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× · · · ×SRn→SRn.
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, where q=
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 ()
for allx∈/mj=suppfjandf, . . . ,fm∈S(Rn), where
K(y,y, . . . ,ym)≤
A
(ml,k=|yl–yk|)mn
()
and
K(y, . . . ,yj, . . . ,ym) –K
y, . . . ,yj, . . . ,ym≤
A|yj–yj| (ml,k=|yl–yk|)mn+
()
for some> and all ≤j≤m, whenever|yj–yj| ≤max≤k≤m|yj–yk|.
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 satisfying q =
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 allω∈A
P:
Tb(f)Lq(ν
ω)≤CbBMO
m m
j=
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 allω∈A(,...,),
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;ω∈AP∩(A∞)mwith
p=
p
+· · ·+
pm
and <pj<∞,j= , . . . ,m;andb∈BMOm.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< ,ω∈A(,...,)∩ (A∞)m,andb∈BMOm.Then,for anyλ> and cube Q,there exists a constant C such that
νω(Q)k
νω
x∈Q: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
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)dx≤Cinf
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
xω(x))–. As remarked in [],
m
j=Apj is strictly contained inAP, moreover, in generalω∈APdoes not implyωj∈Lloc for anyj, but instead
ω∈AP ⇔
⎧ ⎨ ⎩
(νω)p∈Amp,
ω–p
j
j ∈Ampj, j= , . . . ,m,
where the conditionω–p
j
j ∈Ampj in the casepj= is understood asω/mj ∈A.
Definition .(Weighted Morrey spaces) Let <p<∞, <k< , andωbe a weight
function onRn. The weighted Morrey space is defined by
Lp,k(ω) =f ∈Lploc:fLp,k(ω)<∞
,
where
fLp,k(ω)=sup Q
ω(Q)k
Q
f(x)pω(x)
/p .
The weighted weak Morrey space is defined by
WLp,k(ω) =f measurable :fWLp,k(ω)<∞
where
fWLp,k(ω)=sup Q
inf
λ> λ ω(Q)k/pω
x∈Q:|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)≤Cω(Q) holds for any cubeQ. Ifω∈Apwith ≤p<∞, we know thatω(λQ)≤λnp[ω]Apω(Q) for allλ> ; thenω∈ .
Lemma .([]) Supposeω∈ ,then there exists a constant D> such that
ω(Q)≥Dω(Q)
for any cube.
Lemma .([]) Ifωj∈A∞,then for any cube Q,we have
Q m
j=
ωθjj(x)dx≥
m
j=
Qωj(x)dx [ωj]∞
θj
,
wheremj=θj= , ≤θj≤.
Lemma .([]) Supposeω∈A∞,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)
From the fact|bjQ–bQ| ≤CjbBMOand Lemma ., we deduce that|bjQ,ω–bQ,ω| ≤
CjbBMO. The following lemma is the multilinear version of the Fefferman-Stein type
inequality.
Lemma .(Theorem . []) Assume thatωiis a weight in A for 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 thatb∈ BMOm.Then there exist Cω(independent ofb)and Cω,bsuch that
Rn
Tb(f)(x)ω(x)dx≤Cω
m
j=
bjBMO
RnM(
f)(x)pω(x)dx
and
sup
t> (m)(
t)
ωy∈Rn:Tb(f)(y)>tm
≤Cω,bsup t>
(m)(
t)
ωy∈Rn:M(f)(y)>tm
for allf = (f, . . . ,fm)bounded with compact support.
Lemma .(Theorem . []) Letω∈A(,...,).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;ω∈AP
with
p=
p
+· · ·+
pm
and <pj<∞,j= , . . . ,m;andb∈BMOm.Then there exists a constant C such that
Tb(f)Lp(ν
ω)≤C
m
j=
bjBMO
m
j=
Lemma .(Theorem . []) Let T be an m-linear Calderón-Zygmund operator;ω∈
A(,...,),andb∈BMOm.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=fjχQandfj∞=
fj–fj,j= , . Then we only need to verify the following inequalities:
I=
νω(Q)k
Q
Tb
f,f(x)pνω(x)dx
/p ≤C j= bjBMO j= fj
Lpj,k(ωj),
II=
νω(Q)k
Q
Tb
f,f∞(x)pνω(x)dx
/p ≤C j= bjBMO j= fj
Lpj,k(ωj),
III=
νω(Q)k
Q
Tb
f∞,f(x)pνω(x)dx
/p ≤C j= bjBMO j= fjLpj,k
(ωj),
IV=
νω(Q)k
Q
Tb
f∞,f∞(x)pνω(x)dx
/p ≤C j= bjBMO j= fj
Lpj,k(ωj).
From Lemma . and Lemma ., we get
I≤C
νω(Q)k/p j= 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)
Using the size condition () ofK, Definition ., and Lemma ., we deduce that for any
x∈Q,
Tf,f∞(x)
≤C
Q
Rn\Q
|f(y)f(y)|
(|x–y|+|x–y|)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) –λ p
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) –λpνω(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
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) –λ p
ω(y)–p
/pdy
/p
≤C ∞ l= l j= |l+Q|
l+Q
fj(yj) pj
ωj(yj)dyj
/pj l+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) –λ p
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) –λ p
ω(y)–p
dy
j
× l+Q
f(y) p
ω(y)dy
/p
l+Q
ω(y)–p
/pdy
/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 termIIis 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)|
(|x–y|+|x–y|)ndydy
≤C
∞
l=
(l+Q)\(lQ)
|f(y)f(y)|
(|x–y|+|x–y|)ndydy
≤C
∞
l= |l+Q|
(l+Q)
j=
fj(yj)dyj
≤C
j= fjLpj,k
(ωj) ∞
l= νω
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=bBMO=bBMO= and we only need to prove that
νω
x∈Q:Tb(f,f)(x)>
≤Cνω(Q)k
j=
fjL(),k(ω j)
/ .
To prove the above inequality, we can write
νω
x∈Q:Tb(f,f)(x)>
≤νω
x∈Q:Tb
f,f(x)> /+νω
x∈Q:Tb
f,f∞(x)> / +νω
x∈Q:Tb
f∞,f(x)> /+νω
x∈Q:Tb
f∞,f∞(x)> /
=V+VI+VII+VIII
for any cubeQ. Employing Lemma . and Lemma ., we have
V≤C
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
x∈Q:Tb
f,f∞(x)> / ≤sup
t> (m)(
t) νω
x∈Q:Tb
f,f∞(x)>t
≤Cνω,bsup t>
(m)(
t) νω
y∈Q:Mf,f∞(y)>t
≤Cνω,bsup t>
(m)(
t) νω
y∈Q:M
f(y)M
f∞(y)>t
≤Cνω,b
t
Rn
f(y)M(χQω)(y)dy
Rn
f∞(y)M(χQω)(y)dy
/
≤Cνω,b
t ωj(Q)
kfjL
,k(ω j)
where the last inequality holds by the (.) and (.) in []. Then from Lemma . and the fact thatt(t) > , we have
VI≤Cνω(Q)ωj(Q)kfjL,k(ω j)
/ .
A similar statement follows:
VII≤Cνω(Q)ωj(Q)kfjL,k(ω j)
/ ;
VIII≤Cνω(Q)ωj(Q)kfjL,k(ω j)
/ .
Thus we complete the proof of Theorem ..
4 A problem
FixN∈N. Letm∈CL(RNn\{}), for some positive integerL, satisfying the following
con-dition:
∂α ξ · · ·∂
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)dξ· · ·dξN ()
for allf, . . . ,fN∈S(Rn), wheref = (f, . . . ,fN). IfF–mis an integrable function, then this can also be written as
Tm(f)(x) =
(RNn)F
–m(x–y
, . . . ,x–yN)f(y)· · ·f(yN)dy· · ·dyN.
In [], Fujita and Tomita obtained the following theorem.
Theorem . Let <p, . . . ,pN<∞,p+· · ·+p N =
p and
n
<sj≤n for≤j≤N.Assume
pj>n/sj and wj∈Apjsj/nfor≤j≤N.If m∈L∞(R
Nn)satisfies
mkW(s,...,sN)=
RNn
+|ξ|/· · · +|ξN|
/ ˆ
m(ξ)dξ
/ <∞,
then TN is bounded from Lp(ω)× · · · ×LpN(ωN)to Lp(νω),where
mj(ξ) =m
jξ, . . . , jξN
(ξ, . . .ξN),
whereis the Schwarz function and satisfies
supp⊂ξ∈RNn: /≤ |ξ| ≤,
k∈Z
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
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Cite this article as:Wang and Jiang:Commutators for multilinear singular integrals on weighted Morrey spaces.