# Multilinear Singular and Fractional Integral Operators on Weighted Morrey Spaces

Authors: Hua Wang; Wentan Yi
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We will study the boundedness properties of multilinear Calderón-Zygmund operators and multilinear fractional integrals on products of weighted Morrey spaces with multiple weights.


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## Body

## 1. Introduction and Main Results

Multilinear Calderón-Zygmund theory is a natural generalization of the linear case. The initial work on the class of multilinear Calderón-Zygmund operators was done by Coifman and Meyer in [1] and was later systematically studied by Grafakos and Torres in [2–4]. Let ℝn be the n-dimensional Euclidean space, and let (ℝn)m=ℝn×⋯×ℝn be the m-fold product space (m∈ℕ). We denote by 𝒮(ℝn) the space of all Schwartz functions on ℝn and by 𝒮′(ℝn) its dual space, the set of all tempered distributions on ℝn. Let m≥2 and T be an m-linear operator initially defined on the m-fold product of Schwartz spaces, and taking values into the space of tempered distributions,
(1)T:𝒮(ℝn)×⋯×𝒮(ℝn)⟶𝒮′(ℝn).
Following [2], for given f→=(f1,…,fm), we say that T is an m-linear Calderón-Zygmund operator if, for some q1,…,qm∈[1,∞) and q∈(0,∞) with 1/q=∑k=1m‍1/qk, it extends to a bounded multilinear operator from Lq1(ℝn)×⋯×Lqm(ℝn) into Lq(ℝn) and if there exists a kernel function K(x,y1,…,ym) in the class m-CZK(A,ε), defined away from the diagonal x=y1=⋯=ym in (ℝn)m+1 such that
(2)T(f→)(x)=T(f1,…,fm)(x)=∫(ℝn)m‍K(x,y1,…,ym)f1(y1)⋯fm(ym)dy1⋯dym,
whenever f1,…,fm∈𝒮(ℝn) and x∉⋂k=1msuppfk. We say that K(x,y1,…,ym) is a kernel in the class m-CZK(A,ε) if it satisfies the size condition
(3)|K(x,y1,…,ym)|≤A(|x-y1|+⋯+|x-ym|)mn,
for some A>0 and all (x,y1,…,ym)∈(ℝn)m+1 with x≠yk for some 1≤k≤m. Moreover, for some ε>0, it satisfies the regularity condition that
(4)|K(x,y1,…,ym)-K(x′,y1,…,ym)|≤A·|x-x′|ε(|x-y1|+⋯+|x-ym|)mn+ε
whenever |x-x′|≤(1/2)max1≤k≤m|x-yk| and also that, for each fixed k with 1≤k≤m,
(5)|K(x,y1,…,yk,…,ym)-K(x,y1,…,yk′,…,ym)|≤A·|yk-yk′|ε(|x-y1|+⋯+|x-ym|)mn+ε
whenever |yk-yk′|≤(1/2)max1≤i≤m|x-yi|. In recent years, many authors have been interested in studying the boundedness of these operators on function spaces; see, for example, [5–8]. In 2009, the weighted strong and weak type estimates of multilinear Calderón-Zygmund singular integral operators were established in [9] by Lerner et al. New more refined multilinear maximal function was defined and used in [9] to characterize the class of multiple AP→ weights.Theorem A (see [9]).
Letm≥2 and T be an m-linear Calderón-Zygmund operator. If p1,…,pm∈(1,∞) and p∈(0,∞) with 1/p=∑k=1m‍1/pk and w→=(w1,…,wm) satisfies the AP→ condition, then there exists a constant C>0 independent of f→=(f1,…,fm) such that
(6)∥T(f→)∥Lp(νw→)≤C∏i=1m‍∥fi∥Lpi(wi),
where νw→=∏i=1m‍wip/pi.Theorem B (see [9]).
Letm≥2, and let T be an m-linear Calderón-Zygmund operator. If p1,…,pm∈[1,∞), min{p1,…,pm}=1 and p∈(0,∞) with 1/p=∑k=1m‍1/pk, and w→=(w1,…,wm) satisfies the AP→ condition, then there exists a constant C>0 independent of f→=(f1,…,fm) such that
(7)∥T(f→)∥WLp(νw→)≤C∏i=1m‍∥fi∥Lpi(wi),
where νw→=∏i=1m‍wip/pi.Letm≥2, and let 0<α<mn. For given f→=(f1,…,fm), the m-linear fractional integral operator is defined by
(8)Iα(f→)(x)=Iα(f1,…,fm)(x)=∫(ℝn)m‍f1(y1)⋯fm(ym)|(x-y1,…,x-ym)|mn-αdy1⋯dym.
For the boundedness properties of multilinear fractional integrals on various function spaces, we refer the reader to [10–16]. In 2009, Moen [17] considered the weighted norm inequalities for multilinear fractional integral operators and constructed the class of multiple AP→,q weights (see also [18]).Theorem C (see [17, 18]).
Letm≥2, 0<α<mn, and let Iα be an m-linear fractional integral operator. If p1,…,pm∈(1,∞), 1/p=∑k=1m‍1/pk and 1/q=1/p-α/n, and w→=(w1,…,wm) satisfies the AP→,q condition, then there exists a constant C>0 independent of f→=(f1,…,fm) such that
(9)∥Iα(f→)∥Lq((νw→)q)≤C∏i=1m‍∥fi∥Lpi(wipi),
where νw→=∏i=1m‍wi.Theorem D (see [17, 18]).
Letm≥2, 0<α<mn, and let Iα be an m-linear fractional integral operator. If p1,…,pm∈[1,∞), min{p1,…,pm}=1, 1/p=∑k=1m‍1/pk and 1/q=1/p-α/n, and w→=(w1,…,wm) satisfies the AP→,q condition, then there exists a constant C>0 independent of f→=(f1,…,fm) such that
(10)∥Iα(f→)∥WLq((νw→)q)≤C∏i=1m‍∥fi∥Lpi(wipi),
where νw→=∏i=1m‍wi.On the other hand, the classical Morrey spacesℒp,λ were originally introduced by Morrey in [19] to study the local behavior of solutions to second-order elliptic partial differential equations. For the boundedness of the Hardy-Littlewood maximal operator, the fractional integral operator, and the Calderón-Zygmund singular integral operator on these spaces, we refer the reader to [20–22]. For the properties and applications of classical Morrey spaces, one can see [23–25] and the references therein.In 2009, Komori and Shirai [26] first defined the weighted Morrey spaces Lp,κ(w) which could be viewed as an extension of weighted Lebesgue spaces and studied the boundedness of the above classical operators in Harmonic Analysis on these weighted spaces. Recently, in [27–34], we have established the continuity properties of some other operators and their commutators on the weighted Morrey spaces Lp,κ(w).The main purpose of this paper is to establish the boundedness properties of multilinear Calderón-Zygmund operators and multilinear fractional integrals on products of weighted Morrey spaces with multiple weights. We now formulate our main results as follows.Theorem 1.
Letm≥2, and let T be an m-linear Calderón-Zygmund operator. If p1,…,pm∈(1,∞) and p∈(0,∞) with 1/p=∑k=1m‍1/pk and w→=(w1,…,wm)∈AP→ with w1,…,wm∈A∞, then for any 0<κ<1, there exists a constant C>0 independent of f→=(f1,…,fm) such that
(11)∥T(f→)∥Lp,κ(νw→)≤C∏i=1m‍∥fi∥Lpi,κ(wi),
where νw→=∏i=1m‍wip/pi.Theorem 2.
Letm≥2 and T be an m-linear Calderón-Zygmund operator. If p1,…,pm∈[1,∞), min{p1,…,pm}=1 and p∈(0,∞) with 1/p=∑k=1m‍1/pk, and w→=(w1,…,wm)∈AP→ with w1,…,wm∈A∞, then for any 0<κ<1, there exists a constant C>0 independent of f→=(f1,…,fm) such that
(12)∥T(f→)∥WLp,κ(νw→)≤C∏i=1m‍∥fi∥Lpi,κ(wi),
where νw→=∏i=1m‍wip/pi.Theorem 3.
Letm≥2, let 0<α<mn, and let Iα be an m-linear fractional integral operator. If p1,…,pm∈(1,∞), 1/p=∑k=1m‍1/pk, 1/qk=1/pk-α/mn and 1/q=∑k=1m‍1/qk=1/p-α/n, and w→=(w1,…,wm)∈AP→,q with w1q1,…,wmqm∈A∞, then for any 0<κ<p/q, there exists a constant C>0 independent of f→=(f1,…,fm) such that
(13)∥Iα(f→)∥Lq,κq/p((νw→)q)≤C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi),
where νw→=∏i=1m‍wi.Theorem 4.
Letm≥2, let 0<α<mn, and let Iα be an m-linear fractional integral operator. If p1,…,pm∈[1,∞), min{p1,…,pm}=1, 1/p=∑k=1m‍1/pk, 1/qk=1/pk-α/mn and 1/q=∑k=1m‍1/qk=1/p-α/n, and w→=(w1,…,wm)∈AP→,q with w1q1,…,wmqm∈A∞, then, for any 0<κ<p/q, there exists a constant C>0 independent of f→=(f1,…,fm) such that
(14)∥Iα(f→)∥WLq,κq/p((νw→)q)≤C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi),
where νw→=∏i=1m‍wi.


## 2. Notations and Definitions

The classicalAp weight theory was first introduced by Muckenhoupt in the study of weighted Lp boundedness of Hardy-Littlewood maximal functions in [35]. A weight w is a nonnegative, locally integrable function on ℝn; B=B(x0,rB) denotes the ball with the center x0 and radius rB. For 1<p<∞, a weight function w is said to belong to Ap if there is a constant C>0 such that, for every ball B⊆ℝn,
(15)(1|B|∫B‍w(x)dx)(1|B|∫B‍w(x)-1/(p-1)dx)p-1≤C,
where |B| denotes the Lebesgue measure of B. For the case p=1, w∈A1, if there is a constant C>0 such that for every ball B⊆ℝn,
(16)1|B|∫B‍w(x)dx≤C·essinfx∈Bw(x).
A weight function w∈A∞ if it satisfies the Ap condition for some 1<p<∞. We also need another weight class Ap,q introduced by Muckenhoupt and Wheeden in [36]. A weight function w belongs to Ap,q for 1<p<q<∞ if there is a constant C>0 such that, for every ball B⊆ℝn,
(17)(1|B|∫B‍w(x)qdx)1/q(1|B|∫B‍w(x)-p′dx)1/p′≤C.
When p=1, w is in the class A1,q with 1<q<∞ if there is a constant C>0 such that, for every ball B⊆ℝn,
(18)(1|B|∫B‍w(x)qdx)1/q(esssupx∈B1w(x))≤C.Now let us recall the definitions of multiple weights. Form exponents p1,…,pm, we will write P→ for the vector P→=(p1,…,pm). Let p1,…,pm∈[1,∞), and let p∈(0,∞) with 1/p=∑k=1m‍1/pk. Given w→=(w1,…,wm), set νw→=∏i=1m‍wip/pi. We say that w→ satisfies the AP→ condition if it satisfies
(19)supB(1|B|∫B‍νw→(x)dx)1/p∏i=1m‍(1|B|∫B‍wi(x)1-pi′dx)1/pi′<∞.
When pi=1, ((1/|B|)∫B‍wi(x)1-pi′dx)1/pi′ is understood as (infx∈Bwi(x))-1.Letp1,…,pm∈[1,∞), let  1/p=∑k=1m‍1/pk, and let q>0. Given w→=(w1,…,wm), set νw→=∏i=1m‍wi. We say that w→ satisfies the AP→,q condition if it satisfies
(20)supB(1|B|∫B‍νw→(x)qdx)1/q×∏i=1m‍(1|B|∫B‍wi(x)-pi′dx)1/pi′<∞.
When pi=1, ((1/|B|)∫B‍wi(x)-pi′dx)1/pi′ is understood as (infx∈Bwi(x))-1.Given a ballB and λ>0, λB denotes the ball with the same center as B whose radius is λ times that of B. For a given weight function w and a measurable set E, we also denote the Lebesgue measure of E by |E| and the weighted measure of E by w(E), where w(E)=∫E‍w(x)dx.Lemma 5 (see [37]).
Letw∈Ap with 1≤p<∞. Then, for any ball B, there exists an absolute constant C>0 such that
(21)w(2B)≤Cw(B).Lemma 6 (see [38]).
Letw∈A∞. Then for all balls B⊆ℝn, the following reverse Jensen inequality holds:
(22)∫B‍w(x)dx≤C|B|·exp(1|B|∫B‍logw(x)dx).Lemma 7 (see [37]).
Letw∈A∞. Then, for all balls B and all measurable subsets E of B, there exists δ>0 such that
(23)w(E)w(B)≤C(|E||B|)δ.Lemma 8 (see [9]).
Letp1,…,pm∈[1,∞), and let  1/p=∑k=1m‍1/pk. Then w→=(w1,…,wm)∈AP→ if and only if
(24)νw→∈Amp,wi1-pi′∈Ampi′,i=1,…,m,
where νw→=∏i=1m‍wip/pi and the condition wi1-pi′∈Ampi′ in the case pi=1 is understood as wi1/m∈A1.Lemma 9 (see [17, 18]).
Let0<α<mn, and p1,…,pm∈[1,∞), let 1/p=∑k=1m‍1/pk, and let 1/q=1/p-α/n. If w→=(w1,…,wm)∈AP→,q, then
(25)(νw→)q∈Amq,wi-pi′∈Ampi′,i=1,…,m,
where νw→=∏i=1m‍wi.Given a weight functionw on ℝn, for 0<p<∞, the weighted Lebesgue space Lp(w)   is defined as the set of all functions f such that
(26)∥f∥Lp(w)=(∫ℝn‍|f(x)|pw(x)dx)1/p<∞.
We also denote by WLp(w) the weighted weak space consisting of all measurable functions f such that
(27)∥f∥WLp(w)=supλ>0λ·w({x∈ℝn:|f(x)|>λ})1/p<∞.In 2009, Komori and Shirai [26] first defined the weighted Morrey spaces Lp,κ(w) for 1≤p<∞. In order to deal with the multilinear case m≥2, we will define Lp,κ(w) for all 0<p<∞.Definition 10.
Let0<p<∞, let 0<κ<1, and let w be a weight function on ℝn. Then the weighted Morrey space is defined by
(28)Lp,κ(w)={f∈Llocp(w):∥f∥Lp,κ(w)<∞},
where
(29)∥f∥Lp,κ(w)=supB(1w(B)κ∫B‍|f(x)|pw(x)dx)1/p
and the supremum is taken over all balls B in ℝn.Definition 11.
Let0<p<∞, let 0<κ<1, and let w be a weight function on ℝn. Then the weighted weak Morrey space is defined by
(30)WLp,κ(w)={fmeasurable:∥f∥WLp,κ(w)<∞},
where
(31)∥f∥WLp,κ(w)=supBsupλ>01w(B)κ/pλ·w({x∈B:|f(x)|>λ})1/p.Furthermore, in order to deal with the fractional order case, we need to consider the weighted Morrey spaces with two weights.Definition 12.
Let0<p<∞ and 0<κ<1. Then for two weights u and v, the weighted Morrey space is defined by
(32)Lp,κ(u,v)={f∈Llocp(u):∥f∥Lp,κ(u,v)<∞},
where
(33)∥f∥Lp,κ(u,v)=supB(1v(B)κ∫B‍|f(x)|pu(x)dx)1/p.Throughout this paper, we will useC to denote a positive constant, which is independent of the main parameters and not necessarily the same at each occurrence. Moreover, we will denote the conjugate exponent of p>1 by p′=p/(p-1).


## 3. Proofs of Theorems1 and 2

Before proving the main theorems of this section, we need to establish the following lemma.Lemma 13.
Letm≥2, let  p1,…,pm∈[1,∞), and let p∈(0,∞) with 1/p=∑k=1m‍1/pk. Assume that w1,…,wm∈A∞ and νw→=∏i=1m‍wip/pi; then, for any ball B, there exists a constant C>0 such that
(34)∏i=1m‍(∫B‍wi(x)dx)p/pi≤C∫B‍νw→(x)dx.Proof.
Sincew1,…,wm∈A∞, then, by using Lemma 6, we have
(35)∏i=1m‍(∫B‍wi(x)dx)p/pi≤C∏i=1m‍(|B|·exp(1|B|∫B‍logwi(x)dx))p/pi=C∏i=1m‍(|B|p/pi·exp(1|B|∫B‍logwi(x)p/pidx))=C·(|B|)∑i=1m‍p/pi·exp(∑i=1m‍1|B|∫B‍logwi(x)p/pidx).
Note that ∑i=1m‍p/pi=1 and νw→(x)=∏i=1m‍wi(x)p/pi. Then, by Jensen inequality, we obtain
(36)∏i=1m‍(∫B‍wi(x)dx)p/pi≤C·|B|·exp(1|B|∫B‍logνw→(x)dx)≤C∫B‍νw→(x)dx.
We are done.Proof of Theorem1.
For any ballB=B(x0,rB)⊆ℝn and letting  fi=fi0+fi∞, where fi0=fiχ2B, i=1,…,m, and χ2B denotes the characteristic function of 2B, then we write
(37)∏i=1m‍fi(yi)=∏i=1m‍(fi0(yi)+fi∞(yi))=∑α1,…,αm∈{0,∞}‍f1α1(y1)⋯fmαm(ym)=∏i=1m‍fi0(yi)+∑′f1α1(y1)⋯fmαm(ym),
where each term of ∑′ contains at least one αi≠0. Since T is an m-linear operator, then we have
(38)1νw→(B)κ/p(∫B‍|T(f1,…,fm)(x)|pνw→(x)dx)1/p≤1νw→(B)κ/p(∫B‍|T(f10,…,fm0)(x)|pνw→(x)dx)1/p+∑′1νw→(B)κ/p(∫B‍|T(f1α1,…,fmαm)(x)|pνw→(x)dx)1/p=I0+∑′Iα1,…,αm.
In view of Lemma 8, we have that νw→∈Amp. Applying Theorem A and Lemmas 13 and 5, we get
(39)I0≤C·1νw→(B)κ/p∏i=1m‍(∫2B‍|fi(x)|piwi(x)dx)1/pi≤C∏i=1m‍∥fi∥Lpi,κ(wi)·∏i=1mwi(2B)κ/piνw→(B)κ/p≤C∏i=1m‍∥fi∥Lpi,κ(wi)·νw→(2B)κ/pνw→(B)κ/p≤C∏i=1m‍∥fi∥Lpi,κ(wi).
For the other terms, let us first consider the case when α1=⋯=αm=∞. By the size condition, for any x∈B, we obtain
(40)|T(f1∞,…,fm∞)(x)|≤C∫(ℝn)m∖(2B)m‍|f1(y1)⋯fm(ym)|(|x-y1|+⋯+|x-ym|)mndy1⋯dym≤C∑j=1∞‍∫(2j+1B)m∖(2jB)m‍|f1(y1)⋯fm(ym)|(|x-y1|+⋯+|x-ym|)mndy1⋯dym≤C∑j=1∞‍∏i=1m‍∫2j+1B∖2jB‍|fi(yi)||x-yi|ndyi≤C∑j=1∞‍∏i=1m‍1|2j+1B|∫2j+1B‍|fi(yi)|dyi,
where we have used the notation Em=E×⋯×E. Furthermore, by using Hölder’s inequality, the multiple AP→ condition, and Lemma 13, we deduce that
(41)|T(f1∞,…,fm∞)(x)|≤C∑j=1∞‍∏i=1m‍1|2j+1B|(∫2j+1B‍|fi(yi)|piwi(yi)dyi)1/pi×(∫2j+1B‍wi(yi)1-pi′dyi)1/pi′≤C∑j=1∞‍1|2j+1B|m·|2j+1B|1/p+∑i=1m(1-1/pi)νw→(2j+1B)1/p×∏i=1m‍(∥fi∥Lpi,κ(wi)wi(2j+1B)κ/pi)≤C∏i=1m‍∥fi∥Lpi,κ(wi)·∑j=1∞‍(∏i=1mwi(2j+1B)κ/piνw→(2j+1B)1/p)≤C∏i=1m‍∥fi∥Lpi,κ(wi)·∑j=1∞‍νw→(2j+1B)(κ-1)/p.
Since νw→∈Amp⊂A∞, then it follows directly from Lemma 7 that
(42)νw→(B)νw→(2j+1B)≤C(|B||2j+1B|)δ.
Hence,
(43)I∞,…,∞≤νw→(B)(1-κ)/p|T(f1∞,…,fm∞)(x)|≤C∏i=1m‍∥fi∥Lpi,κ(wi)·∑j=1∞‍νw→(B)(1-κ)/pνw→(2j+1B)(1-κ)/p≤C∏i=1m‍∥fi∥Lpi,κ(wi)·∑j=1∞‍(|B||2j+1B|)δ(1-κ)/p≤C∏i=1m‍∥fi∥Lpi,κ(wi),
where the last inequality holds since 0<κ<1 and δ>0. We now consider the case where exactly ℓ of the αi are ∞ for some 1≤ℓ<m. We only give the arguments for one of these cases. The rest are similar and can easily be obtained from the arguments below by permuting the indices. Using the size condition again, we deduce that, for any x∈B,
(44)|T(f1∞,…,fℓ∞,fℓ+10,…,fm0)(x)|≤C∫(ℝn)ℓ∖(2B)ℓ‍∫(2B)m-ℓ‍|f1(y1)⋯fm(ym)|(|x-y1|+⋯+|x-ym|)mn×dy1⋯dym≤C∏i=ℓ+1m‍∫2B‍|fi(yi)|dyi×∑j=1∞‍1|2j+1B|m×∫(2j+1B)ℓ∖(2jB)ℓ‍|f1(y1)⋯fℓ(yℓ)|dy1⋯dyℓ≤C∏i=ℓ+1m‍∫2B‍|fi(yi)|dyi×∑j=1∞‍1|2j+1B|m∏i=1ℓ‍∫2j+1B∖2jB‍|fi(yi)|dyi≤C∑j=1∞‍∏i=1m‍1|2j+1B|∫2j+1B‍|fi(yi)|dyi,
and we arrive at the expression considered in the previous case. So for any x∈B, we also have
(45)|T(f1∞,…,fℓ∞,fℓ+10,…,fm0)(x)|≤C∏i=1m‍∥fi∥Lpi,κ(wi)·∑j=1∞‍νw→(2j+1B)(κ-1)/p.
Therefore, by the inequality (42) and the above pointwise inequality, we have
(46)Iα1,…,αm≤νw→(B)(1-κ)/p|T(f1∞,…,fℓ∞,fℓ+10,…,fm0)(x)|≤C∏i=1m‍∥fi∥Lpi,κ(wi)·∑j=1∞‍νw→(B)(1-κ)/pνw→(2j+1B)(1-κ)/p≤C∏i=1m‍∥fi∥Lpi,κ(wi)·∑j=1∞‍(|B||2j+1B|)δ(1-κ)/p≤C∏i=1m‍∥fi∥Lpi,κ(wi).
Combining the above estimates and then taking the supremum over all balls B⊆ℝn, we complete the proof of Theorem 1.Proof of Theorem2.
For any ballB=B(x0,rB)⊆ℝn and decomposing fi=fi0+fi∞, where fi0=fiχ2B, i=1,…,m, then, for any given λ>0, we can write
(47)νw→({x∈B:|T(f1,…,fm)|>λ})1/p≤νw→({x∈B:|T(f10,…,fm0)|>λ2m})1/p+∑′νw→({x∈B:|T(f1α1,…,fmαm)|>λ2m})1/p=I*0+∑′I*α1,…,αm,
where each term of ∑′ contains at least one αi≠0. By Lemma 8 again, we know that νw→∈Amp with 1≤mp<∞. Applying Theorem B and Lemmas 13 and 5, we have
(48)I*0≤Cλ∏i=1m‍(∫2B‍|fi(x)|piwi(x)dx)1/pi≤C·∏i=1mwi(2B)κ/piλ∏i=1m‍∥fi∥Lpi,κ(wi)≤C·νw→(2B)κ/pλ∏i=1m‍∥fi∥Lpi,κ(wi)≤C·νw→(B)κ/pλ∏i=1m‍∥fi∥Lpi,κ(wi).
In the proof of Theorem 1, we have already showed the following pointwise estimate (see (40) and (44)). Consider
(49)|T(f1α1,…,fmαm)(x)|≤C∑j=1∞‍∏i=1m‍1|2j+1B|∫2j+1B‍|fi(yi)|dyi.
Without loss of generality, we may assume that p1=⋯=pℓ=min{p1,…,pm}=1 and pℓ+1,…,pm>1. Using Hölder’s inequality, the multiple AP→ condition, and Lemma 13, we obtain
(50)|T(f1α1,…,fmαm)(x)|≤C∑j=1∞‍∏i=1ℓ‍1|2j+1B|∫2j+1B‍|fi(yi)|dyi×∏i=ℓ+1m‍1|2j+1B|∫2j+1B‍|fi(yi)|dyi≤C∑j=1∞‍∏i=1ℓ‍1|2j+1B|∫2j+1B‍|fi(yi)|wi(yi)dyi×(infyi∈2j+1Bwi(yi))-1×∏i=ℓ+1m‍1|2j+1B|(∫2j+1B‍|fi(yi)|piwi(yi)dyi)1/pi×(∫2j+1B‍wi(yi)1-pi′dyi)1/pi′≤C∏i=1m‍∥fi∥Lpi,κ(wi)∑j=1∞‍νw→(2j+1B)(κ-1)/p.
Observe that νw→∈Amp with 1≤mp<∞. Thus, it follows from the inequality (42) that, for any x∈B,
(51)|T(f1α1,…,fmαm)(x)|=C∏i=1m‍∥fi∥Lpi,κ(wi)·1νw→(B)(1-κ)/p∑j=1∞‍νw→(B)(1-κ)/pνw→(2j+1B)(1-κ)/p≤C∏i=1m‍∥fi∥Lpi,κ(wi)·1νw→(B)(1-κ)/p∑j=1∞‍(|B||2j+1B|)δ(1-κ)/p≤C∏i=1m‍∥fi∥Lpi,κ(wi)·1νw→(B)(1-κ)/p.
If {x∈B:|T(f1α1,…,fmαm)(x)|>λ/2m}=Ø, then the inequality
(52)I*α1,…,αm≤C·νw→(B)κ/pλ∏i=1m‍∥fi∥Lpi,κ(wi)
holds trivially. Now, if instead we suppose that {x∈B:|T(f1α1,…,fmαm)(x)|>λ/2m}≠Ø, then, by the pointwise inequality (51), we have
(53)λ<C∏i=1m‍∥fi∥Lpi,κ(wi)·1νw→(B)(1-κ)/p,
which is equivalent to
(54)νw→(B)1/p≤C·νw→(B)κ/pλ∏i=1m‍∥fi∥Lpi,κ(wi).
Therefore,
(55)I*α1,…,αm≤νw→(B)1/p≤C·νw→(B)κ/pλ∏i=1m‍∥fi∥Lpi,κ(wi).
Summing up all the above estimates and then taking the supremum over all balls B⊆ℝn and all λ>0, we complete the proof of Theorem 2.By using Hölder’s inequality, it is easy to check that if eachwi is in Api, then
(56)∏i=1m‍Api⊂AP→,
and this inclusion is strict (see [9]). Thus, as direct consequences of Theorems 1 and 2, we immediately obtain the following.Corollary 14.
Letm≥2, and let T be an m-linear Calderón-Zygmund operator. If p1,…,pm∈(1,∞) and p∈(0,∞) with 1/p=∑k=1m‍1/pk and w→=(w1,…,wm)∈∏i=1m‍Api, then, for any 0<κ<1, there exists a constant C>0 independent of f→=(f1,…,fm) such that
(57)∥T(f→)∥Lp,κ(νw→)≤C∏i=1m‍∥fi∥Lpi,κ(wi),
where νw→=∏i=1m‍wip/pi.Corollary 15.
Letm≥2 and let  T be an m-linear Calderón-Zygmund operator. If p1,…,pm∈[1,∞), min{p1,…,pm}=1 and p∈(0,∞) with 1/p=∑k=1m‍1/pk, and w→=(w1,…,wm)∈∏i=1m‍Api, then, for any 0<κ<1, there exists a constant C>0 independent of f→=(f1,…,fm) such that
(58)∥T(f→)∥WLp,κ(νw→)≤C∏i=1m‍∥fi∥Lpi,κ(wi),
where νw→=∏i=1m‍wip/pi.


## 4. Proofs of Theorems3 and 4

Following along the same lines as those of Lemma13, we can also show the following result, which plays an important role in our proofs of Theorems 3 and 4.Lemma 16.
Letm≥2, q1,…,qm∈[1,∞) and q∈(0,∞) with 1/q=∑k=1m‍1/qk. Assume that w1q1,…,wmqm∈A∞ and νw→=∏i=1m‍wi; then, for any ball B, there exists a constant C>0 such that
(59)∏i=1m‍(∫B‍wiqi(x)dx)q/qi≤C∫B‍νw→(x)qdx.Proof of Theorem3.
Arguing as in the proof of Theorem1, fix a ball B=B(x0,rB)⊆ℝn and decompose fi=fi0+fi∞, where fi0=fiχ2B, i=1,…,m. Since Iα is an m-linear operator, then we have
(60)1νw→q(B)κ/p(∫B‍|Iα(f1,…,fm)(x)|qνw→(x)qdx)1/q≤1νw→q(B)κ/p(∫B‍|Iα(f10,…,fm0)(x)|qνw→(x)qdx)1/q+∑′1νw→q(B)κ/p(∫B‍|Iα(f1α1,…,fmαm)(x)|qνw→(x)qdx)1/q=J0+∑′Jα1,…,αm,
where each term of ∑′ contains at least one αi≠0. In view of Lemma 9, we can see that (νw→)q∈Amq. Using Theorem C and Lemmas 16 and 5, we get
(61)J0≤C·1νw→q(B)κ/p∏i=1m‍(∫2B‍|fi(x)|piwi(x)pidx)1/pi≤C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi)·∏i=1mwiqi(2B)κq/pqiνw→q(B)κ/p=C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi)·(∏i=1mwiqi(2B)q/qi)κ/pνw→q(B)κ/p≤C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi)·νw→q(2B)κ/pνw→q(B)κ/p≤C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi).
For the other terms, let us first deal with the case when α1=⋯=αm=∞. By the definition of Iα, for any x∈B, we obtain
(62)|Iα(f1∞,…,fm∞)(x)|=∫(ℝn)m∖(2B)m‍|f1(y1)⋯fm(ym)|(|x-y1|+⋯+|x-ym|)mn-αdy1⋯dym=∑j=1∞‍∫(2j+1B)m∖(2jB)m‍|f1(y1)⋯fm(ym)|(|x-y1|+⋯+|x-ym|)mn-α×dy1⋯dym≤C∑j=1∞‍∏i=1m‍∫2j+1B∖2jB‍|fi(yi)||x-yi|n-α/mdyi≤C∑j=1∞‍∏i=1m‍1|2j+1B|1-α/mn∫2j+1B‍|fi(yi)|dyi.
Moreover, by using Hölder’s inequality, the multiple AP→,q condition, and Lemma 16, we deduce that
(63)|Iα(f1∞,…,fm∞)(x)|≤C∑j=1∞‍∏i=1m‍1|2j+1B|1-α/mn×(∫2j+1B‍|fi(yi)|piwi(yi)pidyi)1/pi×(∫2j+1B‍wi(yi)-pi′dyi)1/pi′≤C∑j=1∞‍1|2j+1B|m-α/n·|2j+1B|1/q+∑i=1m(1-1/pi)νw→q(2j+1B)1/q×∏i=1m‍(∥fi∥Lpi,κpiq/pqi(wipi,wiqi)wiqi(2j+1B)κq/pqi)≤C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi)·∑j=1∞‍[(∏i=1mwiqi(2j+1B)q/qi)κ/pνw→q(2j+1B)1/q]≤C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi)·∑j=1∞‍νw→q(2j+1B)κ/p-1/q.
Since (νw→)q∈Amq⊂A∞, then it follows immediately from Lemma 7 that
(64)νw→q(B)νw→q(2j+1B)≤C(|B||2j+1B|)δ′.
Hence,
(65)J∞,…,∞≤νw→q(B)1/q-κ/p|Iα(f1∞,…,fm∞)(x)|≤C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi)·∑j=1∞‍νw→q(B)1/q-κ/pνw→q(2j+1B)1/q-κ/p≤C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi)·∑j=1∞‍(|B||2j+1B|)δ′(1/q-κ/p)≤C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi),
where in the last inequality we have used the fact that 0<κ<p/q and δ′>0. We now consider the case where exactly ℓ of the αi are ∞ for some 1≤ℓ<m. We only give the arguments for one of these cases. The rest are similar and can easily be obtained from the arguments below by permuting the indices. Using the definition of Iα again, we can see that, for any x∈B,
(66)|Iα(f1∞,…,fℓ∞,fℓ+10,…,fm0)(x)|=∫(ℝn)ℓ∖(2B)ℓ‍∫(2B)m-ℓ‍|f1(y1)⋯fm(ym)|(|x-y1|+⋯+|x-ym|)mn-α×dy1⋯dym≤C∏i=ℓ+1m‍∫2B‍|fi(yi)|dyi×∑j=1∞‍1|2j+1B|m-α/n×∫(2j+1B)ℓ∖(2jB)ℓ‍|f1(y1)⋯fℓ(yℓ)|dy1⋯dyℓ≤C∏i=ℓ+1m‍∫2B‍|fi(yi)|dyi×∑j=1∞‍1|2j+1B|m-α/n∏i=1ℓ‍∫2j+1B∖2jB‍|fi(yi)|dyi≤C∑j=1∞‍∏i=1m‍1|2j+1B|1-α/mn∫2j+1B‍|fi(yi)|dyi,
and we arrive at the expression considered in the previous case. Thus, for any x∈B, we also have
(67)|Iα(f1∞,…,fℓ∞,fℓ+10,…,fm0)(x)|≤C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi)·∑j=1∞‍νw→q(2j+1B)κ/p-1/q.
Therefore, by the inequality (64) and the above pointwise inequality, we obtain
(68)Jα1,…,αm≤νw→q(B)1/q-κ/p|Iα(f1∞,…,fℓ∞,fℓ+10,…,fm0)(x)|≤C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi)·∑j=1∞‍νw→q(B)1/q-κ/pνw→q(2j+1B)1/q-κ/p≤C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi)·∑j=1∞‍(|B||2j+1B|)δ′(1/q-κ/p)≤C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi).
Summarizing the estimates derived above and then taking the supremum over all balls B⊆ℝn, we finish the proof of Theorem 3.Proof of Theorem4.
As before, fix a ballB=B(x0,rB)⊆ℝn and split fi into fi=fi0+fi∞, where fi0=fiχ2B, i=1,…,m. Then for each fixed λ>0, we can write
(69)νw→q({x∈B:|Iα(f1,…,fm)|>λ})1/q≤νw→q({x∈B:|Iα(f10,…,fm0)|>λ2m})1/q+∑′νw→q({x∈B:|Iα(f1α1,…,fmαm)|>λ2m})1/q=J*0+∑′J*α1,…,αm,
where each term of ∑′ contains at least one αi≠0. By Lemma 9 again, we know that (νw→)q∈Amq with 1<mq<∞. Using Theorem D and Lemmas 16 and 5, we have
(70)J*0≤Cλ∏i=1m‍(∫2B‍|fi(x)|piwi(x)pidx)1/pi≤C·∏i=1mwiqi(2B)κq/pqiλ∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi)≤C·νw→q(2B)κ/pλ∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi)≤C·νw→q(B)κ/pλ∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi).
In the proof of Theorem 3, we have already proved the following pointwise estimate (see (62) and (66)). Consider
(71)|Iα(f1α1,…,fmαm)(x)|≤C∑j=1∞‍∏i=1m‍1|2j+1B|1-α/mn∫2j+1B‍|fi(yi)|dyi.
Without loss of generality, we may assume that p1=⋯=pℓ=min{p1,…,pm}=1 and pℓ+1,…,pm>1. By using Hölder’s inequality, the multiple AP→,q condition, and Lemma 16, we obtain
(72)|Iα(f1α1,…,fmαm)(x)|≤C∑j=1∞‍∏i=1ℓ‍1|2j+1B|1-α/mn∫2j+1B‍|fi(yi)|dyi×∏i=ℓ+1m‍1|2j+1B|1-α/mn∫2j+1B‍|fi(yi)|dyi≤C∑j=1∞‍∏i=1ℓ‍1|2j+1B|1-α/mn×∫2j+1B‍|fi(yi)|wi(yi)dyi(infyi∈2j+1Bwi(yi))-1×∏i=ℓ+1m‍1|2j+1B|1-α/mn(∫2j+1B‍|fi(yi)|piwi(yi)pidyi)1/pi×(∫2j+1B‍wi(yi)-pi′dyi)1/pi′≤C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi)·∑j=1∞‍νw→q(2j+1B)κ/p-1/q.
Note that (νw→)q∈Amq with 1<mq<∞. Hence, it follows from the inequality (64) that, for any x∈B,
(73)|Iα(f1α1,…,fmαm)(x)|=C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi)·1νw→q(B)1/q-κ/p∑j=1∞‍νw→q(B)1/q-κ/pνw→q(2j+1B)1/q-κ/p≤C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi)·1νw→q(B)1/q-κ/p∑j=1∞‍(|B||2j+1B|)δ′(1/q-κ/p)≤C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi)·1νw→q(B)1/q-κ/p.
If {x∈B:|Iα(f1α1,…,fmαm)(x)|>λ/2m}=Ø, then the inequality
(74)J*α1,…,αm≤C·νw→q(B)κ/pλ∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi)
holds trivially. Now, if instead we assume that {x∈B:|Iα(f1α1,…,fmαm)(x)|>λ/2m}≠Ø, then, by the pointwise inequality (73), we get
(75)λ<C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi)·1νw→q(B)1/q-κ/p,
which in turn gives that
(76)νw→q(B)1/q≤C·νw→q(B)κ/pλ∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi).
Therefore,
(77)J*α1,…,αm≤νw→q(B)1/q≤C·νw→q(B)κ/pλ∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi).
Collecting all the above estimates and then taking the supremum over all balls B⊆ℝn and all λ>0, we conclude the proof of Theorem 4.By using Hölder’s inequality, it is easy to verify that if1≤pi<qi, 1/q=∑k=1m‍1/qk, and each wi is in Api,qi, then we have
(78)∏i=1m‍Api,qi⊂AP→,q,
and this inclusion is strict (see [17]). Also, recall that w∈Ap,q if and only if wq∈A1+q/p′⊂A∞ (see [36]). Thus, as straightforward consequences of Theorems 3 and 4, we finally obtain the following.Corollary 17.
Letm≥2, let  0<α<mn, and let  Iα be an m-linear fractional integral operator. If p1,…,pm∈(1,∞), 1/p=∑k=1m‍1/pk, 1/qk=1/pk-α/mn and 1/q=∑k=1m‍1/qk=1/p-α/n, and w→=(w1,…,wm)∈∏i=1m‍Api,qi, then, for any 0<κ<p/q, there exists a constant C>0 independent of f→=(f1,…,fm) such that
(79)∥Iα(f→)∥Lq,κq/p((νw→)q)≤C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi),
where νw→=∏i=1m‍wi.Corollary 18.
Letm≥2, let 0<α<mn, and let  Iα be an m-linear fractional integral operator. If p1,…,pm∈[1,∞), min{p1,…,pm}=1, 1/p=∑k=1m‍1/pk, 1/qk=1/pk-α/mn and 1/q=∑k=1m‍1/qk=1/p-α/n, and w→=(w1,…,wm)∈∏i=1m‍Api,qi, then, for any 0<κ<p/q, there exists a constant C>0 independent of f→=(f1,…,fm) such that
(80)∥Iα(f→)∥WLq,κq/p((νw→)q)≤C∏i=1m‍∥fi∥Lpi,κpiq/pqi(wipi,wiqi),
where νw→=∏i=1m‍wi.



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