# Multilinear Singular Integrals and their Commutators with Nonsmooth Kernels on Weighted Morrey Spaces

Authors: Songbai Wang; Yinsheng Jiang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Some multilinear maximal functions and the generalized Calderón-Zygmund operators and their commutators with nonsmooth kernels are studied. The purpose of this paper is to establish that these operators are bounded on certain product Morrey spacesLp,k(ℝn). Based on the boundedness of these operators from Lp1(ωi)×⋯×Lpm(ωm) to Lp(∏j=1m‍ωjp/pj), we obtained that they are also bounded from Lp1,k(ωi)×⋯×Lpm,k(ωm) to Lp,k(∏j=1m‍ωjp/pj) with 0<k<1, 1<pj<∞, 1/p=1/p1,…,pm and ωj∈Apj, j=1,…,m.


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## Body

## 1. Introduction

Let𝒮(ℝn) and 𝒮′(ℝn) be the Schwartz spaces of all rapidly decreasing functions and tempered distributions, respectively. Let T be a multilinear 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 [1], the m-multilinear Calderón-Zygmund operator T satisfies the following conditions:(S1)
there existqi<∞(i=1,…,m), it extends to such that a bounded multilinear operator from Lq1×⋯×Lqm to Lq, where
(2)1q=1q1+⋯+1qm;(S2)
there exists a functionK, defined off the diagonal x=y1=⋯=ym in (ℝn)m+1, satisfying
(3)T(f→)(x)=T(f1,…,fm)(x)=∫(ℝm)n‍K(x,y1,…,ym)f1(y1)⋯=∫(ℝm)n‍fm(ym)dy1⋯dym,
for all x∉∩j=1msuppfj and f1,…,fm∈𝒮(ℝn), where
(4)|K(y0,y1,…,ym)|≤A(∑l,k=0m|yl-yk|)mn;(5)|K(y0,…,yj,…,ym)-K(y0,…,yj,…,ym)|≤A|yj-yj′|ϵ(∑l,k=0m|yl-yk|)mn+ϵ,
for some ϵ>0 and all 0≤j≤m, whenever |yj-yj|≤(1/2)max0≤k≤m|yj-yk|.We also take some notation following [2]. Given a locally integrable vector function b=(b1,…,bm)∈(BMO)m. The commutator of b and the m-linear Calderón-Zygmund operator T, denoted here by TΣb, was introduced by Pérez and Torres in [3] and is defined via
(6)TΣb→(f→)=∑j=1m‍Tbjj(f→),
where
(7)Tbjj(f→)=bjT(f→)-T(f1,…,bjfj,…,fm).And the iterated commutatorsTΠb are defined by
(8)TΠb→(f→)=[b1,…,[bm-1,[bm,T]m]m-1…]1(f→).To clarify the notations, ifT is associated in the usual way with a Calderón-Zygmund kernel K, then at a formal level
(9)TΣb→(f→)(x)=∫(ℝn)m‍∑j=1m(bj(x)-bj(yj))×K(x,y1,…,ym)f1(y1)⋯=×wiefm(ym)dy1⋯dym,TΠb→(f→)(x)=∫(ℝn)m‍∏j=1m‍(bj(x)-bj(yj))×K(x,y1,…,ym)f1(y1)⋯=×weifm(ym)dy1⋯dym.In this paper, we will considerT to be associated with the kernel satisfying a weaker regularity conditions introduced by [4, 5]. A special example is the mth Calderón commutator.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 given by kernels at(x,y) in the sense that
(10)Atf(x)=∫ℝn‍at(x,y)f(y)dy,
for all f∈∪p∈[1,∞]Lp and x∈ℝn, and the kernels at(x,y) satisfy the following conditions:
(11)|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(12)limr→∞rn+ηh(rs)=0.Recall that thejth transpose T*,j of the m-linear operator T is defined via
(13)〈T*,j(f1,…,fm),g〉=〈T*,j(f1,…,fj-1,g,fj+1,…,fm),fj〉,
for all f1,…,fm,g in 𝒮(ℝn). It is seen that the kernel K*,j of T*,j is related to the kernel K of T via the identity
(14)K*,j(x,y1,…,yj-1,yj,yj+1,…,fm)=K(yj,y1,…,yj-1,x,yj+1,…,fm).If anm-linear operator T maps a product of Banach spaces X1×⋯×Xm into another Banach space X, then the transpose T*,j maps X1×⋯×Xj-1×X×Xj+1×⋯×Xm to 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 1.
Assume that for eachi=1,…,m there exist operators {At(i)}t>0 with kernels at(i)(x,y) that satisfy conditions (11) and (12) with constants s and η and that, for every j=0,1,2,…,m, there exist kernel K*,j,(i)(x,y1,…,ym) such that
(15)〈T*,j(f1,…,At(i)fi,…,fm),g〉=∫ℝn‍∫(ℝn)m‍Kt*,j,(i)(x,y1,…,fm)f1(y1)⋯=∫ℝn‍∫(ℝn)m‍fm(ym)g(x)dy1⋯dymdx,
for all f1,…,fm,g in 𝒮(ℝn) with ∩k=1msuppfk∩suppg=∅. Also assume that there exist a function ϕ∈𝒞(ℝ) with suppϕ⊂[-1,1] and constants ϵ>0 and A so that for every j=0,1,2,…,m and every i=1,2,…,m, we have
(16)|K*,j(x,y1,…,ym)-Kt*,j,(i)(x,y1,…,ym)|≤A(∑k=1m|x-yk|)mn∑k=1,k≠im‍ϕ(|yi-yk|t1/s)+Atϵ/t(∑k=1m|x-yk|)mn+ϵ
whenever t1/s≤|x-yi|/2.IfT satisfies Assumption 1 we will say that T is an m-linear operator with generalized Calderón-Zygmund kernel K. The collection of function K satisfying (15) and (16) with parameters m,A,s,η, and ϵ will be denoted by m-linear GCZK(A,s,η,ϵ). We say that T is of class m-GCZO(A,s,η,ϵ) if T has an associated kernel K in m-GCZK(A,s,η,ϵ). Throughout this paper, we always assume that the m-linear operator T satisfies the following assumption.Assumption 2.
Assume that there exist some1≤q1,…,qm<∞ and some 0<q<∞ with 1/q=1/q1+⋯+1/qm, such that T maps Lq1(ℝn)×⋯×Lqm(ℝn) to Lq,∞(ℝn).Theorem 3 (see [4]).
Assume thatT is a multilinear operator in m-GCZO(A,s,η,ϵ). Let 1/m≤p<∞,1≤pj≤∞ with 1/p=1/p1+⋯+1/pm, all the following statement are valid:(i)
when allpj>1, then T can be extended to be a bounded operator from the m-fold product Lp1(ℝn)×⋯×Lpm(ℝn) to Lp(ℝn);(ii)
when somepj=1, then T can be extended to be a bounded operator from the m-fold product Lp1(ℝn)×⋯×Lpm(ℝn) to Lp,∞(ℝn).
Moreover, there exists a constantC(n,m,pj,qj) such that
(17)∥T∥L1×⋯×L1→L1/m,∞≤C(n,m,pj,qj)(A+∥T∥Lq1×⋯×Lqm→Lq,∞).Assumption 4.
Assume that there exist operators{Bt}t>0 with kernels bt(x,y) that satisfy condition (11) and (12) with constants s and η. Let
(18)Kt(0)(x,y1,…,ym)=∫ℝn‍K(z,y1,…,ym)bt(x,z)dz.
We assume that the kernels Kt(0)(x,y1,…,ym) satisfy the following estimates; there exist a function ϕ∈𝒞(ℝ) with suppϕ⊂[-1,1] and constants ϵ>0 and A such that
(19)|Kt(0)(x,y1,…,ym)|≤A(∑k=1m|x-yk|)mn,
whenever 2t1/s≤min1≤j≤m|x-yj|, and
(20)|K(x,y1,…,ym)-Kt(0)(x′,y1,…,ym)|≤A(∑k=1m|x-yk|)mn∑k=1m‍ϕ(|yi-yk|t1/s)+Atϵ/s(∑k=1m|x-yk|)mn+ϵ,
whenever 2|x-x′|≤t1/s≤max1≤j≤m|x-yj|/2.It is known that condition (16) is weaker than, and indeed a consequence of, the Calderón-Zygmund kernel condition (5) from the proof of Proposition 2.1 in [4]. And also it is pointed out that Assumption 4 is weaker than the condition (5) for K(x,y1,…,ym) in [6].ForT be an m-linear Calderón-Zygmund operator, ω→∈AP→ and νω→=∏j=1m‍ωjp/pj with 1/p=1/p1+⋯+pm and b→∈BMOm, Lerner et al. [7] proved that T and TΣb→ bounded from Lp1(ω1)×⋯×Lpm(ωm) to Lp(νω→) and Pérez et al. [2] extended the result to TΠb→ when all 1<pj<∞, in the case of the endpoint, that is, some pj=1, weak type estimates have been established; for some details refer [2, 7]. To obtain the same results for the multilinear singular integral operators T in m-GCZO(A,s,η,ϵ) with kernel satisfying Assumption 4, some authors have done so much work. Duong et al. [5] obtained that T maps Lp1(ω)×⋯×Lpm(ω) to Lp(ω), where ω∈Ap0 with p0=min(p1,…,pm)>1. Grafakos et al. [8] proved that T maps Lp1(ω1)×⋯×Lpm(ωm) to Lp(νω→) where all pj>1 and ω→∈AP→, and maps Lp1(ω1)×⋯×Lpm(ωm) to Lp,∞(νω→) with some pj=1. For ω→∈∏j=1m‍Apj with pj>1,j=1,…,m, Anh and Duong [6] established that TΣb→ are of boundedness from Lp1(ω1)×⋯×Lpm(ωm) to Lp(νω→); after that, Chen and Wu [9] extended the results of Lerner et al. [7] and Pérez et al. [2] to the multilinear singular integral operators T in m-GCZO(A,s,η,ϵ) without the endpoint case.Definition 5.
Some multilinear maximal function used in Theorem6 will be listed in the following, which are introduced by Lerner et al. [7] and Grafakos et al. [8]:
(21)ℳ(f→)(x)=supQ∋x∏j=1m‍1|Q|∫Q‍|fj(yj)|dyj,ℳr(f→)(x)=supQ∋x∏j=1m‍1|Q|(∫Q‍|fj(yj)|rdyj)1/r,ℳLlogL(f→)(x)=supQ∋x∏j=1m‍∥fj∥LlogL,Q.The following relationship with the above three maximal functions is easy to check:(22)ℳ(f→)(x)≤ℳLlogL(f→)(x)≤Cℳr(f→)(x).Letr>1,1≤l<m,σ={j1,…,jl}⊂{1,…,m}, and σ′={1,…,m}∖σ. We define the following multilinear maximal functions:
(23)ℳσ(f→)(x)=supQ∋x∑k=0∞‍2-knl∏j∈σ‍1|Q|∫Q‍|fj(yj)|dyj=supQ∋x∑k=0∞‍2-knl×∏j∈σ′‍1|2kQ||fj(yj)|dyj,ℳσ(f→)(x)=supQ∋x∑k=0∞‍2-knl∏j∈σ‍(1|Q|∫Q‍|fj(yj)|dyj)1/r=supQ∋x∑k=0∞‍2-knl×∏j∈σ′‍(1|2kQ||fj(yj)|rdyj)1/r,ℳσ,LlogL(f→)(x)=supQ∋x∑k=0∞‍2-knl∏j∈σ‍∥fj∥LlogL,Q∏j∈σ′∥fj∥LlogL,2kQ‍.We have that(24)ℳσ(f→)(x)≤ℳσ,LlogL(f→)(x)≤Cℳσ,r(f→)(x).The following statements are our main results.Theorem 6.
Let0<k<1,1≤p1,…,pm<∞,1/p=1/p1+⋯+1/pm, and ω→∈∏i=1m‍Api. Let 1≤j<m, σ={i1,…,ij}⊂{1,…,m}, and for some r>1 (r depending only on ω→), if all pi>1, then ℳr and ℳσ,r are bounded from Lp1,k(ω1)×⋯×Lpm,k(ωm) to Lp,k(νω→), and or else, bounded from Lp1,k(ω1)×⋯×Lpm,k(ωm) to WLp,k(νω→).Corollary 7.
Under the same assumptions as in Theorem6. ℳ,ℳLlogL,ℳσ,ℳσ,LlogL are bounded from Lp1,k(ω1)×⋯×Lpm,k(ωm) to Lp,k(νω→) or WLp,k(νω→).Theorem 8.
Assume thatT is a multilinear operator in m-GCZO(A,s,η,ϵ) with kernel K satisfying Assumption 4. Let 0<k<1, 1/m≤p<∞, 1≤pj≤∞ with 1/p=1/p1+⋯+1/pm, and ωj∈Apj,j=1,…,m. Then we have the following:(i)
when allpj>1, there exists a constant C such that
(25)∥T(f→)∥Lp,k(νω→)≤C∥fj∥Lpj,k(ωj),(ii)
when somepj=1, there exists a constant C such that
(26)∥T(f→)∥WLp,k(νω→)≤C∥fj∥Lpj,k(ωj),where νω→=∏j=1m‍ωjp/pj.Theorem 9.
Assume thatT is a multilinear operator in m-GCZO(A,s,η,ϵ) with kernel K satisfying Assumption 4. Let 0<k<1, ω→=(ω1,…,ωm)∈∏j=1m‍Apj, and νω→=∏j=1m‍ωjp/pj with 1/p = 1/p1+⋯+pm and 1<pj<∞,j=1,…,m and b→=(b1,…,bm)∈BMOm. Then, there exists a constant C such that
(27)∥TΣb→(f→)∥Lp,k(νω→)≤C∥fj∥Lpj,k(ωj),∥TΠb→(f→)∥Lp,k(νω→)≤C∥fj∥Lpj,k(ωj).Following [2], for positive integers m and j with 1≤j≤m, we denote by Cjm the family of all finite subsets σ={σ(1),…,σ(j)} of {1,…,m} of j different elements, where we always take σ(l)<σ(k) if l<k. For any σ∈Cjm, the associated complementary sequence σ′∈Cm-jm is given by σ′={1,…,m}∖σ with the convention C0m=∅. Given an m-tuple of functions b→ and σ∈Cjm, we also use the notation b→σ for the j-tuple obtained from b→ given by (bσ(1),…,σ(j)). Similar to TΠb→, we define for T in m-GCZO(A,s,η,ϵ),σ∈Cjm and b→σ=(bσ(1),…,bσ(j)) in BMOj, the jth order iterated commutator
(28)TΠb→σ(f→)=[bσ(1),[bσ(2),…,[bσ(j),T]σ(j)…]σ(2)]σ(1)(f→);
that is, formally
(29)TΠb→σ(f→)(x)∫(ℝn)m‍(∏i=1j‍(bσ(i)(x)-bσ(i)(yσ(i))))TΠb→σ(f→)(x)∫(ℝn)m‍×K(x,y1,…,ym)∏i=1m‍fi(yi)dy→.
Clearly, TΠb→σ=TΠb→ when σ={1,…,m}, and TΠb→σ=Tbjj when σ={j}. We have the following general forms of Theorem 9 without the proof.Theorem 10.
Assume thatT is a multilinear operator in m-GCZO(A,s,η,ϵ) with kernel K satisfying Assumption 4. Let 1≤j≤m, σ∈Cjm, ω→=(ω1,…,ωm)∈∏i=1m‍Api, and νω→=∏i=1m‍ωip/pi with 1/p=1/p1+⋯+pm and 1<pi<∞,i=1,…,m and b→σ=(bσ(1),…,bσ(j))∈BMOj. Then, there exists a constant C such that
(30)∥TΠb→σ(f→)∥Lp,k(νω→)≤C∏i=1j‍∥bσ(i)∥∏i=1m‍∥fi∥Lpi,k(ωi).


## 2. Some Definitions and Results

In this section, we introduce some definitions and results used be later on.Definition 11 (Ap weights).
A weightω is a nonnegative, locally integrable function on ℝn. Let 1<p<∞; we call that a weight function ω that belongs to the class Ap, if there is a constant C such that, for any cube Q,
(31)(1|Q|∫Q‍ω(x)dx)(1|Q|∫Q‍ω(x)1-p′dx)p-1≤C,
and ω belongs to the class A1, if there is a constant C such that, for any cube Q,
(32)1|Q|∫Q‍ω(x)dx≤Cinfx∈Qω(x).
We denote A∞=∪p>1Ap.Definition 12 (see [7]).
Form exponents p1,…,pm∈[1,∞), we often write p for the number given by p=∑j=1m‍pj and denote P→ by the vector P→=(p1,…,pm). A multiple weight ω→=(ω1,…,ωm) is said to satisfy the AP→ condition if for
(33)νω→=∏j=1m‍ωp/pj,
it holds that
(34)supQ(1|Q|∫Q‍νω→(x)dx)1/p∏j=1m‍(1|Q|∫Q‍ωj(x)1-pj′dx)1/pj′<∞,
when pj=1, ((1/|Q|)∫Q‍ωj(x)1-pj′dx)1/pj′ is understood as (infxω(x))-1.As remarked in [7], ∏j=1m‍Apj is strictly contained in AP→; moreover, in general ω→∈AP→ does not imply ωj∈Lloc1 for any j, but instead
(35)ω→∈AP→⟺{νω→∈Amp,ωj1-pj′∈Ampj′,j=1,…,m,
where the condition ωj1-pj′∈Ampj′ in the case pj=1 is understood as ωj1/m∈A1.Definition 13 (see [10]).
Let1≤p<∞, 0<k<1, and ω be a weight function on ℝn. The weighted Morrey space is define by
(36)Lp,k(ω)={f∈Llocp:∥f∥Lp,k(ω)<∞},
where
(37)∥f∥Lp,k(ω)=supQ(1ω(Q)k∫Q‍|f(x)|pω(x))1/p.
The weighted weak Morrey space is defined by
(38)WLp,k(ω)={fmeasurable:∥f∥WLp,k(ω)<∞},
where
(39)∥f∥WLp,k(ω)=supQsupλ>0λω(Q)k/pω({x∈Q:|f|(x)>λ})1/p.We say that a weightω satisfies the doubling condition, denoting ω∈Δ2, if there is a constant C>0 such that ω(2Q)≤Cω(Q) holds for any cube Q. If ω∈Ap with 1≤p<∞, we know that ω(λQ)≤λnp[ω]Apω(Q) for all λ>1, then ω∈Δ2.Lemma 14 (see [10]).
Supposeω∈Δ2, then there exists a constant D>1 such that
(40)ω(2Q)≥Dω(Q),
for any cube.Lemma 15 (see [11]).
Ifωj∈A∞, then for any cube Q, we have
(41)∫Q‍∏j=1m‍ωjθj(x)dx≥∏j=1m‍(∫Q‍ωj(x)dx[ωj]∞)θj,
where ∑j=1m‍θj=1,0≤θj≤1.Lemma 16 (see [12]).
Supposeω∈A∞, then ∥b∥BMO(ω)≈∥b∥BMO. Here
(42)BMO(ω)={∫Q‍b:∥b∥BMO(ω)BMO(ω)=w=supQ1ω(Q)∫Q‍|b(x)-bQ,ω|ω(x)dx<∞},bQ,ω=1ω(Q)∫Q‍b(x)ω(x)dx.From the fact|b2jQ-bQ|≤Cj∥b∥BMO and Lemma 16, we can deduce that |b2jQ,ω-bQ,ω|≤Cj∥b∥BMO.Lemma 17 (see [8]).
Assume thatT is a multilinear operator in m-GCZO(A,s,η,ϵ) with kernel K satisfying Assumption 4. Let 1/m≤p<∞, 1≤pj≤∞ with 1/p=1/p1+⋯+1/pm and ωj∈Apj,j=1,…,m. Then we have the following:(i)
T extends to a bounded operators from Lp1(ω1)×⋯×Lpm(ωm) to Lp(νω→) if all the exponents pj are strictly greater than 1;(ii)
T extends to a bounded operators from Lp1(ω1)×⋯×Lpm(ωm) to Lp,∞(νω→) if some exponents pj are equal to 1.
In either case, the norm ofT is bounded by C(A+∥T∥Lq1×⋯×qm→Lq), where C is a positive constant depending on A,s,η,ϵ, and [w]AP→.Lemma 18 (see [6]).
Assume thatT is a multilinear operator in m-GCZO(A,s,η,ϵ) with kernel K satisfying Assumption 4. Let b→∈BMOm with ∥b→∥=1 and 1/p=1/p1+⋯1/pm with 1<pj<∞, j=1,…,m. Then we have the following:(i)
there exists a constantC such that
(43)∥TΣb→(f→)∥Lp(νω→)≤C∏j=1m‍∥fj∥Lpj(Mωj);(ii)
ifωj∈Apj, then there exists a constant C such that
(44)∥TΣb→(f→)∥Lp(νω→)≤C∏j=1m‍∥fj∥Lpj(ωj),where νω→=∏j=1m‍ωjp/pj.Lemma 19 (see [9]).
Assume thatT is a multilinear operator in m-GCZO(A,s,η,ϵ) with kernel K satisfying Assumption 4. Let b→∈BMOm with ∥b→∥=1 and 1/p=1/p1+⋯1/pm with 1<pj<∞, j=1,…,m. If ωj∈AP→ with P→=(p1,…,pm), then there exists a constant C such that
(45)∥TΠb→(f→)∥Lp(νω→)≤C∏j=1m‍∥fj∥Lpj(ωj),
where νω→=∏j=1m‍ωjp/pj.


## 3. Proof of Theorems

Proof of Theorem6.
Here, we only prove the boundedness ofℳσ,r. From [9], there exists some t∈(0,1) only depend on ω→ such that
(46)ℳ(f→)(x)≤C∏j=1m‍{Mνω→c((|fj|pjωj/νω→)t)(x)}1/tpj,
where Mνω→c is the weighted centered maximal operator. Then by the Hölder inequality,
(47)∥ℳσ,r(f→)(x)∥Lp,k(νω→)≤C∥∏i=1m‍{Mνω→([|fi|piωi/νω→]t)}1/tpi∥Lp,k(νω→)≤C∏i=1m‍∥{Mνω→([|fi|piωi/νω→]t)}1/tpi∥Lpi,k(νω→)≤C∏i=1m‍∥{Mνω→([|fi|piωi/νω→]t)}1/tpi∥L1/t,k(νω→)1/tpi≤C∏i=1m‍∥(|fi|piωi/νω→)t∥L1/t,k(νω→)1/tpi≤C∏i=1m‍∥fi∥Lpi,k(ωi).
The weak version is a very similar process by the Hölder inequality for the weak spaces. We omit the details.Proof of Theorem8.
For anyB=B(xB,rB)⊂ℝn, we split fi=fi0+fi∞ where fi0=fiχB*,i=1,2,…,m, and B*=8B; then
(48)∏i=1m‍fi(yi)=∏i=1m‍(fi0(yi)+fi∞(yi))=∑α1,…,αm∈{0,∞}‍∏i=1m‍fiαi(yi)=∏i=1m‍fi0(yi)+∑‍′f1α1(y1)⋯fmαm(ym),
where each term of ∑‍′ contains at least one αi≠0. Write then
(49)1νω→(B)k/p(∫B‍|T(f1,…,fm)(x)|pνω→(x)dx)1/p≤1νω→(B)k/p(∫B‍|T(f10,…,fm0)(x)|pνω→(x)dx)1/p+∑‍′1νω→(B)k/p(∫B‍|T(f1α1,…,fmαm)(x)|pνω→(x)dx)1/p=I0,…,0+∑‍′Iα1,…,αm.
From Definition 12, Lemma 17, we can get
(50)I0,…,0≤Cνω→(B)k/p∏i=1m‍(∫B*‍|fi0(x)|piωi(x)dx)1/pi≤C∏i=1mωi(B*)k/piνω→(B)k/p∏i=1m‍∥fi∥Lp,k(ωi)≤C∏i=1m‍∥fi∥Lp,k(ωi).
The last inequality holds by Lemma 15. For ∑‍′Iα1,…,αm, we first consider the case when α1=⋯=αm=∞. Taking t=(2rB)s, since x∈B and yi∈ℝn∖8B, we get
(51)|yi-x|>7rB>2t1/s,forallj=1,…,m;
hence, h(|yi-x|/t1/s)=0. By Assumption 4, we have that
(52)|K(x,y1,…,ym)-Kt(0)(x,y1,…,ym)|≤Atϵ/t(∑k=1m|x-yk|)mn+ϵ≤A(∑k=1m|x-yk|)mn.
For any x∈B, then by Assumption 4,
(53)|T(f1∞,…,fm∞)(x)|≤∫(ℝn∖B*)m‍|K(x,y1,…,ym)-Kt(0)(x,y1,…,ym)|≤∫(ℝn∖B*)m‍×∏i=1m‍|fi∞(yi)|dy→+∫(ℝn∖B*)m‍|Kt(0)(x,y1,…,ym)|∏i=1m‍|fi∞(yi)|dy→≤C∫(ℝn∖B*)m‍A(|x-y1|+⋯+|x-ym|)mn≤C∫(ℝn∖B*)m‍×∏i=1m‍|fi∞(yi)|dy→≤C∑l=1∞‍∏i=1m‍∫8l+1B∖8jB|fi(yi)||x-yi|ndyi≤C∑l=1∞‍∏i=1m‍1|8l+1B|(∫8l+1B‍|fi(yi)|piωi(yi)dyi)1/pi≤C∑l=1∞‍×(∫8l+1B‍ωi(yi)1-pi′dyi)1/pi′≤C∑l=1∞‍∏i=1m‍ωi(8l+1B)k/pi|8l+1B|∥fi∥Lpi,k(ωi)|8l+1B|ωi(8l+1B)1/pi≤C∑l=1∞‍νω→(8l+1B)(k-1)/p∏i=1m‍∥fi∥Lpi,k(ωi).
Since νω→∈Amp, then there is a positive δ such that
(54)νω(B)νω(8l+1B)≤C(|B||8l+1B|)δ.
Hence
(55)I∞,…,∞≤νω→(B)(1-k)/psupx∈B|T(f1∞,…,fm∞)(x)|≤C∑l=1∞‍(|B||8l+1B|)δ(1-k)/p∏i=1m‍∥fi∥Lpi,k(ωi)≤C∏i=1m‍∥fi∥Lpi,k(ωi).
It remains to estimate the terms with αi1=⋯=αij=0 for some {i1,…,ij}⊂{1,…,m} and 1≤j<m. We have
(56)|T(f1α1,…,fmαm)(x)|≤∫(ℝn∖B*)m‍|K(x,y1,…,ym)≤∫(ℝn∖B*)m‍-Kt(0)(x,y1,…,ym)|∏i=1m‍|fiαi(yi)|dy→+∫(ℝn∖B*)m‍|Kt(0)(x,y1,…,ym)|∏i=1m‍|fiαi(yi)|dy→≤C∏i∈{i1,…,ij}‍∫B*‍|fi(yi)|dyi≤C∏i∈{i1,…,ij}‍∫B*‍e×[∫(ℝn∖B*)m-j‍tϵ/s∏i∉{i1,…,ij}|fi(yi)|dyi(∑i∉{i1,…,ij}|x-yi|)mn+ϵ≤C∏i∈{i1,…,ij}‍∫B*‍|fiw|+∫(ℝn∖B*)m-j‍∏i∉{i1,…,ij}|fi(yi)|dyi(∑i∉{i1,…,ij}|x-yi|)mn]≤C1|8j+1B|m∏i∈{i1,…,ij}‍∫B*‍|fi(yi)|dyi×∑l=1∞‍∏i∉{i1,…,ij}‍∫8j+1B∖8jB‍|fi(yi)|dyi≤C∑l=1∞‍νω→(8l+1B)(k-1)/p∏i=1m‍∥fi∥Lpi,k(ωi).
Therefore, we also have
(57)Iα1,…,αm≤C∏i=1m‍∥fi∥Lpi,k(ωi).
Combining the above estimates and then taking the supermum over all balls B in ℝn, we have proved the previous part of Theorem 8.
Next, we turn to complete the proof of the weak inequality. For anyλ>0, we can write
(58)νω→({x∈B:|T(f1,…,fm)(x)|>λ})1/p≤νω→({x∈B:|T(f1∞,…,fm∞)(x)|>λ})1/p+∑‍′νω→({x∈B:|T(f1α1,…,fmαm)(x)|>λ})1/p=II0,…,0+∑‍′IIα1,…,αm.
By Lemmas 17 and 15, we can easily check that
(59)II0,…,0≤Cλ∏i=1m‍(∫B*‍fi0(yi)piωi(yi)dyi)1/pi≤Cνω→(B)k/pλ∏i=1m‍∥fi∥Lpi,k(ωi).
From the proof of (53) and (56), we have the following pointwise estimate:
(60)|T(f1α1,…,fmαm)(x)|≤C∑l=1∞‍∏i=1m‍1|8l+1B|∫8l+1B‍|fi(yi)|dyi.
Since at least one pi=1, we can assume that {i1,…,ij}⊂{1,…,m} such that pi1=⋯=pij=1 and others greater than 1. Then,
(61)|T(f1α1,…,fmαm)(x)|≤C∑l=1∞‍∏i∈{i1,…,ij}‍1|8l+1B|∫8l+1B‍|fi(yi)|ωi(yi)dyi≤C∑l=1∞‍∏i∈{i1,…,ij}‍1|8l+1B|∫8l+1B‍w×(infyi∈8l+1Bωi(yi))-1×∏i∉{i1,…,ij}‍1|8l+1B|(∫8l+1B‍|fi(yi)|piωi(yi)dyi)1/pi×∏i∉{i1,…,ij}‍ww×(∫8l+1B‍ωi(yi)1-pi′dyi)1/pi′≤Cνω→(B)(1-k)/p∏i=1m‍∥fi∥Lpi,k(ωi).
Suppose that {x∈B:|T(f1α1,…,fmαm)(x)|>λ}≠∅; then we have that
(62)νω→(B)1/p≤Cνω→(B)k/pλ∏i=1m‍∥fi∥Lpi,k(ωi);
therefore,
(63)IIα1,…,αm≤νω→(B)1/p≤Cνω→(B)k/pλ∏i=1m‍∥fi∥Lpi,k(ωi).
Taking the supremum over all balls B⊂ℝn and all λ>0, we complete the proof of Theorem 6.Proof of Theorem9.
We will show the proof forTΠb→ because the proof for TΣb→ is very similar but easier. Moreover, for simplicity of the expansion, we only present the case m=2.
For any cubeB, we also split fi as fi=fi0+fi∞ with fi0=fiχB⋆ and fi∞=fi-fi0. Then it remains only to verify the following inequalities:
(64)I=(1νω→(Q)k∫Q‍|TΠb(f10,f20)(x)|pνω→(x)dx)1/p≤C∏j=12‍∥bj∥BMO∏j=12‍∥fj∥Lpj,k(ωj),II=(1νω→(Q)k∫Q‍|TΠb(f10,f2∞)(x)|pνω→(x)dx)1/p≤C∏j=12‍∥bj∥BMO∏j=12‍∥fj∥Lpj,k(ωj),III=(1νω→(Q)k∫Q‍|TΠb(f1∞,f20)(x)|pνω→(x)dx)1/p≤C∏j=12‍∥bj∥BMO∏j=12‍∥fj∥Lpj,k(ωj),IV=(1νω→(Q)k∫Q‍|TΠb(f1∞,f2∞)(x)|pνω→(x)dx)1/p≤C∏j=12‍∥bj∥BMO∏j=12‍∥fj∥Lpj,k(ωj).
From Lemma 19, Lemma 15, and Hölder's inequality, we can get
(65)I≤C1νω→(Q)k/p∏j=12‍∥bj∥BMO(∫ℝn‍|fj0(x)|pjωj(x)dx)1/pj≤C1νω→(Q)k/p∏j=12‍[∥bj∥BMOωj(2Q)k/pj∥fj∥Lpj,k(ωj)]≤C∏j=12‍[∥bj∥BMO∥fj∥Lpj,k(ωj)].
Since II and III are symmetric, we only estimate II. Taking λj=(bj)B,ωj, TΠb→ can be divided into four part:
(66)TΠb(f10,f2∞)(x)=(b1(x)-λ1)(b2(x)-λ2)T(f10,f2∞)(x)-(b1(x)-λ1)T(f10,(b2-λ2)f2∞)(x)-(b2(x)-λ2)T((b1-λ1)f10,f2∞)(x)+T((b1-λ1)f10,(b2-λ2)f2∞)(x)=II1+II2+II3+II4.
From the proof of Theorem 8 we know that, for any x∈B,
(67)|T(f10,f2∞)(x)|≤C∑l=1∞‍νω→(8l+1B)(k-1)/p∏i=1m‍∥fj∥Lpj,k(ωj).Applying (67), Hölder's inequality and Lemma 16, we have
(68)(1νω→(Q)k∫Q‍|II1|pνω→(x)dx)1/p≤1νω→(Q)k/p(∫Q‍|(b1(x)-λ1)(b2(x)-λ2)|p≤1νω→(Q)k/pwwi×νω→(x)dx∫Q‍)1/p×∏j=12‍∥fj∥Lpj,k∑l=1∞‍νω→(2l+1Q)(k-1)/p≤νω→(Q)1/pνω→(Q)k/p∏j=12‍(1νω→(Q)∫Q|(bj(x)-λ1)|2p‍νω→(x)dx)1/2p×∏j=12‍∥fj∥Lpj,k∑l=1∞‍νω→(2l+1Q)(k-1)/p≤∏j=12‍∥bj∥BMO∥fj∥Lpj,k(ωj).
The last inequality is obtained by the property of A∞: there is a constant δ>0 such that
(69)νω→(Q)νω→(2l+1Q)≤C(|Q||2l+1Q|)δ.
For II2, by the Assumption 4, Lemma 15, and Lemma 16, it follows that
(70)|T(f10,(b2-λ2)f2∞)(x)|≤∫(ℝn)2‍|K(x,y1,y2)-Kt(0)(x,y1,y2)|≤∫(ℝn)2‍×|f10(y1)(b2(y2)-λ2)f2∞(y2)|dy1dy2+∫(ℝn)2‍|Kt(0)(x,y1,y2)||f10(y1)(b2(y2)-λ2)+∫(ℝn)2‍wwwwwwwwwwww×f2∞(y2)|dy1dy2≤C∫8B‍|f1(y1)|dy1∑l=1∞‍1|8lB|2≤C∫8B‍w×∫2l+1Q∖2lQ‍|(b2(y2)-λ2)f2(y2)|dy2≤C∑l=1∞‍∥b2∥BMO|8l+1B|2(∫8l+1B‍|f1(y1)|p1ωj(y1)dy1)1/p1≤C∑l=1∞‍ww×(∫2l+1Q‍ω1(y1)1-p1′dyj)1/p1′×(∫2l+1Q‍|f2(y2)|p2ω2(y2)dy2)1/p2×(∫2l+1Q‍|b2(y2)-λ2|p2′ω2(y2)-p2′/p2dy2)1/p2′≤C∑l=1∞‍l∏j=12‍∥b2∥BMO|8l+1B|(∫8l+1B‍|fj(yj)|pjωj(yj)dyj)1/pj≤C∑l=1∞‍l∏j=12‍ww×(∫2l+1Q‍ωj(yj)1-pj′dyj)1/pj′≤C∥b2∥BMO∏j=12‍∥fj∥Lpj,k(ωj)∑l=1∞‍lνω→(8l+1B)(k-1)/p.
Hölder's inequality and Lemma 16 tell us that
(71)(1νω→(Q)k∫Q‍|II2|pνω→(x)dx)1/p≤C1νω→(Q)k/p(∫Q‍|b1(x)-λ1|pνω→(x)dx)1/p×∏j=12‍∥fj∥Lpj,k∑l=1∞‍lνω→(2l+1Q)(k-1)/p≤Cνω→(Q)1/pνω→(Q)k/p∏j=12‍∥fj∥Lpj,k∑l=1∞‍lνω→(2l+1Q)(k-1)/p≤C∏j=12‍∥bj∥BMO∥fj∥Lpj,k(ωj).
Similarly, we also have that
(72)(1νω→(Q)k∫Q‍|II3|pνω→(x)dx)1/p≤C∏j=12‍∥bj∥BMO∥fj∥Lpj,k(ωj).
By Assumption 4, Lemma 15, and Lemma 16, a similar way deduces that
(73)|T((b1-λ1)f10,(b2-λ2)f2∞)(x)|≤C∥b1∥BMO∥b2∥BMO∏j=12‍∥fj∥Lpj,k(ωj)∑l=1∞‍l2νω→(8l+1B)(k-1)/p,
and so,
(74)(1νω→(Q)k∫Q‍|II4|pνω→(x)dx)1/p≤C∏j=12‍∥bj∥BMO∥fj∥Lpj,k(ωj).
Finally, we still decompose TΠb(f1∞,f2∞)(x) into four terms:
(75)TΠb(f1∞,f2∞)(x)=(b1(x)-λ1)(b2(x)-λ2)T(f1∞,f2∞)(x)-(b1(x)-λ1)T(f1∞,(b2-λ2)f2∞)(x)-(b2(x)-λ2)T((b1-λ1)f1∞,f2∞)(x)+T((b1-λ1)f1∞,(b2-λ2)f2∞)(x)=IV1+IV2+IV3+IV4.
Because each term of IVj is completely analogous to IIj, j=1,2,3,4 with a bit difference, so we get the following estimate without details:
(76)(1νω→(Q)k∫Q‍|IV|pνω→(x)dx)1/p≤C∏j=12‍∥bj∥BMO∥fj∥Lpj,k(ωj).
To this, we end the proof of Theorem 9.



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*Source: 280401-2013-12-30.xml*