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R E S E A R C H

Open Access

Commutators for multilinear singular

integrals on weighted Morrey spaces

Songbai Wang and Yinsheng Jiang

*

*Correspondence:

ysjiang@xju.edu.cn

College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, P.R. China

Abstract

In this paper we study the iterated commutators for multilinear singular integrals on weighted Morrey spaces. A strong type estimate and a weak endpoint estimate for the commutators are obtained. In the last section we present a problem for the multilinear Fourier multiplier with limited smooth condition.

MSC: 42B20; 42B25

Keywords: multilinear singular integrals; multiple weights; commutators

1 Introduction

As an important direction of harmonic analysis, the theory of multilinear Calderón-Zygmund singular integral operators has attracted more and more attention, which orig-inated from the work of Coifman and Meyer [], and it systematically was studied by Grafakos and Torres [, ]. The literature of the standard theory of multilinear Calderón-Zygmund singular integrals is by now quite vast, for example see [, –]. In , the au-thors [] introduced the new multiple weights and new maximal functions and obtained some weighted estimates for multilinear Calderón-Zygmund singular integrals. They also resolved some problems opened up in [] and [].

LetS(Rn) and S(Rn) be the Schwartz spaces of all rapidly decreasing functions and

tempered distributions, respectively. Having fixedm∈N, letTbe a multilinear operator

initially defined on them-fold product of Schwartz spaces and taking values into the space

of tempered distributions,

T:SRn× · · · ×SRnSRn.

Following [], them-multilinear Calderón-Zygmund operator T satisfies the following

conditions:

(S) there existqi<∞(i= , . . . ,m), it extends to a bounded multilinear operator from

Lq× · · · ×LqmtoLq, whereq=

q+· · ·+

qm;

(S) there exists a functionK, defined off the diagonalx=y=· · ·=ymin (Rn)m+, satis-fying

T(f)(x) =T(f, . . . ,fm)(x) =

(Rm)n

K(x,y, . . . ,ym)f(y)· · ·fm(ym)dy· · ·dym ()

(2)

for allx∈/mj=suppfjandf, . . . ,fmS(Rn), where

K(y,y, . . . ,ym)≤

A

(ml,k=|ylyk|)mn

()

and

K(y, . . . ,yj, . . . ,ym) –K

y, . . . ,yj, . . . ,ym

A|yjyj| (ml,k=|ylyk|)mn+

()

for some>  and all ≤jm, whenever|yjyj| ≤max≤k≤m|yjyk|.

We also use some notation following []. Given a locally integrable vector function

b= (b, . . . ,bm)∈(BMO)m, the commutator ofband them-linear Calderón-Zygmund

op-eratorT, denoted here byTb, was introduced by Pérez and Torres in [] and is defined

via

Tb(f) = m

j=

Tbj

j(f),

where

Tbj

j(f) =bjT(f) –T(f, . . . ,bjfj, . . . ,fN).

The iterated commutatorTbis defined by

Tb(f) = b, . . . , bm–, [bm,T]m

m–· · ·

(f).

To clarify the notations, ifT is associated in the usual way with a Calderón-Zygmund

kernelK, then at a formal level

Tb(f)(x) =

(Rn)m m

j=

bj(x) –bj(yj)

K(x,y, . . . ,ym)f(y)· · ·fm(ym)dy· · ·dym

and

Tb(f)(x) =

(Rn)m m

j=

bj(x) –bj(yj)

K(x,y, . . . ,ym)f(y)· · ·fm(ym)dy· · ·dym.

It was shown in [] that ifq=q

+· · ·+

qm, then anm-linear Calderón-Zygmund operator

T maps fromLq× · · · ×Lqm toLq, when  <q

j<∞for allj= , . . . ,m; and fromLq× · · · ×Lqm toLq,∞, when q

j<∞for allj= , . . . ,m, andmin≤j≤mqj= . The weighted

strong and weakLqboundedness ofTis also true for weights in the classAPwhich will be

introduced in next section (see Corollary . []). It was proved in [] thatTbis bounded

fromLq× · · · ×Lqm toLqfor all indices satisfyingq =

q +· · ·+

qm withq>  andqj> ,

j= , . . . ,m. The result was extended in [] to allq> /m. In fact, the authors obtained the

weightedLq-version bounds as follows, for allωA

P:

Tb(f)Lq(ν

ω)≤CbBMO

m m

j=

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As may be expected from the situation in the linear case,Tbis not bounded fromL× · · · ×LtoL,∞. However, a sharp weak-type estimate in a very general sense was obtained in [], that is, for allωA(,...,),

νω

x∈Rn:Tb(f)(x)>tm

C

m

j=

Rn

|

fj(x)|

t

ωj(x)dx

/m ,

where(t) =t( +log+t). Whenm= , the above endpoint estimate was obtained in [].

The same as forTb, the strong type bound and the endpoint estimate forTbwere also

established by Pérezet al.in [].

The weighted Morrey spacesLp,k(w) was introduced by Komori and Shirai [].

More-over, they showed that some classical integral operators and corresponding commuta-tors are bounded in weighted Morrey spaces. Some other authors have been interested in this space for sublinear operators, see [–]. In [], Ye proved two results similar to Pérez and Trujillo-González [] for the multilinear commutators of the normal Calderón-Zygmund operators on weighted Morrey spaces. Wang and Yi [] considered the multilin-ear Calderón-Zygmund operators on weighted Morrey spaces and obtained some results similar to weighted Lebesgue spaces.

We will prove the following strong type bound forTbon weighted Morrey spaces.

Theorem . Let T be an m-linear Calderón-Zygmund operator;ωAP∩(A∞)mwith

p=

p

+· · ·+ 

pm

and <pj<∞,j= , . . . ,m;andbBMOm.Then,for any <k< ,there exists a constant

C such that

Tb(f)Lp,k(ν

ω)≤C

m

j=

bjBMO

m

j= fj

Lpj,k(ωj).

The following endpoint estimate will also be proved.

Theorem . Let T be an m-linear Calderón-Zygmund operator;  <k< ,ωA(,...,)∩ (A∞)m,andbBMOm.Then,for anyλ> and cube Q,there exists a constant C such that

νω(Q)k

νω

xQ:Tb(f)(x)>λ

C

m

j=

fj/λL(m),k(ω j)

/m ,

where(m)= m

◦ · · · ◦,(t) =t( +log+t)andfL(m),k(ω)=(m)(|f|)L,k(ω).

Remark . Here we remark that the above estimate is also valid forTb.

2 Some definitions and results

In this section, we introduce some definitions and results used later.

Definition . (Ap weights) A weight ω is a nonnegative, locally integrable function

onRn. Let  <p<, a weight functionωis said to belong to the classA

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constantCsuch that for any cubeQ,

 |Q|

Q ω(x)dx

 |Q|

Q

ω(x)–pdx

p–C,

and to the classA, if there is a constantCsuch that for any cubeQ,

 |Q|

Q

ω(x)dxCinf

x∈Qω(x).

We denoteA∞=p>Ap.

Definition .(Multiple weights) Formexponentsp, . . . ,pm∈[,∞), we often writep for the number given byp=mj=pjand denote byP the vector (p, . . . ,pm). A multiple weightω= (ω, . . . ,ωm) is said to satisfy theAPcondition if for

νω=

m

j= ωp/pj,

we have

sup

Q

 |Q|

Q

νω(x)dx

/p m

j=

 |Q|

Q ωj(x)–p

jdx

/pj <∞,

when pj= , (|Q|

Qωj(x) –pj

dx)/pj is understood as (inf

(x))–. As remarked in [],

m

j=Apj is strictly contained inAP, moreover, in generalωAPdoes not implyωjLloc for anyj, but instead

ωAP

⎧ ⎨ ⎩

(νω)pAmp,

ω–p

j

jAmpj, j= , . . . ,m,

where the conditionω–p

j

jAmpj in the casepj=  is understood asω/mjA.

Definition .(Weighted Morrey spaces) Let  <p<∞,  <k< , andωbe a weight

function onRn. The weighted Morrey space is defined by

Lp,k(ω) =fLploc:fLp,k(ω)<∞

,

where

fLp,k(ω)=sup Q

ω(Q)k

Q

f(x)(x)

/p .

The weighted weak Morrey space is defined by

WLp,k(ω) =f measurable :fWLp,k(ω)<∞

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where

fWLp,k(ω)=sup Q

inf

λ> λ ω(Q)k/pω

xQ:|f|(x) >λ/p.

Definition .(Maximal function) For(t) =t( +log+t) and a cubeQinRnwe will

consider the averagef,Qof a functionf given by the Luxemburg norm

f,Q=inf

λ>  :  |Q|

Q

|

f(x)| λ

dx≤

,

and the corresponding maximal is naturally defined by

Mf(x) =sup

Qxf,Q,

and the multilinear maximal operatorM,Qis given by

M(f)(x) =sup

Qx m

j= fj,Q.

The following pointwise equivalence is very useful:

Mf(x)≈Mf(x),

whereMis the Hardy-Littlewood maximal function. We refer reader to [, ] and their

references for details.

We say that a weightωsatisfies the doubling condition, simply denotedω∈ , if there is

a constantC>  such thatω(Q)≤(Q) holds for any cubeQ. IfωApwith ≤p<∞, we know thatω(λQ)≤λnp[ω]Apω(Q) for allλ> ; thenω∈ .

Lemma .([]) Supposeω∈ ,then there exists a constant D> such that

ω(Q)≥(Q)

for any cube.

Lemma .([]) IfωjA∞,then for any cube Q,we have

Q m

j=

ωθjj(x)dx

m

j=

Qωj(x)dx [ωj]∞

θj

,

wheremj=θj= , ≤θj≤.

Lemma .([]) SupposeωA∞,thenbBMO(ω)≈ bBMO.Here

BMO(ω) =

b:bBMO(ω)=sup Q

ω(Q)

Q

b(x) –bQ,ωω(x)dx<∞

,

and bQ,ω=ω(Q)

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From the fact|bjQbQ| ≤CjbBMOand Lemma ., we deduce that|bjQ,ωbQ,ω| ≤

CjbBMO. The following lemma is the multilinear version of the Fefferman-Stein type

inequality.

Lemma .(Theorem . []) Assume thatωiis a weight in Afor all i= , . . . ,m,and

setν= (mi=ωi)/m.Then

m

j=

M(fj)

Lp,∞(ν) ≤

m

j=

fjL(Mω

j).

Lemma .(Proposition . []) Letp =p

+· · ·+

pm.If≤pj≤ ∞,j= , . . . ,m,then

M(f)Lp,∞(νω)≤ m

j=

fjLpj(Mω j).

Lemma .(Theorem . []) Let p> and letωbe a weight in A∞.Suppose thatbBMOm.Then there exist Cω(independent ofb)and Cω,bsuch that

Rn

Tb(f)(x)ω(x)dx

m

j=

bjBMO

RnM(

f)(x)(x)dx

and

sup

t>(m)(

t)

ωy∈Rn:Tb(f)(y)>tm

,bsup t>

(m)(

t)

ωy∈Rn:M(f)(y)>tm

for allf = (f, . . . ,fm)bounded with compact support.

Lemma .(Theorem . []) LetωA(,...,).Then there exists a constant C such that

νω

x∈Rn:MLlogL(f)(x)>tm

C

m

j=

Rn (m)

|

fj(x)|

t

ωj(x)dx

/m .

By the above two inequalities, Pérez and Trujillo-González obtained the following re-sults.

Lemma .(Theorem . []) Let T be an m-linear Calderón-Zygmund operator;ωAP

with

p=

p

+· · ·+ 

pm

and <pj<∞,j= , . . . ,m;andbBMOm.Then there exists a constant C such that

Tb(f)Lp(ν

ω)≤C

m

j=

bjBMO

m

j=

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Lemma .(Theorem . []) Let T be an m-linear Calderón-Zygmund operator;ω

A(,...,),andbBMOm.Then,for anyλ> and cube Q,there exists a constant C such that

νω

x∈Rn:Tb(f)(x)>λ

C m j= Rn (m) |

fj(x)|

t

ωj(x)dx

/m ,

where(t) =t( +log+t)and(m)= m

◦ · · · ◦.

3 Proofs of theorems

We only present the casem=  for simplicity, but, as the reader will immediately notice, a

complicated notation and a similar procedure can be followed to obtain the general case. Our arguments will be standard.

Proof of Theorem. For any cubeQ, we splitfjintofj+fj∞, wherefj=fjχQandfj∞=

fjfj,j= , . Then we only need to verify the following inequalities:

I=

νω(Q)k

Q

Tb

f,f(x)pνω(x)dx

/p ≤Cj= bjBMOj= fj

Lpj,k(ωj),

II=

νω(Q)k

Q

Tb

f,f∞(x)pνω(x)dx

/p ≤Cj= bjBMOj= fj

Lpj,k(ωj),

III=

νω(Q)k

Q

Tb

f∞,f(x)pνω(x)dx

/p ≤Cj= bjBMOj= fjLpj,k

(ωj),

IV=

νω(Q)k

Q

Tb

f∞,f∞(x)pνω(x)dx

/p ≤Cj= bjBMOj= fj

Lpj,k(ωj).

From Lemma . and Lemma ., we get

IC

νω(Q)k/pj= bjBMO Rn

fj(x)pj ωj(x)dx

/pj

C

νω(Q)k/p

j=

bjBMOωj(Q)k/pjfjLpj,k (ωj)

C

j=

bjBMOfj

Lpj,k(ωj)

.

SinceIIandIIIare symmetric we only estimateII. Takingλj= (bj)Q,ωj, the operatorTb

can be divided into four parts:

Tb

f,f∞(x) =b(x) –λ

b(x) –λ

Tf,f∞(x) –b(x) –λ

Tf, (b–λ)f∞

(x)

b(x) –λ

T(b–λ)f,f∞

(x) +T(b–λ)f, (b–λ)f∞

(x)

(8)

Using the size condition () ofK, Definition ., and Lemma ., we deduce that for any

xQ,

Tf,f∞(x)

C

Q

Rn\Q

|f(y)f(y)|

(|xy|+|xy|)ndydy

C

Q

f(y)dy ∞

l=  |lQ|

l+Q\lQ

f(y)dy

C

l=

j=  |l+Q|

l+Q

fj(yj)dyj

C

l=

j=  |l+Q|

l+Q

fj(yj)pjωj(yj)dyj

/pj

×

l+Q

ωj(yj)–p

jdy j

/pj

C

l=  |l+Q|

|l+Q|

p+p+

p

νω(l+Q)

j= fj

Lpj,k(ωj)ωj

l+Qk/pj

C

j= fjLpj,k

(ωj) ∞

l= νω

l+Q(k–)/p.

Taking the above estimate together with Hölder’s inequality and Lemma ., we have

νω(Q)k

Q

|II|pνω(x)dx

/p

≤ 

νω(Q)k/p

Q

b(x) –λ

b(x) –λp

νω(x)dx

/p

× 

j= fjLpj,k

l= νω

l+Q(k–)/p

νω(Q)/p

νω(Q)k/p

j=

νω(Q)

Q

bj(x) –λpνω(x)dx

/p

× 

j= fjLpj,k

l= νω

l+Q(k–)/p

≤ 

j=

bjBMOfj

Lpj,k(ωj),

where the last inequality is obtained by the property ofA∞: there is a constantδ>  such

that

νω(Q)

νω(l+Q)≤

C

|

Q|

|l+Q|

δ

(9)

ForII, from the size condition () ofK, theAPcondition, Lemma ., and Lemma ., it follows that

T

f, (b–λ)f∞

(x)

C

Q

f(y)dy ∞

l=  |lQ|

l+Q\lQ

b(y) –λ

f(y)dy

C

l=  |l+Q|

l+Q

f(y)pωj(y)dy

/p

l+Q

ω(y)–p

dyj

/p

×

l+Q

f(y)pω(y)dy

/p

×

l+Q

b(y) –λp

ω(y)–p

/pdy

/p

Cl= lj=  |l+Q|

l+Q

fj(yj) pj

ωj(yj)dyj

/pjl+Q

ωj(yj)–p

jdy j

/pj

C

j= fj

Lpj,k(ωj) ∞

l=

lνω

l+Q(k–)/p.

The third inequality can be deduced by the fact that

ω(j+Q)

l+Q

b(y) –bQ,ω

p ω(y)dy

/p

ClbBMO(ω).

Hölder’s inequality and Lemma . tell us

νω(Q)k

Q

|II|pνω(x)dx

/p

C

νω(Q)k/p

Q

b(x) –λp

νω(x)dx

/p

j= fjLpj,k

l=

lνω

l+Q(k–)/p

Cνω(Q)

/p

νω(Q)k/p

j= fj

Lpj,k

l=

lνω

l+Q(k–)/p

C

j=

bjBMOfj

Lpj,k(ωj).

Similarly, we get

Tf, (b–λ)f∞

(x)

C

l=  |l+Q|

l+Q

f(y) p

ωj(y)dy

/p

×

l+Q

b(y) –λp

ω(y)–p

dy

j

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× l+Q

f(y) p

ω(y)dy

/p

l+Q

ω(y)–p

/pdy

/p

C

j= fjLpj,k

(ωj) ∞

l=

lνω

l+Q(k–)/p,

and so

νω(Q)k

Q

|II|pνω(x)dx

/p

C

j=

bjBMOfjLpj,k (ωj).

The termIIis estimated in a similar way and we deduce

T(b–λ)f, (b–λ)f∞

(x)

C

l=  |l+Q|

j=

l+Q

fj(yj) pjω

j(yj)dyj

/pj

×

l+Q

bj(yj) –λj pj

ωj(yj)–p

j/pj

dyj

/pj

C

j= fjLpj,k

(ωj) ∞

l=

lνω

l+Q(k–)/p.

So,

νω(Q)k

Q

|II|pνω(x)dx

/p

C

j=

bjBMOfjLpj,k (ωj).

Finally, we still splitTb(f∞,f∞)(x) into four terms:

Tb

f∞,f∞(x) =b(x) –λ

b(x) –λ

Tf∞,f∞(x) –b(x) –λ

Tf∞, (b–λ)f∞

(x)

b(x) –λ

T(b–λ)f∞,f∞+T

(b–λ)f∞, (b–λ)f∞

(x)(x) =IV+IV+IV+IV.

Because each term ofIVjis completely analogous toIIj,j= , , ,  with a small difference,

we only estimateIV:

Tf∞,f∞(x)≤C

(Rn)\(Q)

|f(y)f(y)|

(|xy|+|xy|)ndydy

C

l=

(l+Q)\(lQ)

|f(y)f(y)|

(|xy|+|xy|)ndydy

C

l=  |l+Q|

(l+Q)

j=

fj(yj)dyj

C

j= fjLpj,k

(ωj) ∞

l= νω

(11)

Hence,

νω(Q)k

Q

|IV|pνω(x)dx

/p

C

j=

bjBMOfj

Lpj,k(ωj).

Combining all estimates, we complete the proof of Theorem ..

We now turn to the proof of Theorem ..

Proof of Theorem. By homogeneity, we may assume thatλ=bBMO=bBMO=  and we only need to prove that

νω

xQ:Tb(f,f)(x)> 

Cνω(Q)k

j=

fjL(),k(ω j)

/ .

To prove the above inequality, we can write

νω

xQ:Tb(f,f)(x)> 

νω

xQ:Tb

f,f(x)> /+νω

xQ:Tb

f,f∞(x)> / +νω

xQ:Tb

f∞,f(x)> /+νω

xQ:Tb

f∞,f∞(x)> /

=V+VI+VII+VIII

for any cubeQ. Employing Lemma . and Lemma ., we have

VC

j=

Rn

(m)fj(x)ωj(x)dx

/

C

j=

ωj(Q)kfjL(m),k(ω j)

/

Cνω(Q)k

j= fj

L(m),k(ω j)

/ .

From Lemma . and Lemma ., we deduce that

νω

xQ:Tb

f,f∞(x)> / ≤sup

t>(m)(

t) νω

xQ:Tb

f,f∞(x)>t

ω,bsup t>

(m)(

t) νω

yQ:Mf,f∞(y)>t

ω,bsup t>

(m)(

t) νω

yQ:M

f(y)M

f∞(y)>t

ω,b

t

Rn

f(y)M(χQω)(y)dy

Rn

f∞(y)M(χQω)(y)dy

/

ω,b

t ωj(Q)

kfjL

,k(ω j)

(12)

where the last inequality holds by the (.) and (.) in []. Then from Lemma . and the fact thatt(t) > , we have

VICνω(Q)ωj(Q)kfjL,k(ω j)

/ .

A similar statement follows:

VIICνω(Q)ωj(Q)kfjL,k(ω j)

/ ;

VIIICνω(Q)ωj(Q)kfjL,k(ω j)

/ .

Thus we complete the proof of Theorem ..

4 A problem

FixN∈N. LetmCL(RNn\{}), for some positive integerL, satisfying the following

con-dition:

∂αξ · · ·

αN

ξN m(ξ, . . . ,ξN)≤Cα,...,αN

|ξ|+· · ·+|ξN||α| ()

for all|α| ≤sandξ∈RNn\{}, whereα= (α

, . . . ,αN) andξ= (ξ, . . . ,ξN). The multilinear

Fourier multiplier operatorTN is defined by

Tm(f)(x) =  (π)Nn

(RNn)e

ix(ξ+···+ξN)m(ξ

, . . . ,ξN)fˆ(ξ)· · · ˆfN(ξN)· · ·dξN ()

for allf, . . . ,fNS(Rn), wheref = (f, . . . ,fN). IfF–mis an integrable function, then this can also be written as

Tm(f)(x) =

(RNn)F

–m(xy

, . . . ,xyN)f(y)· · ·f(yN)dy· · ·dyN.

In [], Fujita and Tomita obtained the following theorem.

Theorem . Let <p, . . . ,pN<∞,p+· · ·+pN =

p and

n

 <sjn for≤jN.Assume

pj>n/sj and wjApjsj/nfor≤jN.If mL∞(R

Nn)satisfies

mkW(s,...,sN)=

RNn

 +|ξ|/· · · +|ξN|

/ ˆ

m(ξ)

/ <∞,

then TN is bounded from Lp(ω)× · · · ×LpN(ωN)to Lp(νω),where

mj(ξ) =m

, . . . , jξN

(ξ, . . .ξN),

whereis the Schwarz function and satisfies

suppξ∈RNn: /≤ |ξ| ≤,

k∈Z

(13)

A natural problem is whether the Lebesgue spacesLpj(ω

j) andLp(νω) can be replaced by

Lpj,k(ω) andLp,k(ν

ω). It should be pointed out that the method in this paper may not be

suitable to address this problem.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the referee for some very valuable suggestions. This research was supported by NSF of China (no. 11161044, no. 11261055) and by XJUBSCX-2012004.

Received: 6 September 2013 Accepted: 18 February 2014 Published:04 Mar 2014

References

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3. Grafakos, L, Torres, R: On multilinear singular integrals of Calderón-Zygmund type. Publ. Mat.Extra, 57-91 (2002) 4. Duong, XT, Grafakos, L, Yan, L-X: Multilinear operators with non-smooth kernels and commutators of singular

integrals. Trans. Am. Math. Soc.362, 2089-2113 (2010)

5. Duong, XT, Gong, R, Grafakos, L, Li, J, Yan, L-X: Maximal operator for multilinear singular integrals with non-smooth kernels. Indiana Univ. Math. J.58, 2517-2541 (2009)

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10. Pérez, C, Pradolini, G, Torres, RH, Trujillo-González, RT: End-point estimates for iterated commutators for multilinear singular integrals. arXiv:1004.4976v2 [math.CA] (2011)

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12. Ding, Y, Yang, D-C, Zhou, Z: Boundedness of sublinear operators and commutators onLpω(Rn). Yokohama Math. J.46,

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Cite this article as:Wang and Jiang:Commutators for multilinear singular integrals on weighted Morrey spaces.

References

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