K satisfies Assumption (H2). If there exist 1≤p1,…,pm<∞, with 1/p=1/p1+⋯+1/pm and  ω→=(ω1,…,ωm)∈AP→ with P→=(p1,…,pm) then(i)
Tq can be extended to a bounded operator from Lp1(ω1)×⋯×Lpm(ωm) to Lp(νω→) if all exponents pj are strictly greater than 1;(ii)
Tq can be extended to a bounded operator from Lp1(ω1)×⋯×Lpm(ωm) to Lp,∞(νω→) if 1≤pj<∞, j=1,…,m, and at least one of the pj=1.Theorem 2.
Assume thatTq is a vector-valued multilinear operator that satisfies the assumptions in Theorem 1, and the multilinear commutator TΣb→,q is defined as (12). Let b→∈BMOm, 1<p1,…,pm<∞,1<p<∞ with 1/p=1/p1+⋯+1/pm, and ω→=(ω1,…,ωm)∈AP→ with P→=(p1,…,pm). Then there exists a constant C>0 such that
(23)∥TΣb→,q(f→)∥Lp(νω→)≤C∑i=1m‍∥bi∥BMO∏j=1m‍∥|fj|qj∥Lpj(ωj).Theorem 3.
Assume thatTq is a vector-valued multilinear operator that satisfies the assumptions in Theorem 1, and the iterated commutator TΠb→,q is defined as (13). Let b→∈BMOm, 1<p1,…,pm<∞,  1<p<∞ with 1/p=1/p1+⋯+1/pm  ω→=(ω1,…,ωm)∈AP→ with P→=(p1,…,pm). Then there exists a constant C>0 such that
(24)∥TΠb→,q(f→)∥Lp(νω→)≤C∏i=1m‍∥bi∥BMO∏j=1m‍∥|fj|qj∥Lpj(ωj),
where νω→=∏j=1m‍ωjp/pj, 1<q1,…,qm<∞, 1/m<q<∞,  and 1/q=1/q1+⋯+1/qm.The rest of this paper is organized as follows. In Section2, we recall some standard definitions and lemmas. In Section 3, we introduce a class of new maximal functions and prove some useful estimates which will play key roles in the proofs of our theorems. In Section 4, it is devoted to the proof our theorems. Throughout this paper, we use the letter C to denote a positive constant that varies from line to line, but it is independent of the essential variable.


## 2. Some Preliminaries

Lemma 4 (see [1, Lemma 6.1]).
Assume thatω→=(ω1,…,ωm) satisfies AP→ condition, then there exists a finite constant r>1 such that ω→∈AP→/r.We recall the definition of the Hardy-Littlewood maximal function and the sharp maximal function:(25)M(f)(x)=supQ∋x1|Q|∫Q‍|f(y)|dy,M♯(f)(x)=supQ∋xinfx1|Q|∫Q‍|f(y)-c|dy≈supQ∋x1|Q|∫Q‍|f(y)-fQ|dy,
and their variants Mδ(f)(x)=M(|f|δ)1/δ(x) and Mδ♯(f)(x)=M♯(|f|δ)1/δ(x).We will use the following inequality (see [9]):
(26)∫ℝn‍(Mδ(f)(x))pω(x)dx≤C∫ℝn‍(Mδ♯(f)(x))pω(x)dx,
all functions f for which the left-hand side is finite, and where 0<p,  δ<∞,  ω∈A∞.Now we introduce some facts on Orlicz spaces. LetΦ: [0,∞)→[0,∞) be a Young function, that is, a continuous, convex, increasing function with Φ(0)=0 and such that Φ(t)→∞ as t→∞. In this paper, any Young function Φ will be doubling, namely Φ(2t)≤CΦ(t) for t>0. We define the Φ-average of function f over a cube Q by
(27)∥f∥Φ,Q=inf{λ>0:1|Q|∫Q‍Φ(|f(x)|λ)dx≤1}.
It is a simple but important observation that
(28)∥f∥Φ,Q>1iff1|Q|∫Q‍Φ(|f(x)|λ)dx>1.A particular case of interest, and especially in this paper, is the Young functionΦ(t)=t(1+log+t), the average ∥f∥Φ,Q of a function f given by the Luxemburg norm ∥f∥L(logL),Q.Associated with this average, we have a maximal function(29)ML(logL)f(x)=supQ∋x∥f∥L(logL),Q,
where the supremum is taken over all the cubes containing x.By the generalized Hölder’s inequality, we also get(30)1|Q|∫Q‍|f(x)g(x)|dx≤C∥f∥BMO∥g∥L(logL),Q.


## 3. New Maximal Functions

In this section, we will introduce certain variant multilinear maximal functions and establish the multiple weighted estimates for such functions, which are one of the main parts in this paper.Recall the definitions of these maximal functions, which are introduced by Lerner et al. in [1]
(31)ℳ(f→)(x)=supQ∋x∏j=1m‍1|Q|∫Q‍|fj(y)|dy,ℳr(f→)(x)=supQ∋x∏j=1m‍(1|Q|∫Q‍|fj(y)|rdy)1/r,ℳL(logL)(f→)(x)=supQ∋x∏j=1m‍∥fj∥L(logL),Q.The fact thatr>1, there exists a constant C>0 such that
(32)∥f∥L(logL),Q≤C(1|Q|∫Q‍|fj(y)|rdy)1/r,
so it is easy to check that
(33)ℳ(f→)(x)≤ℳL(logL)(f→)(x)≤ℳr(f→)(x).Characterizations of the multiple weights in terms ofℳ were proven in Theorems 3.3 and 3.7 in [1].Lemma 5.
Let1≤p1,…,pm<∞, 1/p=1/p1+⋯+1/pm, and P→=(p1,…,pm),(i)
If1<p1,…,pm<∞, then ℳ is bounded from Lp1(ω1)×⋯×Lpm(ωm) to Lp(νω→) if and only if ω→=(ω1,…,ωm)∈AP→;(ii)
If1≤p1,…,pm<∞, then ℳ is bounded from Lp1(ω1)×⋯×Lpm(ωm) to Lp,∞(νω→) if and only if ω→=(ω1,…,ωm)∈Ap→.In the following, we introduce the modified multilinear maximal functions.Letr>1, 1≤l<m, ϱ={j1,…,jl}⊆{1,…,m} and ϱ′={1,…,m}∖ϱ. We defined the following multilinear maximal functions:
(34)ℳϱ(|f→|q)(x)≔supQ∋x∑ν=0∞‍2-νnl∏j∈ϱ‍1|Q|∫Q‍|fj|qjdyj×∏j∈ϱ′‍1|2νQ|∫2νQ‍|fj|qjdyj,(35)ℳϱ,L(logL)(|f→|q)(x)≔supQ∋x∑ν=0∞‍2-νnl∏j∈ϱ‍∥|fj|qj∥L(logL),Q×∏j∈ϱ′‍∥|fj|qj∥L(logL),2νQ,(36)ℳϱ,r(|f→|q)(x)≔supQ∋x∑ν=0∞‍2-νnl∏j∈ϱ‍(1|Q|∫Q‍|fj|qjrdyj)1/r×∏j∈ϱ′‍(1|2νQ|∫2νQ‍|fj|qjrdyj)1/r,where|fj|qj=(∑k=1∞‍|fjk(yj)|qj)1/qj.We remark that whenϱ=1,…,l and f→=(f1,…,fm), ℳϱ was first introduced by Grafakos el al. in [10] and denoted ℳϱ by ℳl. Chen and Wu [2] proved the multiple weighted norm inequality for ℳϱ,r. Similarly to (33), for any r>0, we have
(37)ℳϱ(|f→|q)≤ℳϱ,L(logL)(|f→|q)≤ℳϱ,r(|f→|q).Lemma 6.
Let1<p1,…,pm<∞, 1/p=1/p1+⋯+1/pm, P→=(p1,…,pm), ω→∈Ap→, and ϱ={j1,…,jl}⊆{1,…,m}, 1≤l<m. Then for some r>1 (r depending only on ω→), ℳr and ℳϱ,r defined by (36) are bounded from Lp1(ω1)×⋯×Lpm(ωm) to Lp(νω→). ℳ, ℳL(logL), ℳϱ, and ℳϱ,r are bounded from Lp1(ω1)×⋯×Lpm(ωm) to Lp(νω→).


## 4. Weighted Inequalities for Vector-Valued Singular Integral and Its Commutators

To prove our theorems, we first give two Lemmas about vector-valued operatorTq associated with T in m-GCZO(A,s,η,ϵ) which were obtained by Duong et al. in [5].Lemma 7.
LetT be a multilinear operator in m-GCZO(A,s,η,ϵ) with K satisfying Assumption (H2). And let 1/m<p<∞, 1<p1,…,pm<∞ with 1/p=1/p1+⋯+1/pm,  1/m<q<∞, and 1<q1,…,qm<∞ with 1/q=1/q1+⋯+1/qm. Then there exists a constant C>0 such that
(38)∥T(f→)∥Lp(ℝn)≤C∏j=1m‍∥f∥Lpj(ℝn).Lemma 8.
LetT be a multilinear operator in m-GCZO(A,s,η,ϵ) with K satisfying Assumption (H2). And let 1/m<p<∞, 1≤p1,…,pm<∞ with 1/p=1/p1+⋯+1/pm, and 1/m<q<∞, 1<q1,…,qm<∞ with 1/q=1/q1+⋯+1/qm. Then there exists a constant C>0 such that
(39)∥T(f→)∥Lp,∞(ℝn)≤C∏j=1m‍∥f∥Lpj(ℝn).Before proving Theorem1, we first present the estimates on the pointwise estimates for sharp Fefferman-stein maximal function action on Tq.Proposition 9.
LetT be an m-linear operator in GCZO(A,s,η,ϵ) and Tq satisfy the assumption in Theorem 1. Assume that ϱ0={1,…,m} and 0<δ<1/m. For any {f1m,…,fmk}k=1∞ in the product spaces Lp1(lq1)×⋯×Lpm(lqm) with 1<pj,  qj<∞ for j=1,…,m, then these exists a constant C>0 such that
(40)Mδ♯(Tq(f→))(x)≤C(ℳ(|f→|q)(x)+ℳϱ(|f→|q)(x)).
The ideas and arguments used in the proof are similar to those in [1] with some modifications. For completeness, we give the proof as follows.Proof of Proposition9.
For a pointx and a cube Q∋x, since ||α|r-|β|r|≤|α-β|β, for 0<r<1, to obtain (40), it suffices to prove for 0<δ<1/m(41)(1|Q|∫Q‍|Tq(f→)(z)-CQ|δdz)1/δ≤C(ℳ(|f→|q)(x)+ℳϱ(|f→|q)(x)),
for some constant CQ to be determined later.
Let{f1k,…,fmk}k=1∞ be any smooth vector-valued functions. Set each f→k=f→k0+f→k∞, where {f→k0}k=1∞={f→kχQ*}k=1∞={f1kχQ*,…,fmkχQ*}k=1∞ and Q*=(8n+4)Q, then we can write
(42)∏j=1m‍fjk(yj)=∏j=1m‍fjk(yj)(fjk0(yj)+fjk∞(yj))=∑α1,…,αmf1kα1(y1)⋯fmkαm(ym)=∏j=1m‍fjk0(yj)+∑′f1kα1(y1)⋯fmkαm(ym),
where each term of ∑‍′ contains at least one αj≠0.
Hence, we can write(43)Tq(f→)(z)=Tq(f→0)(z)+∑′Tq(f1α1,…,fmαm)(z).
Applying Kolmogorov’s inequality to the term
(44)Tq(f→0)(z)=Tq(f1k0,…,fmk0)(z),
with p=δ and q=1/m, we derive
(45)(1|Q|∫Q‍|Tq(f→0)(z)|δdz)1/δ≤C∥Tq(f→0)∥L1/m,∞(Q,dx/|Q|)≤C∏j=1m‍1|Q*|∫Q*‍|fj|qjdyj≤Cℳ(|f→|q)(x),
since T:L1×⋯×L1→L1/m,∞.
To estimate the remaining terms, we now setCQ=∑′Tq(f1kα1,…,fmkαm)(x) and will show that, for any z∈Q,
(46)∑′|Tq(f1kα1,…,fmkαm)(z)-Tq(f1kα1,…,fmkαm)(x)|≤Cℳ(|f→|q)(x).
Consider the case whenα1=⋯=αm=∞ and define
(47)Tq(f→∞)=Tq(f1k∞,…,fmk∞).
So
(48)|Tq(f1k∞,…,fmk∞)(z)-Tq(f1k∞,…,fmk∞)(x)|≤(∑k=1∞‍|∫(ℝn∖Q*)m‍K(z,y→)f1k(y1)⋯fmk(ym)dy→∫(ℝn/Q*)m‍|q)1/q-(∑k=1∞‍|∫(ℝn∖Q*)m‍K(x,y→)f1k(y1)⋯fmk(ym)dy→∫(ℝn/Q*)m‍|q∑k=1∞‍)1/q≤[∑k=1∞‍(∫(ℝn∖Q*)m‍|K(z,y→)-K(x,y→)|×|f1k(y1)⋯fmk(ym)|dy→∫(ℝn/Q*)m‍)q∫(ℝn/Q*)m‍]1/q≤[∑k=1∞‍(∫(ℝn∖Q*)m‍|K(z,y→)-Kt(0)(z,y→)|+|Kt(0)(z,y→)-K(x,y→)|×|f1k(y1)⋯fmk(ym)|dy→∫(ℝn/Q*)m‍)q∑k=1∞‍∫(ℝn/Q*)m‍]1/q≤(∑k=1∞‍|∑ν=0∞‍∫(2ν+1Q*∖2νQ*)m‍|K(z,y→)-Kt(0)(z,y→)|+|Kt(0)(z,y→)-K(x,y→)|×|f1k(y1)⋯fmk(ym)|dy→∑ν=0∞‍∫(2ν+1Q*∖2νQ*)m‍|q∑k=1∞‍|∑ν=0∞‍∫(2ν+1Q*∖2νQ*)m‍|K(z,y→)-Kt(0)(z,y→)|)1/q:=I.
Since x,z∈Q,  yj∈ℝn∖(8n+4)Q, and t=(2nl(Q))s, then |x-yj|~|z-yj|, |z-yj|>(4n+1)l(Q)>2t1/s, |x-yj|>2t1/s, and |x-z|<2t1/s. Hence,ϕ(|yj-z|/t1/s)=0 for j=1,2. By Assumption (H2), we can infer
(49)|K(z,y→)-Kt(0)(z,y→)|≤Atϵ/s(∑i=1m|z-yj|)mn+ϵ,|Kt(0)(z,y→)-K(x,y→)|≤Atϵ/s(∑i=1m|x-yj|)mn+ϵ.
Thus,
(50)I≤C(∑k=1∞‍(∑ν=0∞‍∫(2ν+1Q*∖2νQ*)m‍Atϵ/s(∑i=1m|z-yj|)mn+ϵ×∏j=1m‍|fjk(yj)|dy→)q)1/q≤C(∑k=1∞‍(∏j=1m‍|fjk(yj)|∑ν=0∞‍|Q*|ϵ/n(2n|Q*|1/n)mn+ϵ×∫(2ν+1Q*)m‍∏j=1m‍|fjk(yj)|dy→(∏j=1m‍|fjk(yj)|∑ν=0∞‍|Q*|ϵ/n(2n|Q*|1/n)mn+ϵ)q(∏j=1m‍|fjk(yj)|∑ν=0∞‍|Q*|ϵ/n(2n|Q*|1/n)mn+ϵ)1/q≤C(∑k=1∞‍(∑ν=0∞‍12νϵ1(2ν+1|Q*|1/n)mn×∫(2ν+1Q*)m‍∏j=1m‍|fjk(yj)|dy→(∏j=1m‍|fjk(yj)|∑ν=0∞‍|Q*|ϵ/n(2n|Q*|1/n)mn+ϵ)q(∏j=1m‍|fjk(yj)|∑ν=0∞‍|Q*|ϵ/n(2n|Q*|1/n)mn+ϵ)1/q≤C∑ν=1∞‍12νϵ∏j=1m‍[∑k=1∞‍(1(2ν+1|Q*|1/n)n×∫2ν+1Q*‍|fjk(yj)|dyj1(2ν+1|Q*|1/n)n)qj]1/qj≤C∑ν=1∞‍12νϵ∏j=1m‍1(2ν+1|Q*|1/n)n∫2ν+1Q*‍|fj|qjdyj≤Cℳ(|f→|q)(x).It remains to estimate the terms in (46) with αj1=⋯=αjl=0, for some ϱ={j1,…,jl}⊂{1,…,m}=ϱ0 for 1≤l<m and ϱ′=ϱ0∖ϱ, by Assumption (H2), we have
(51)|Tq(f1α1,…,fmαm)(z)-Tq(f1α1,…,fmαm)(x)|≤[∑k=1∞‍(∫(ℝn)m‍|K(z,y→)-K(x,y→)|×|f1α1(y1)⋯fmαm(ym)|dy→∫(ℝn)m‍)q∑k=1∞‍(∫(ℝn)m‍|K(z,y→)-K(x,y→)|]1/q≤[∑k=1∞‍(∫(ℝn)m‍|K(z,y→)-Kt(0)(z,y→)|+|Kt(0)(z,y→)-K(x,y→)|×|f1α1(y1)⋯fmαm(ym)|dy→∫(ℝn)m‍)q∑k=1∞‍(∫(ℝn)m‍|K(z,y→)-Kt(0)(z,y→)|]1/q≤C{×∏j∈ρ′‍|fjk(yj)|dyj)]q∑k=1∞‍[∏j∈ϱ‍∫Q*‍|fjk(yj)|dyj×(∫(ℝn∖Q*)m‍(tϵ/t(∑j∈ϱ′|z-yj|)mn+ϵ+1(∑j∈ϱ′|z-yj|)mn)×∏j∈ϱ′‍|fjk(yj)|dyj∫(ℝn∖Q*)m‍(tϵ/t(∑j∈ϱ′|z-yj|)mn+ϵ)]q}1/q≤C{∫(2ν+1Q*)m‍∏j∈ρ′‍|fjk(yj)|dyj)]∑k=1∞‍[∏j∈ϱ‍∫Q*‍|fjk(yj)|dyj×(∫(2ν+1Q*)m‍∏j∈ρ′‍∑ν=0∞‍(|Q*|ϵ/n(2ν|Q*|1/n)mn+ϵ+1(2ν|Q*|1/n)mn)×∫(2ν+1Q*)m‍∏j∈ϱ′‍|fjk(yj)|dyj∫(ℝn∖Q*)m‍(tϵ/t(∑j∈ϱ′|z-yj|)mn+ϵ)]q}1/q≤C∑ν=0∞‍(|Q*|ϵ/n(2ν|Q*|1/n)mn+ϵ+1(2ν|Q*|1/n)mn)×∏j∈ϱ‍[∑k=1∞‍(∫Q*‍|fjk(yj)|dyj)qj]1/qj×∏j∈ϱ′‍[∑k=1∞‍(∫2ν+1Q*‍|fjk(yj)|dyj)qj]1/qj≤C∑ν=0∞‍12νϵ1(2ν|Q*|1/n)mn∏j∈ϱ‍∫Q*‍|fj|qjdyj×∏j∈ϱ′‍∫2ν+1Q*‍|fj|qjdyj+C∑ν=0∞‍1(2ν|Q*|1/n)mn∏j∈ϱ‍∫Q*‍|fj|qjdyj×∏j∈ϱ′‍∫2ν+1Q*‍|fj|qjdyj≤C∑ν=0∞‍12νϵ1(2ν+1|Q*|1/n)mn∏j=1m‍∫2ν+1Q*‍|fj|qjdyj+C∑ν=0∞‍12νnl∏j∈ϱ‍1|Q*|∫Q*‍|fj|qjdyj×∏j∈ϱ′‍1(|2ν+1Q*|1/n)n∫2ν+1Q*‍|fj|qjdyj≤C(ℳ(|f→|q)(x)+ℳϱ(|f→|q)(x)).
This finishes the proof of Proposition9.Now we restore to prove Theorem1.Proof of Theorem1.
By the definition ofω→∈AP→ implies that νω→∈A∞. Using Proposition 9 and the Fefferman-Stein inequality (26) and observing that 0<δ<1/m, we have
(52)∥Tq(f→)∥Lp(νω)≤∥Mδ(Tq(f→))∥Lp(νω)≤∥Mδ♯(Tq(f→))∥Lp(νω)≤∥ℳ(|f→|q)∥Lp(νω)+∑ϱ⊂ϱ0‍∥ℳϱ(|f→|q)∥Lp(νω).
Then following from (52) and Lemma 6, Theorem 1 can be proved.
We are left to check that∥Mδ(Tq(f→))∥Lp(νω) is finite. The method of proof is similar to that of Theorem 3.19 in [2]. So we omit it here.Here is a crucial proposition on commutatorTΣb→,q to prove Theorem 2.Proposition 10.
LetTq be a vector-valued multilinear operator associated with an m-linear operator T in GCZO(A,s,η,ϵ) whose kernels satisfie the Assumption (H2). Suppose that TΣb→,q is the corresponding commutator of Tq with b→∈BMOm. Let 0<δ<min{ϵ,1/m}, r>1. Then there exists a constant C>0 depending on δ and ϵ, such that
(53)Mδ♯(TΣb→,q(f→))(x)≤C∑i=1m‍(∑∅≠ρ⊂ρ0‍ℳρ,L(logL)(|f→|q)Mϵ(Tq(f→))(x)+ℳL(logL)(|f→|q)(x)+∑∅≠ϱ⊂ϱ0‍ℳϱ,L(logL)(|f→|q)(x))
holds for all bounded measurable vector functions {f→k}k=1∞={f1k,…,fmk}k=1∞.Proof of Proposition10.
By linearity, it suffices to consider the operator with only symbols. Without loss of generality, we only consider the case:j=1 and denote b1 by b for convenience.
Note that for any constantλ we have
(54)TΣb→,q(f→)(z)=(b(z)-λ)Tq(f1,…,fm)(z)-Tq(bf1,…,fm)(z).
Fix x, for any cube Q centered at x and a constant c determined later, we have
(55)(1|Q|∫Q‍|Tb,q(f→)(z)|δ-|c|δdz)1/δ≤(1|Q|∫Q‍|Tb,q(f→)(z)-c|δdz)1/δ≤(C|Q|∫Q‍|(b(z)-λ)Tq(f→)(z)|δdz)1/δ+(C|Q|∫Q‍|Tq((b-λ)f1,…,fm)(z)-c|δdz)1/δ≔I+II.
We analyze each term separately. Recall that Q*=(8n+4)Q and λ=bQ*  and thanks to Hölder’s inequality and note that 1<q<ϵ/δ, it follows that
(56)I≤C(1|Q|∫Q‍|b(z)-λ|δq′dz)1/δq′×(1|Q|∫Q‍|Tq(f→)(z)|δqdz)1/δq≤C∥b∥BMOMϵ(Tq(f→))(x).
To estimateII, we split again each f→k into f→k0+f→k∞, where f→k0=f→kχQ*={f1kχQ*,…,fmkχQ*}. This yields
(57)∏j=1m‍fjk(yj)=∑{α1,…,αm}∈{0,∞}‍fα1(y1)⋯fmkαm(ym)=∏j=1m‍fjk0(yj)+∑′f1kα1(y1)⋯fmkαm(ym),
where each term in ∑‍′ contains at least one αj≠0.
Choosec=∑‍′Cα1,…,αm with Cα1,…,αm=Tq((b-λ)f1kα1,…,fmkαm)(x), then
(58)II≤C(1|Q|∫Q‍|Tq((b-λ)f1k0,…,fmk0)(z)|δdz)1/δ+∑′(1|Q|∫Q‍|Tq((b-λ)f1kα1,…,fmkαm)(z)-Cα1,…,αm|δdz)1/δ≔II0+∑′IIα1,…,αm.
Noting that δ<1/m and Tq:L1×⋯×L1→L1/m,∞ and using Hölder’ inequality, we obtain
(59)II0=C(1|Q|∫Q‍|Tq((b-λ)f1k0,…,fmk0)(z)|δdz)1/δ≤C∥Tq((b-λ)f1k0,…,fmk0)∥L1/m,∞(Q,dx/|Q|)≤C1|Q|∫Q‍|b(y1)-λ||f10|qdy1∏j=2m‍1|Q|∫Q‍|fj0|qjdyj≤C1|Q|∫Q*‍|b(y1)-λ||f1|qdy1∏j=2m‍1|Q|∫Q*‍|fj|qjdyj≤C∥b∥BMO∥|f1|q∥L(logL),Q*∏j=2m‍(|fj|qj)Q*≤C∥b∥BMOℳL(logL)(|f→|q)(x).
Now consider the termII∞,…,∞. By Hölder’s and Minkoswki’s inequalities as well as Assumptions (H1) and (H2), we obtain
(60)II∞,…,∞=(1|Q|∫Q‍|Tq((b-λ)f1k∞,…,fmk∞)(z)-Tq((b-λ)f1k∞,…,fmk∞)(x)|δdz1|Q|∫Q‍)1/δ≤C1|Q|∫Q‍[∑k=1∞‍(∏j=2m‍|fjk(yj)|dy→∫(ℝn∖Q*)m‍|K(z,y→)-K(x,y→)|×|b(y1)-λ||f1k(y1)|×∏j=2m‍|fjk(yj)|dy→)q]1/qdz≤C1|Q|∫Q‍[∑k=1∞‍(∏j=2m‍|fjk(yj)|∫(ℝn∖Q*)m‍(|K(z,y→)-Kt(0)(z,y→)|+|Kt(0)(z,y→)-K(x,y→)|)×|b(y1)-λ||f1k(y1)|×∏j=2m‍|fjk(yj)|dy→)q]1/qdz≔II∞,…,∞′.
Sincex,z∈Q,yj∈ℝn∖(8n+4)Q, and t=(2nl(Q))s, then |x-yj|~|z-yj|, |z-yj|>(4n+1)l(Q)>2t1/s, |x-yj|>2t1/s and |x-z|<2t1/s. Hence,ϕ(|yj-z|/t1/s)=0 for j=1,2.
Thus from the Assumption (H2), we can follow that(61)II∞,…,∞′≤C|Q|∫Q‍[∑k=1∞‍(∏j=2m‍∫(ℝn∖Q*)m‍Atϵ/s(∑i=1m|z-yj|)mn+ϵ×|b(y1)-λ||f1k(y1)|×∏j=2m‍|fjk(yj)|dy→)q]1/qdz≤C[∑k=1∞‍(∏j=1m‍∑ν=0∞‍|Q*|ϵ/n(2ν|Q*|1/n)mn+ϵ×∫(2ν+1Q*)m‍|b1(y1)-λ||f1(y1)|×∏j=1m‍|fj(yj)|dy→∑ν=0∞‍|Q*|ϵ/n(2ν|Q*|1/n)mn+ϵ)q]1/q≤C∑ν=0∞‍12νϵ1(2ν+1|Q*|1/n)mn×[∑k=1∞‍(∑k=1∞‍∫2ν+1Q*‍|b1(y1)-λ|×|f1(y1)|dy1∑k=1∞‍)q1]1/q1×∏j=2m‍[∑k=1∞‍(∫2ν+1Q*‍|fj(yj)|dyj)qj]1/qj≤C∑ν=0∞‍12νϵ1(2ν+1|Q*|1/n)mn×∫2ν+1Q*‍|b1(y1)-λ||f1|q1dy1×∏j=2m‍∫2ν+1Q*‍|fj|qjdyj≤C∑ν=0∞‍ν2νϵ∥b∥BMO∥|f1|q1∥L(logL),2ν+1Q*×∏j=2m‍∥|fj|qj∥L(logL),2ν+1Q*≤C∥b∥BMOℳL(logL)(|f→|q)(x),
where we use the fact that∥b-λ∥expL,2ν+1Q≤Cν∥b∥expL,Q.
What remains to be considered is the termIIα1,…,αm such that αj1=⋯=αjl=0, for some ϱ={j1,…,jl}⊂{1,…,m} for 1≤l<m and denote ϱ′=ϱ0∖ϱ. We consider only the case 1∈ϱ, by Assumption (H2), we have
(62)IIα1,…,αm=(1|Q|∫Q‍|Tq((b-λ)f1kα1,…,fmkαm)(z)-Tq((b-λ)f1kα1,…,fmkαm)(x)|δdz1|Q|)1/δ≤1|Q|∫Q‍|Tq((b-λ)f1kα1,…,fmkαm)(z)-Tq((b-λ)f1kα1,…,fmkαm)(x)|dz≤1|Q|∫Q‍[∑k=1∞‍(∏j=2m‍∫(ℝn)m‍|K(z,y→)-K(x,y→)|×|b(y1)-λ||f1k0(y1)|×∏j=2m‍|fjkαj(yj)|dy→)q]1/qdz≤1|Q|∫Q‍[∑k=1∞‍(∏j=2m‍∫(ℝn)m‍(|K(z,y→)-Kt(0)(x,y→)|+|Kt(0)(z,y→)-K(x,y→)|)×|b(y1)-λ||f1k0(y1)|×∏j=2m‍|fjkαj(yj)|dy→)q]1/qdz≤C|Q|∫Q‍[∑k=1∞‍(tϵ/t(∑j∈ϱ′|z-yj|)mn+ϵ∫Q*‍|b(y1)-λ||f1k(y1)|dy1×∏j∈ϱ∖{1}‍∫Q*‍|fjk(yj)|dyj×∫(ℝn∖Q*)m-l‍×(tϵ/t(∑j∈ϱ′|z-yj|)mn+ϵ+1(∑j∈ϱ′|z-yj|)mn)×∏j∈ϱ′‍|fjk(yj)|dyj)q]1/qdz≤C|Q|∫Q‍[∏j∈ϱ′‍|fjk(yj)|dyj)q∑k=1∞‍(∏j∈ϱ′‍|fjk(yj)|dyj)q∫Q*‍|b(y1)-λ||f1k(y1)|dy1×∏j∈ϱ∖{1}‍∫Q*‍|fjk(yj)|dyj×∑ν=0∞‍∫(2ν+1Q*∖2νQ*)m-l‍×(tϵ/t(∑j∈ϱ′|z-yj|)mn+ϵ+1(∑j∈ϱ′|z-yj|)mn)×∏j∈ϱ′‍|fjk(yj)|dyj)q]1/qdz≤C[∑k=1∞‍(∏j∈ρ′‍∫Q*‍|b(y1)-λ||f1k(y1)|dy1×∏j∈ϱ∖{1}‍∫Q*‍|fjk(yj)|dyj×∑ν=0∞‍(|Q*|ϵ/n(2ν|Q*|1/n)mn+ϵ+1(2ν|Q*|1/n)mn)×∫(2ν+1Q*)m-l‍∏j∈ϱ′‍|fjk(yj)|dyj)q]1/q≤C[∑k=1∞‍(∫Q*‍|b(y1)-λ||f1k(y1)|dy1)q1]1/q1×∏j∈ϱ∖{1}‍[∑k=1∞‍(∫Q*‍|fjk(yj)|dyj)qj]1/qj×∑ν=0∞‍(|Q*|ϵ/n(2ν|Q*|1/n)mn+ϵ+1(2ν|Q*|1/n)mn)×∏j∈ϱ′‍[∑k=1∞‍(∫2ν+1Q*‍|fjk(yj)|dyj)qj]1/qj≤∫Q*‍|b(y1)-λ||f1|q1dy1×∏j∈ϱ∖{1}‍∫Q*‍|fj|qjdyj×∑ν=0∞‍(12νϵ1(2ν|Q*|1/n)mn+1(2ν|Q*|1/n)mn)×∏j∈ϱ′‍∫2ν+1Q*‍|fj|qjdyj≤C∥b∥BMO∥|f1|q1∥L(logL),Q*|Q*|×∏j∈ϱ∖{1}‍∫Q*‍|fj|qjdyj×[∑ν=0∞‍12νϵ1(2ν|Q*|1/n)mn×∏j∈ϱ′‍∫2ν+1Q*‍|fj|qjdyj+∑ν=0∞‍1(2ν|Q*|1/n)mn×∏j∈ϱ′‍∫2ν+1Q*‍|fj|qjdyj]≤C∥b∥BMO(∑ν=0∞‍12νϵ∏j=1m‍∥|fj|qj∥L(logL),2ν+1Q*+∑ν=0∞‍12νnl∏j∈ϱ‍∥|fj|qj∥L(logL),Q*×∏j∈ϱ′‍∥|fj|qj∥L(logL),2ν+1Q*)≤C∥b∥BMO(ℳL(logL)(|f→|q)(x)+ℳϱ,L(logL)(|f→|q)(x)).
This finishes the proof of Proposition 10.The next paragraph we will present the proof of Theorem2.Proof of Theorem2.
By Lemma4, ω→∈AP→ implies that νω→∈A∞. For simplicity, we may assume that ∑i=1m‍∥bi∥BMO=1. Using Proposition 10 and the Fefferman-Stein inequality (26), with 0<δ<ϵ<1/m, we have
(63)∥TΣb→,q(f→)∥Lp(νω)≤C∥Mδ(TΣb→,q(f→))∥Lp(νω)≤C∥Mδ♯(TΣb→,q(f→))∥Lp(νω)≤C(∑ρ⊂ρ0‍∥Mϵ(Tq(f→))∥Lp(νω)+∥ℳL(logL)(|f→|q)∥Lp(νω)+∑ϱ⊂ϱ0‍∥ℳϱ,L(logL)(|f→|q)∥Lp(νω))≤C(∑ρ⊂ρ0‍∥Mϵ♯(Tq(f→))∥Lp(νω)+∥ℳL(logL)(|f→|q)∥Lp(νω)+∑ϱ⊂ϱ0‍∥ℳϱ,L(logL)(|f→|q)∥Lp(νω))≤C(∑ρ⊂ρ0‍∥ℳ(|f→|q)∥Lp(νω)+∥ℳϱ(|f→|q)∥Lp(νω)+∥ℳL(logL)(|f→|q)∥Lp(νω)+∑ϱ⊂ϱ0‍∥ℳϱ,L(logL)(|f→|q)∥Lp(νω)).
Then following from (63) and Lemma 5, Theorem 2 is proved.The reminder that we are left to do is similar to the proof of Theorem1. Here we omit it.The proof of Theorem3 heavily relied on Proposition 11.Proposition 11.
LetTq be a vector-valued multilinear operator associated with an m-linear operator T in GCZO(A,s,η,ϵ) whose kernel satisfies the Assumption (H2). Suppose that TΠb→,q is the corresponding iterated commutator of Tq with b→∈BMO2. Let 0<δ<min{ϵ,1/6}, r>1, and ϱ⊂ϱ0={1,…,m}. Then there exists a constant C>0 depending on δ and ϵ, such that
(64)Mδ♯(TΠb→(f→))(x)≤C∥b1∥BMO∥b2∥BMO×(Mϵ(Tq(f→))(x)+ℳL(logL)(|f→|q)(x)+ℳ{1},L(logL)(|f→|q)(x)+ℳ{2},L(logL)(|f→|q)(x)).The proof of Proposition11 may be omitted since it is easy to get by the similar arguments of the proof of Proposition 10 and [3].Finally we will give the proof of Theorem3.Proof of Theorem3.
By Lemma4, ω→∈AP→ implies that νω→∈A∞. For simplicity, we may assume that ∑i=1m‍∥bi∥BMO=1. Using Proposition 11 and the Fefferman-Stein inequality (26) with 0<δ<ϵ<1/m, we have
(65)∥TΠb→,q(f→)∥Lp(νω)≤C∥Mδ(TΠb→,q(f→))∥Lp(νω)≤C∥Mδ♯(TΠb→,q(f→))∥Lp(νω)≤C(∑ρ⊂ρ0‍∥Mϵ(Tq(f→))∥Lp(νω)+∥ℳL(logL)(|f→|q)∥Lp(νω)+∑ϱ⊂ϱ0‍∥ℳϱ,L(logL)(|f→|q)∥Lp(νω))≤C(∑ρ⊂ρ0‍∥Mϵ♯(Tq(f→))∥Lp(νω)+∥ℳL(logL)(|f→|q)∥Lp(νω)+∑ϱ⊂ϱ0‍∥ℳϱ,L(logL)(|f→|q)∥Lp(νω))≤C(∑ρ⊂ρ0‍∥ℳ(|f→|q)∥Lp(νω)+∥ℳϱ(|f→|q)∥Lp(νω)+∥ℳL(logL)(|f→|q)∥Lp(νω)+∑ϱ⊂ϱ0‍∥ℳϱ,L(logL)(|f→|q)∥Lp(νω)).
Then following from (65) and Lemma 5, Theorem 3 is proved.How to check that∥Mδ(TΣb→,q(f→))∥Lp(νω) and ∥Mϵ(Tq(f→))∥Lp(νω) are finite is similar to the proof of Theorem  3.19 in [2]. We omit it here, too.



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*Source: 363916-2013-12-19.xml*