# Multilinear Singular Integral Operators on Generalized Weighted Morrey Spaces

Authors: Yue Hu; Zhang Li; Yueshan Wang
Journal: Journal of Function Spaces
 (2014)

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## Abstract

The purpose of this paper is to discuss the boundedness properties of multilinear Calderón-Zygmund operator and its commutator on the generalized weighted Morrey spaces.


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## Body

## 1. Introduction and 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]. In 2009, the weighted estimates of multilinear Calderón-Zygmund singular integral operator and its commutator were established in [5] by Lerner et al. In 2013, the results of [5] were extended to the weighted Morrey space (see [6, 7]). In this paper, we will discuss the boundedness properties of multilinear Calderón-Zygmund operator and its commutator on the generalized weighted Morrey spaces.LetRn 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 denote by S′(Rn) its dual space of the set of all tempered distributions on Rn.We say that a locally integrable functionK(y0,y1,…,ym) defined away from the diagonal y0=y1=⋯=ym in (Rn)m+1 is a kernel in the class m-CZK(A,ϵ) if it satisfies the size estimate
(1)|K(y0,y1,…,ym)|≤A(∑k,l=0m|yl-yk|)mn
for some A>0 and all (y0,y1,…,ym)∈(Rn)m+1 with yl≠yk for some l, k. Moreover, assume that for some ϵ>0 and all 0≤j≤m it satisfies the smoothness estimates
(2)|K(y0,…,yj,…,ym)-K(y0,…,yj′,…,ym)|≤A|yj-yj′|ϵ(∑k,l=0m|yl-yk|)mn+ϵ
whenever
(3)|yj-yj′|≤12max⁡0≤k≤m|yj-yk|.LetT be a multilinear operator initially defined on the m-fold product of Schwartz spaces, and, by taking values into the space of tempered distributions,
(4)T:S(Rn)×⋯×S(Rn)⟶S′(Rn).
Following [2], we say that T is an m-linear Calderón-Zygmund operator if, for some q1,…,qm∈[1,∞) and q∈(0,∞) with 1/q=∑i=1m1/qi, 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, satisfying
(5)T(f→)(x)=T(f1,…,fm)(x)=∫(Rn)mK(x,y1,…,ym)f1(y1)⋯f(ym)dy1⋯dym
for all x∉⋂j=1msupp⁡fj and f1,…,fm are C∞ functions with compact support.Theorem 1 (see [5]).
Letm≥2 and let T be an m-linear Calderón-Zygmund operator. Suppose that 1/p=∑i=1m1/pi, ω→=(ω1,…,ωm) satisfies the Ap→ condition, and νω→=∏i=1mωip/pi.(1)
If1<pi<∞, i=1,…,m, then
(6)∥T(f→)∥Lp(νω→)≤C∏i=1m∥fi∥Lpi(ωi).(2)
If1≤pi<∞, i=1,…,m, and at least one of the pi=1, then
(7)∥T(f→)∥WLp(νω→)≤C∏i=1m∥fi∥Lpi(ωi).The purpose of this paper is to discuss the boundedness properties ofm-linear Calderón-Zygmund operator and its commutator on the generalized weighted Morrey spaces Mφp(ω).The classical Morrey spacesLp,λ were originally introduced by Morrey in [8] to study the local behavior of solutions to the 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 readers to [9–11]. Moreover, various Morrey spaces are defined in the process of study. Mizuhara [12] introduced the generalized Morrey space Lp,φ; Komori and Shirai [13] defined the weighted Morrey spaces Lp,κ(ω); Guliyev [14] gave a concept of generalized weighted Morrey space Mφp(ω) which could be viewed as extension of both Lp,φ and Lp,κ(ω). The boundedness of some operators on these Morrey spaces can be seen in [12–16].Our first result can be formulated as follows.Theorem 2.
Letm≥2, let T be an m-linear Calderón-Zygmund operator, and let φ→k=(φk1,…,φkm), k=1,2. Suppose that 1/p=∑i=1m1/pi, ω→=(ω1,…,ωm) satisfies the Ap→ condition with ω1,…,ωm∈A∞, and (φ→1,φ→2) satisfies the condition
(8)∫s∞essinf⁡r<t<∞⁡∏i=1mφ1i(x,t)ωi(B(x,t))1/pi∏i=1mωi(B(x,r))1/pidrr≤Cφ2(x,s).(1)
If1<pi<∞, i=1,…,m, then
(9)∥T(f→)∥Mφ2p(νω→)≤C∏i=1m∥fi∥Mφ1ipi(ωi).(2)
If1≤pi<∞, i=1,…,m, and at least one of the pi=1, then
(10)∥T(f→)∥WMφ2p(νω→)≤C∏i=1m∥fi∥Mφ1ipi(ωi),
where φ2=∏i=1mφ2i and νω→=∏i=1mωip/pi.Given a locally integrable vector functionb→=(b1,…,bm)∈(BMO)m, the commutator of b→ and the m-linear Calderón-Zygmund operator T, denoted here by TΣb, was introduced in [5] and is defined via
(11)TΣb(f→)=∑i=1mTbii(f→),
where
(12)Tbii(f→)=biT(f→)-T(f1,…,bifi,…,fm).
And the iterated commutator TΠb was introduced in [17] and is defined by
(13)TΠb(f→)=[b1,…,[bm-1,[bm,T]m]m-1⋯]1(f→).To clarify the notation, ifT is associated in the usual way with a Calderón-Zygmund kernel K, then, at a formal level,
(14)TΣb(f→)(x)=∫(Rn)m∑i=1m(bi(x)-bi(yi))K(x,y1,…,ym)mmmmmm×f1(y1)⋯fm(ym)dy1⋯dym,TΠb(f→)(x)=∫(Rn)m∏i=1m(bi(x)-bi(yi))K(x,y1,…,ym)mmmcmmm×f1(y1)⋯fm(ym)dy1⋯dym.ForT is an m-linear Calderón-Zygmund operator, ω→∈Ap→, and νω→=∏i=1mωip/pi with 1/p=1/p1+⋯+1/pm and b→∈(BMO)m, Lerner et al. [5] proved that TΣb bounded from Lp1(ω1)×⋯×Lpm(ωm) to Lp(νω→) and Pérez et al. [17] extended the result to TΠb. Our second result is to extend their results to the generalized weighted Morrey space.Theorem 3.
Letm≥2 and let T be an m-linear Calderón-Zygmund operator. Suppose that p1,…,pm∈(1,∞), p∈(0,∞) with 1/p=∑i=1m1/pi, ω→=(ω1,…,ωm) satisfies the Ap→ condition with ω1,…,ωm∈A∞, and (φ→1,φ→2) satisfies the condition
(15)∫s∞(1+ln⁡rs)messinf⁡r<t<∞⁡∏i=1mφ1i(x,t)ωi(B(x,t))1/pi∏i=1mωi(B(x,r))1/pidrr≤Cφ2(x,s).
Ifbi∈BMO,b→=(b1,…,bm), then there exist constants C>0 independent of f→=(f1,…,fm) such that
(16)∥TΣb(f→)∥Mφ2p(νω→)≤C∏i=1m∥bi∥*·∏i=1m∥fi∥Mφ1ipi(ωi),∥TΠb(f→)∥Mφ2p(νω→)≤C∏i=1m∥bi∥*·∏i=1m∥fi∥Mφ1ipi(ωi),
where νω→=∏i=1mωip/pi and φ2=∏i=1mφ2i.Example 4.
Letφ1i(x,t)=φ2i(x,t)=ωi(B(x,t))(κ-1)/pi and ωi∈A∞ for i=1,…,m, 0<κ<1. Then, (φ→1,φ→2) satisfies conditions (8) and (15).In fact, from (34), in Section 2, we have a constant δ>0 such that
(17)ωi(B(x,2ks))≥C2δnkωi(B(x,s))
for all i. Then,
(18)∫s∞essinf⁡r<t<∞⁡∏i=1mφ1i(x,t)ωi(B(x,t))1/pi∏i=1mωi(B(x,r))1/pidrr≤∫s∞(1+ln⁡rs)mmmmmm×essinf⁡r<t<∞⁡∏i=1mφ1i(x,t)ωi(B(x,t))1/pi∏i=1mωi(B(x,r))1/pidrr=∑k=0∞∫2ks2k+1s(1+ln⁡rs)m∏i=1mωi(B(x,r))(κ-1)/pidrr≤C∑k=0∞km∏i=1mωi(B(x,2k+1s))(κ-1)/pi≤C∑k=0∞km2(k+1)nδ((κ-1)/p)∏i=1mωi(B(x,s))(κ-1)/pi≤C∏i=1mωi(B(x,s))(κ-1)/pi=C∏i=1mφ2i(x,s).
Hence, we get the following corollaries.Corollary 5 (see [6]).
Letm≥2 and let T be an m-linear Calderón-Zygmund operator. Suppose that 1/p=∑i=1m1/pi, 0<κ<1, ω→=(ω1,…,ωm) satisfies the Ap→ condition with ω1,…,ωm∈A∞, and νω→=∏i=1mωip/pi.(1)
If1<pi<∞, i=1,…,m, then
(19)∥T(f→)∥Lp,κ(νω→)≤C∏i=1m∥fi∥Lpi,κ(ωi).(2)
If1≤pi<∞, i=1,…,m, and at least one of the pi=1, then
(20)∥T(f→)∥WLp,κ(νω→)≤C∏i=1m∥fi∥Lpi,κ(ωi).Corollary 6.
Letm≥2 and let T be an m-linear Calderón-Zygmund operator. Suppose that 1/p=∑i=1m1/pi, 0<κ<1, ω→=(ω1,…,ωm) satisfies the Ap→ condition with ω1,…,ωm∈A∞, and νω→=∏i=1mωip/pi. If bi∈BMO, b→=(b1,…,bm), then there exist constants C>0 independent of f→=(f1,…,fm) such that
(21)∥TΣb(f→)∥Lp,κ(νω→)≤C∏i=1m∥bi∥*·∏i=1m∥fi∥Lpi,κ(ωi),∥TΠb(f→)∥Lp,κ(νω→)≤C∏i=1m∥bi∥*·∏i=1m∥fi∥Lpi,κ(ωi).


## 2. Definitions and Preliminaries

A weightω is a nonnegative, locally integrable function on Rn. Let B=B(x0,rB) denote the ball with the center x0 and radius rB. For any ball B and λ>0, λB denotes the ball concentric with B whose radius is λ times as long. For a given weight function ω and a measurable set E, we also denote the Lebesgue measure of E by |E| and set the weighted measure ω(E)=∫Eω(x)dx.Let1≤p<∞, let φ be a positive measurable function on Rn×(0,∞), and let ω be a nonnegative measurable function on Rn. Following [14], we denote by Mφp(ω) the generalized weighted Morrey space and the space of all functions f∈Lloc⁡p(ω) with finite norm
(22)∥f∥Mφp(w)=sup⁡x∈Rn,r>01φ(x,r)(1w(B(x,r))∥f∥Lp(ω,B(x,r))p)1/p,
where
(23)∥f∥Lp(ω,B(x,r))=∫B(x,r)|f(y)|pw(y)dy.
Furthermore, by WMφp(ω) we denote the weak generalized weighted Morrey space of all functions f∈WMφp(ω) for which
(24)∥f∥WMφp(w)=sup⁡x∈Rn,r>01φ(x,r)(1w(B(x,r))∥f∥WLp(ω,B(x,r))p)1/p,
where
(25)∥f∥WLp(ω,B(x,r))=sup⁡t>0t(ω({y∈B(x,r):|f(y)|>t}))1/p.(1)
Ifω=1 and φ(x,r)=r(λ-n)/p with 0<λ<n, then Mφp(ω)=Lp,λ is the classical Morrey space.(2)
Ifφ(x,r)=ω(B(x,r))(κ-1)/p, then Mφp(ω)=Lp,κ(ω) is the weighted Morrey space.(3)
Ifφ(x,r)=ν(B(x,r))κ/pω(B(x,r))-1/p, then Mφp(ω)=Lp,κ(ν,ω) is the two weighted Morrey space.(4)
Ifω=1, then Mφp(ω)=Mφp is the generalized Morrey space.(5)
Ifφ(x,r)=ω(B(x,r))-1/p, then Mφp(ω)=Lp(ω).A weightω is said to belong to Ap, for 1<p<∞, if there exists a constant C such that, for every ball B⊂Rn,
(26)(1|B|∫Bω(x)dx)(1|B|∫Bω(x)1-p′dx)p-1≤C,
where s′ is the dual of s such that 1/s+1/s′=1. The class A1 is defined by replacing the above inequality with
(27)1|B|∫Bw(y)dy≤C·essinf⁡x∈Bw(x)foreveryballB⊂Rn.A weightω is said to belong to A∞ if there are positive numbers C and 0<δ<1 so that
(28)ω(E)ω(B)≤C(|E||B|)δ
for all balls B and all measurable E⊂B. It is well known that
(29)A∞=⋃1≤p<∞Ap.By (26), we have
(30)(∫Bω(x)dx)(∫Bω(x)1-p′dx)p-1≤C|B|p
for 1<p<∞. Note that
(31)(essinf⁡x∈Ef(x))-1=esssup⁡x∈E1f(x)
is true for any real-valued nonnegative function f and measurable on E (see [18], p.143) and, by (27), we get
(32)∥ω-1∥L∞(B)=esssup⁡x∈B1w(x)=1essinf⁡x∈B⁡w(x)≤C|B|ω(B)-1.The classicalAp weight theory was first introduced by Muckenhoupt in the study of weighted Lp boundedness of Hardy-Littlewood maximal function in [19].Lemma 7 (see [19, 20]).
Suppose thatω∈Ap and the following statements hold.(i)
For any1≤p<∞, there are positive numbers C and 0<δ<1 such that
(33)ω(2B)≤C2npω(B),(34)ω(2B)≥C2nδω(B);(ii)
Ap1⊂Ap2 for any 1≤p1<p2≤∞;(iii)
for any1<p<∞, one has ω1-p′∈Ap′.Now, let us recall the definition of multiple weights. Form exponents p1,…,pm, we write p→=(p1,…,pm). Let p1,…,pm∈[1,∞), and let p∈(0,∞) with 1/p=∑k=1m1/pk, ω→=(ω1,…,ωm). Given ω→=(ω1,…,ωm), set νω→=∏i=1mωip/pi. We say that ω→ satisfies the Ap→ condition if it satisfies
(35)sup⁡B(1|B|∫Bνω→(x)dx)1/p∏i=1m(1|B|∫Bωi(x)1-pi′(x)dx)1/pi′<∞.
When pi=1, (1/|B|)∫Bωi(x)1-pi′(x)dx is understood as (inf⁡x∈Bωi(x))-1.Lemma 8 (see [5]).
Letp1,…,pm∈[1,∞), and let 1/p=∑k=1m1/pk. Then, ω→=(ω1,…,ωm)∈Ap→ if and only if
(36)νω→∈Apm,ωi1-pi′∈Ampi′fori=1,…,m,
where νω→=∏i=1mωip/pi and the condition ωi1-pi′∈Ampi′ in the case pi=1 is understood as ωi1/m∈A1.Lemma 9 (see [6]).
Letp1,…,pm∈[1,∞), and let p∈(0,∞) with 1/p=∑k=1m1/pk. Assume that ω1,…,ωm∈A∞ and νω→=∏i=1mωip/pi. Then, for any ball B, there exists a constant C>0 such that
(37)∏i=1m(∫Bωi(x)dx)p/pi≤C∫Bνω→(x)dx.Let us recall the definition and some properties ofBMO. Following [21], a locally integrable function b is said to be in BMO if
(38)sup⁡B⊂Rn1|B|∫B|b(x)-bB|dx=∥b∥*<∞,
where bB=|B|-1∫Bb(y)dy.Lemma 10 (see [22]).
Suppose thatω∈A∞ and b∈BMO. Then, for any p≥1, one has
(39)(1ω(B)∫B|b(x)-bB|pω(x)dx)1/p≤C∥b∥*.Lemma 11 (see [23]).
Letb∈BMO, 1≤p<∞, and r1,r2>0. Then,
(40)(1|B(x0,r1)|∫B(x0,r1)|b(y)-bB(x0,r2)|pdy)1/p≤C∥b∥*(1+|ln⁡r1r2|),
where C>0 is independent of f, x0, r1, and r2.By Lemmas10 and 11, it is easy to prove the following results.Lemma 12.
Suppose thatω∈A∞ and b∈BMO. Then, for any 1≤p<∞ and r1,r2>0, one has
(41)(1ω(B(x0,r1))∫B(x0,r1)|b(x)-bB(x0,r2)|pω(x)dx)1/p≤C∥b∥*(1+|ln⁡r1r2|).


## 3. Proof of Theorem2

We first prove the following conclusions.Theorem 13.
Letm≥2, let T be an m-linear Calderón-Zygmund operator, and let ω→=(ω1,…,ωm) satisfy the Ap→ condition with ω1,…,ωm∈A∞. If p1,…,pm∈(1,∞) and p∈(0,∞) with 1/p=∑i=1m1/pi, then, for any x0∈Rn,
(42)∥T(f→)∥Lp(νω→,B(x0,s))≤C∏i=1mωi(B(x0,s))1/pi×∫2s∞(∏i=1m∥fi∥Lpi(ωi,B(x0,r))ωi(B(x0,r))-1/pi)drr.
If p1,…,pm∈[1,∞), min⁡{p1,…,pm}=1, and p∈(0,∞) with 1/p=∑i=1m1/pi, then, for any x0∈Rn,
(43)∥T(f→)∥WLp(νω→,B(x0,s))≤C∏i=1mωi(B(x0,s))1/pi×∫2s∞(∏i=1m∥fi∥Lpi(ωi,B(x0,r))ωi(B(x0,r))-1/pi)drr.Proof.
We representfi as fi=fi0+fi∞, where fi0=fiχB(x0,2s), i=1,…,m, and χB(x0,2s) denotes the characteristic function of B(x0,2s). Then,
(44)∏i=1mfi(yi)=∏i=1m(fi0(yi)+fi∞(yi))=∑α1,…,αm∈{0,∞}f1α1(y1)⋯fmαm(ym)=∏i=1mfi0(yi)+Σ′f1α1(y1)⋯fmαm(ym),
where each term of Σ′ contains at least one αi≠0. Since T is an m-linear operator, then
(45)∥T(f→)∥Lp(νω→,B(x0,s))≤C∥T(f10,…,fm0)∥Lp(νω→,B(x0,s))+CΣ′∥T(f1α1,…,fmαm)∥Lp(νω→,B(x0,s))=I0,…,0+Σ′Iα1,…,αm,∥T(f→)∥WLp(νω→,B(x0,s))≤C∥T(f10,…,fm0)∥WLp(νω→,B(x0,s))+CΣ′∥T(f1α1,…,fmαm)∥WLp(νω→,B(x0,s))=J0,…,0+Σ′Jα1,…,αm.
Then, by Theorem 1, if 1<pi<∞, i=1,…,m, we get
(46)I0,…,0≤C∏i=1m∥fi∥Lpi(ωi,B(x0,2s)).
If 1≤pi<∞, min⁡{p1,…,pm}=1, then
(47)J0,…,0≤C∏i=1m∥fi∥Lpi(ωi,B(x0,2s)).
Applying Hölder’s inequality, for 1≤pi<∞, i=1,…,m, we have
(48)|B(x0,2s)|≤ωi(B(x0,2s))1/pi∥ωi-1/pi∥Lpi′(B(x0,2s)).
Thus, for 1≤pi<∞,
(49)∏i=1m∥fi∥Lpi(ωi,B(x0,2s))≤C∏i=1m∥fi∥Lpi(ωi,B(x0,2s))·|B(x0,2s)|m∫2s∞drrmn+1≤C∏i=1m(ωi(B(x0,2s))1/pi∥fi∥Lpi(ωi,B(x0,2s))×∥ωi-1/pi∥Lpi′(B(x0,2s))mmmmmmc×∥ωi-1/pi∥Lpi′(B(x0,2s)))∫2s∞drrmn+1≤C∏i=1mωi(B(x0,2s))1/pimmmmm·∫2s∞(∏i=1m∥fi∥Lpi(ωi,B(x0,r))∥ωi-1/pi∥Lpi′(B(x0,r)))mmmmmmmm×drrmn+1.
From (35) and Lemma 9, we get
(50)∏i=1m∥ωi-1/pi∥Lpi′(B(x0,r))≤C|B(x0,r)|1/p+∑i=1m(1/pi′)(∫B(x0,r)νω→(x)dx)-1/p≤C|B(x0,r)|m∏i=1mωi(B(x0,r))-1/pi.
Then, for 1≤pi<∞, i=1,…,m,
(51)∏i=1m∥fi∥Lpi(ωi,B(x0,2s))≤C∏i=1mωi(B(x0,s))1/pi·∫2s∞(∏i=1m∥fi∥Lpi(ωi,B(x0,r))ωi(B(x0,r))-1/pi)drr.
This gives that I0,…,0 and J0,…,0 are majored by
(52)C∏i=1mωi(B(x0,s))1/pi·∫2s∞(∏i=1m∥fi∥Lpi(ωi,B(x0,r))ωi(B(x0,r))-1/pi)drr.
For the other term, let us first consider the case whenα1=⋯=αm=∞. By the size condition (1), for any x∈B(x0,s), we obtain
(53)|T(f1∞,…,fm∞)(x)|≤C∫(Rn∖B(x0,2s))m|f1(y1)⋯fm(ym)|(|x-y1|+⋯+|x-ym|)mndy1⋯dym≤C∑j=1∞∫(B(x0,2j+1s)∖B(x0,2js))m(|f1(y1)⋯fm(ym)|)mmmcmmm×((|x-y1|+⋯+|x-ym|)mn)-1dy1⋯dym≤C∑j=1∞∏i=1m∫B(x0,2j+1s)∖B(x0,2js)‍|fi(yi)||x-yi|ndyi≤C∑j=1∞∏i=1m((2j+1s)-n∫B(x0,2j+1s)|fi(yi)|dyi).
Applying Hölder’s inequality, it can be found that sup⁡x∈B(x0,s)|T(f1∞,…,fm∞)(x)| is less than
(54)C∑j=1∞∏i=1m((2j+1s)-n∥fi∥Lpi(ωi,B(x0,2j+1s))∥ωi-1/pi∥Lpi′(B(x0,2j+1s))).
Hence,
(55)sup⁡x∈B(x0,s)|T(f1∞,…,fm∞)(x)|≤C∑j=1∞∫2j+1s2j+2s(2j+2s)-nm-1mmccmm×(∏i=1m∥fi∥Lpi(ωi,B(x0,2j+1s))∥ωi-1/pi∥Lpi′(B(x0,2j+1s)))dr≤C∑j=1∞∫2j+1s2j+2s(∏i=1m∥fi∥Lpi(ωi,B(x0,2j+1s))mmmmmmmmm×∥ωi-1/pi∥Lpi′(B(x0,2j+1s))∏i=1m)drrmn+1≤C∫2s∞(∏i=1m∥fi∥Lpi(ωi,B(x0,2j+1s))∥ωi-1/pi∥Lpi′(B(x0,2j+1s)))drrmn+1.
Substituting (50) into above, we obtain
(56)sup⁡x∈B(x0,s)|T(f1∞,…,fm∞)(x)|≤C∫2s∞(∏i=1m∥fi∥Lpi(ωi,B(x0,r))ωi(B(x0,r))-1/pi)drr.
Using Hölder’s inequality,(57)(∫B(x0,s)νω→(x)dx)1/p≤C∏i=1mωi(B(x0,s))1/pi.
From (56) and (57), we know that I∞,…,∞ and J∞,…,∞ are not greater than (52) for 1≤pi<∞, i=1,…,m.
Now, we consider the case where exactlyτ of the αi are ∞ for some 1≤τ<m. We only give the arguments for one of the cases. The rest are similar and can easily be obtained from the arguments below by permuting the indices. Applying the size condition again, we deduce that, for any x∈B(x0,s),
(58)|T(f1∞,…,fτ∞,fτ+10,…,fm0)(x)|≤C∫(Rn∖B(x0,2s))τ∫(B(x0,2s))m-τ(|f1(y1)⋯fm(ym)|)mmmmmmcmmmcm×(+|x-ym|)mn(|x-ym||x-y1|+⋯mmmmmmmmmcmmcmc+|x-ym|)mn)-1dy1⋯dym≤C∏i=τ+1m∫B(x0,2s)|fi(yi)|dyi×∑j=1∞1|B(x0,2j+1s)|m×∫(B(x0,2j+1s)∖B(x0,2js))τ|f1(y1)⋯fτ(yτ)|dy1⋯dyτ≤C∏i=τ+1m∫B(x0,2s)|fi(yi)|dyic·∑j=1∞‍1|B(x0,2j+1s)|m∏i=1τ∫B(x0,2j+1s)∖B(x0,2js)|fi(yi)|dyi≤C∑j=1∞∏i=1m1|B(x0,2j+1s)|∫B(x0,2j+1s)|fi(yi)|dyi.
Similar to the estimates for I∞,…,∞, we get
(59)sup⁡x∈B(x0,s)|T(f1∞,…,fτ∞,fτ+10,…,fm0)(x)|≤C∫2s∞(∏i=1m∥fi∥Lpi(ωi,B(x0,r))ωi(B(x0,r))-1/pi)drr.
Then, I∞,…,∞,0,…,0 and J∞,…,∞,0,…,0 are all less than
(60)C∏i=1m‍ωi(B(x0,s))1/pi·∫2s∞‍(∏i=1m‍∥fi∥Lpi(ωi,B(x0,r))ωi(B(x0,r))-1/pi)drr.
Combining the above estimates, we complete the proof of Theorem 13.Now, we can give the proof of Theorem2. From the definition of generalized Morrey space, the norm of T(f→) on Mφ2p(νω→) equals
(61)sup⁡x∈Rn,r>0(∏i=1mφ2i(x,s))-1(∫B(x,s)1νω→(B(x,s))mmmmmmmmmmmmm×∫B(x,s)|T(f→)(y)|pνω→(y)dy)1/p.
By (37), we get
(62)(∫B(x,s)νω→(x)dx)-1/p≤C∏i=1m(∫B(x,s)ωi(x)dx)-1/pi.
Combining (42) and (62),
(63)(1νω→(B(x,s))∫B(x,s)|T(f→)(y)|pνω→(y)dy)1/p≤C∫s∞(∏i=1m∥fi∥Lpi(ωi,B(x,r))ωi(B(x,r))-1/pi)drr.
Since fi∈Mφ1ipi(ωi), then, by (31) and the fact that ∥fi∥Lpi(B(x,r)) are all nondecreasing function of r, we get
(64)∏i=1m∥fi∥Lpi(ωi,B(x,r))essinf⁡r<t<∞⁡∏i=1mφ1i(x,t)ωi(B(x,t))1/pi≤esssup⁡0<r<t<∞∏i=1m∥fi∥Lpi(ωi,B(x,r))∏i=1mφ1i(x,t)ωi(B(x,t))1/pi≤esssup⁡0<t<∞∏i=1m∥fi∥Lpi(ωi,B(x,t))∏i=1mφ1i(x,t)ωi(B(x,t))1/pi≤C∏i=1m∥fi∥Mφ1ipi(ωi).
Then,
(65)∫s∞(∏i=1m∥fi∥Lpi(ωi,B(x,r))ωi(B(x,r))-1/pi)drr=∫s∞∏i=1m∥fi∥Lpi(ωi,B(x,r))essinf⁡r<t<∞⁡∏i=1mφ1i(x,t)ωi(B(x,t))1/pimmmm×essinf⁡r<t<∞⁡∏i=1mφ1i(x,t)ωi(B(x,t))1/pi∏i=1mωi(B(x,t))1/pidrr≤C∏i=1m∥fi∥Mφ1ipi(ωi)mmmmc×∫s∞essinf⁡r<t<∞⁡∏i=1mφ1i(x,t)ωi(B(x,t))1/pi∏i=1nωi(B(x,t))1/pidrr.
By condition (8), we get
(66)∫s∞(∏i=1m∥fi∥Lpi(ωi,B(x,r))ωi(B(x,r))1/pi)drr≤C∏i=1m∥fi∥Mφ1ipi(ωi)∏i=1mφ2i(x,s).
Combining (61), (63), and (66), the proof of the first part of Theorem 2 is completed.Similarly, the norm ofT(f→) on WMφ2p(νω→) equals
(67)sup⁡x∈Rn,r>0(∏i=1mφ2i(x,s))-1mmmm×(1νω→(B(x,s))∥T(f→)∥WLp(νω→,B(x,s))p)1/p.
Combining (43) and (62),
(68)(1νω→(B(x,s))∥T(f→)∥WLp(νω→,B(x,s))p)1/p≤C∫s∞(∏i=1m∥fi∥Lpi(ωi,B(x,r))ωi(B(x,r))-1/pi)drr.
Substituting (66) into (68),
(69)(1νω→(B(x,s))∥T(f→)∥WLp(νω→,B(x,s))p)1/p≤Cφ2(x,s)∏i=1m∥fi∥Mφ1ipi(ωi).
Then,
(70)∥T(f→)∥WMφ2p(νω→)≤C∏i=1m∥fi∥Mφ1ipi(ωi).
This completes the proof of the second part of Theorem 2.


## 4. Proof of Theorem3

Theorem 14.
Letm≥2, let T be an m-linear Calderón-Zygmund operator, and let ω→=(ω1,…,ωm) satisfy the Ap→ condition with ω1,…,ωm∈A∞. If p1,…,pm∈(1,∞) and p∈(0,∞) with 1/p=∑i=1m1/pi, then, for any x0∈Rn,
(71)∥TΣb(f→)∥Lp(νω→,B(x0,s))≤C∏i=1m(∥bi∥*ωi(B(x0,s))1/pi)×∫2s∞(1+ln⁡rs)mmmmmm×(∏i=1m∥fi∥Lpi(ωi,B(x0,r))ωi(B(x0,r))-1/pi)drr,(72)∥TΠb(f→)∥Lp(νω→,B(x0,s))≤C∏i=1m‍(∥bi∥*ωi(B(x0,s))1/pi)×∫2s∞(1+ln⁡rs)mmmmmmm×(∏i=1m∥fi∥Lpi(ωi,B(x0,r))ωi(B(x0,r))-1/pi)drr.Proof.
We will give 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.
We representfi as fi=fi0+fi∞, where fi0=fiχB(x0,2s), i=1,2, and χB(x0,2s) denotes the characteristic function of B(x0,2s). Then,
(73)∥TΠb(f→)∥Lp(νω→,B(x0,s))≤C(∫B(x0,s)|TΠb(f10,f20)(x)|pνω→(x)dx)1/p+C(∫B(x0,s)|TΠb(f10,f2∞)(x)|pνω→(x)dx)1/p+C(∫B(x0,s)|TΠb(f1∞,f20)(x)|pνω→(x)dx)1/p+C(∫B(x0,s)|TΠb(f1∞,f2∞)(x)|pνω→(x)dx)1/p=I+II+III+IV.
Since TΠb bounded from Lp1(ω1)×Lp2(ω2) to Lp(νω→), we get
(74)(∫B(x0,s)|TΠb(f10,f20)(x)|pνω→(x)dx)1/p≤C∏i=1m∥bi∥*∥fi∥Lpi(ωi,B(x0,2s)).
Then, by (51), we get
(75)I≤C∏i=12(∥bi∥*ωi(B(x0,s))1/pi)·∫2s∞(∏i=12∥fi∥Lpi(ωi,B(x0,r))ωi(B(x0,r))-1/pi)drr.
Owing to the symmetry ofII and III, we only estimate II. Taking λi=(bi)B(x0,s), then
(76)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.
Similar to the estimate of (59), for any x∈B(x0,s), we can deduce
(77)sup⁡x∈B(x0,s)|T(f10,f2∞)(x)|≤C∫2s∞(∏i=12∥fi∥Lpi(ωi,B(x0,r))ωi(B(x0,r))-1/pi)drr.
By Lemma 8, we know that νω→∈A∞. Applying Hölder’s inequality and Lemma 10, we have
(78)(∫B(x0,s)|(b1(x)-λ1)(b2(x)-λ2)|pνω→(x)dx)1/p≤C∏i=12(∫B(x0,s)|bi(x)-λi|2pνω→(x)dx)1/2p≤C∏i=12∥bi∥*·(νω→(B(x0,s)))1/p.
Then, by (77), (78), and (57), we have
(79)(∫B(x0,s)|II1|pνω→(x)dx)1/p≤(∫B(x0,s)|(b1(x)-λ1)(b2(x)-λ2)|pνω→(x)dx)1/p×sup⁡x∈B(x0,s)|T(f10,f2∞)(x)|≤C∏i=12(∥bi∥*ωi(B(x0,s))1/pi)·∫2s∞(∏i=12∥fi∥Lpi(ωi,B(x0,r))ωi(B(x0,r))-1/pi)drr.
Applying the size condition, we deduce that, for anyx∈B(x0,s),
(80)|T(f10,(b2-λ2)f2∞)(x)|≤C∫B(x0,2s)∫Rn∖B(x0,2s)(|f1(y1)(b2(y2)-λ2)f2(y2)|)ccmmmmnmmmm×((|x-y1|+|x-y2|)2n)-1dy1dy2≤C∑j=1∞(2j+1s)-2n∫B(x0,2j+1s)|f1(y1)|dy1mcmm×∫B(x0,2j+1s)|(b2(y2)-λ2)f2(y2)|dy2.
Note that
(81)∫B(x0,2j+1s)|f1(y1)|dy1≤C∥f1∥Lp1(ω1,B(x0,2j+1s))∥ω1-1/p1∥Lp1′(B(x0,2j+1s)),∫B(x0,2j+1s)|(b2(y2)-λ2)f2(y2)|dy2≤C∥f2∥Lp2(ω2,B(x0,2k+1s))∥b2(·)-λ2∥Lp2′(ω21-p2′,B(x0,2k+1s)).
Then,
(82)sup⁡x∈B(x0,s)|T(f10,(b2-λ2)f2∞)(x)|≤C∫2s∞∏i=12∥fi∥Lpi(ωi,B(x0,r))mmmm×∥ω1-1/p1∥Lp1′(B(x0,r))∥b2(·)-λ2∥Lp2′(ω21-p2′,B(x0,r))drr2n+1.
Since ω21-p2′∈A2p2′, then, from Lemma 12, we get
(83)∥b2(·)-λ2∥Lp2′(ω21-p2′,B(x0,r))≤C(∫B(x0,r)|b2(z)-λ2|p2′ω21-p2′(z)dz)1/p2′≤C(1+|ln⁡rs|)∥b2∥*(ω21-p2′(B(x0,r)))1/p2′=C(1+|ln⁡rs|)∥b2∥*∥ω2-1/p2∥Lp2′(B(x0,r)).
By (50), we have
(84)∏i=12∥ωi-1/pi∥Lpi′(B(x0,r))≤C|B(x0,r)|2∏i=12ωi(B(x0,r))-1/pi.
From (82), (83), and (84), we can deduce
(85)sup⁡x∈B(x0,s)|T(f10,(b2-λ2)f2∞)(x)|≤C∥b2∥*∫2s∞(1+ln⁡rs)mmmcmmmm×(∏i=12∥fi∥Lpi(ωi,B(x0,r))ωi(B(x0,r))-1/pi)drr.
Applying Lemma 10 and (57), we have
(86)(∫B(x0,s)|b1(x)-λ1|pνω→(x)dx)1/p≤C∥b1∥*(νω→(B(x0,s)))1/p≤C∥b1∥*∏i=12ωi(B(x0,r))1/pi.
Then, by (85),
(87)(∫B(x0,s)|II2|pνω→(x)dx)1/p≤(∫B(x0,s)|b1(x)-λ1|pνω→(x)dx)1/p×sup⁡x∈B(x0,s)|T(f10,(b2-λ2)f2∞)(x)|≤C∏i=12(∥bi∥*ωi(B(x0,s))1/pi)×∫2s∞(1+ln⁡rs)mmcmmm×(∏i=12∥fi∥Lpi(ωi,B(x0,r))ωi(B(x0,r))-1/pi)drr.
Similarly, we also have that
(88)(∫B(x0,s)|II3|pνω→(x)dx)1/p≤C∏i=12(∥bi∥*ωi(B(x0,s))1/pi)×∫2s∞(1+ln⁡rs)mmmmm×(∏i=12∥fi∥Lpi(ωi,B(x0,r))ωi(B(x0,r))-1/pi)drr.
Applying the size condition again, for anyx∈B(x0,s), with the same method of estimate for (85), we have
(89)|T((b1-λ1)f10,(b2-λ2)f2∞)(x)|≤C∑j=1∞(2j+1s)-2n∏i=12∫B(x0,2j+1s)|(bi(yi)-λi)fi(yi)|dyi≤C∫2s∞(∏i=12∥fi∥Lpi(ωi,B(x0,r))mmmmmmm·∥bi(·)-λi∥Lpi′(ωi1-pi′,B(x0,r))∏i=12)drr2n+1≤C∏i=12∥bi∥*c·∫2s∞(1+ln⁡rs)2mmmm×(∏i=12∥fi∥Lpi(ωi,B(x0,r))ωi(B(x0,r))-1/pi)drr.
Then,
(90)(∫B(x0,s)|II4|pνω→(x)dx)1/p≤C(νω→(B(x0,s)))1/pc×sup⁡x∈B(x0,s)|((b1-λ1)f10,(b2-λ2)f2∞)(x)|≤C∏i=12(∥bi∥*ωi(B(x0,s))1/pi)c·∫2s∞(1+ln⁡rs)2mmmm×(∏i=12∥fi∥Lpi(ωi,B(x0,r))ωi(B(x0,r))-1/pi)drr.
Then, combining (79), (87), (88), and (90), we get
(91)(∫B(x0,s)|II|pνω→(x)dx)1/p≤C∏i=12(∥bi∥*ωi(B(x0,s))1/pi)c×∫2s∞(1+ln⁡rs)2mmmm×(∏i=12∥fi∥Lpi(ωi,B(x0,r))ωi(B(x0,r))-1/pi)drr.
Finally, we still decomposeTΠb(f1∞,f2∞)(x) as follows:
(92)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 IVj is completely analogous to IIj, j=1,2,3,4 with a bit difference, we get the following estimate without details:
(93)(∫B(x0,s)|IV|pνω→(x)dx)1/p≤C∏i=12(∥bi∥*ωi(B(x0,s))1/pi)×∫2s∞(1+ln⁡rs)2mmmm×(∏i=12∥fi∥Lpi(ωi,B(x0,r))ωi(B(x0,r))-1/pi)drr.
Summing up the above estimates, (72) is proved.We give the proof of Theorem3 in the following. From (62) and (72),
(94)(1νω→(B(x,s))∫B(x,s)|TΠb(f→)(y)|pνω→(y)dy)1/p≤C∏i=1m∥b∥*c·∫s∞(1+ln⁡rs)mmmcmm×(∏i=1m∥fi∥Lpi(ωi,B(x,r))ωi(B(x,r))-1/pi)drr.
Since (φ→1,φ→2) satisfies condition (15), ∥fi∥Lpi(B(x,r)) are all nondecreasing function of r, and fi∈Mφ1ipi(ωi), then
(95)∫s∞(1+ln⁡rs)m(∏i=1m∥fi∥Lpi(ωi,B(x,r))ωi(B(x,r))-1/pi)drr=∫s∞∏i=1m∥fi∥Lpi(ωi,B(x,r))essinf⁡r<t<∞⁡∏i=1mφ1i(x,t)ωi(B(x,t))1/pimmcm×(1+ln⁡rs)mmmcm×essinf⁡r<t<∞⁡∏i=1mφ1i(x,t)ωi(B(x,t))1/pi∏i=1nωi(B(x,t))1/pidrr≤C∏i=1m∥fi∥Mφ1ipi(ωi)c×∫s∞(1+ln⁡rs)mmmmcm×essinf⁡r<t<∞⁡∏i=1mφ1i(x,t)ωi(B(x,t))1/pi∏i=1nωi(B(x,t))1/pidrr≤Cφ2(x,s)∏i=1m∥fi∥Mφ1ipi(ωi).
Combining (94) and (95), we have proved that
(96)∥TΠb(f→)∥Mφ2p(νω→)≤C∏i=1m∥bi∥*∥fi∥Mφ1ipi(ωi).



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*Source: 325924-2014-07-17.xml*