R E S E A R C H
Open Access
Weighted estimates for vector-valued
multilinear operators with non-smooth
kernels
Zengyan Si
**Correspondence:
[email protected] School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan 454000, China
Abstract
LetTbe the multilinear Calderón-Zygmund operator with non-smooth kernel and let
T∗be its corresponding maximal operator. In this paper, vector-valued weighted norm inequalities forTandT∗are established. As applications, weighted strong type estimates for vector-valued commutators associated withTandT∗are deduced respectively.
1 Introduction and main results
LetTbe a multilinear operator initially defined on them-fold product of Schwartz spaces and taking values in the space of tempered distributions,
T:SRn× · · · ×SRn→SRn.
Following [], we say thatTis a multilinear Calderón-Zygmund operator if, for some ≤
qj<∞, it extends to a bounded multilinear operator fromLq× · · · ×Lqm toLq, where
q=
q+· · ·+
qm, and if there exists a functionK, defined off the diagonalx=y=· · ·=ym
in (Rn)m+, satisfying
T(f)(x) =T(f, . . . ,fm)(x) =
(Rn)mK(x,y, . . . ,ym)f(y)· · ·fm(ym)dy· · ·dym, (.)
for allx∈/mj=suppfj;
K(y,y, . . . ,ym)≤
A
(mk,l=|yk–yl|)mn
; (.)
and
K(y,y, . . . ,yj, . . . ,ym) –K
y,y, . . . ,yj, . . . ,ym≤
A|yj–yj|
(mk,l=|yk–yl|)mn+
(.)
for some> and all ≤j≤m, whenever |yj–yj| ≤max≤k≤m|yj–yk|. Such kernels
are calledm-linear Calderón-Zygmund kernels, and the collection of such functions is
denoted bym-CZK(A,) in []. As in [], we define the maximal multilinear operator by
T∗(f)(x) =sup δ>
Tδ(f, . . . ,fm)(x),
whereTδis the smooth truncation ofT given by
Tδ(f)(x) =
|x–y|+···+|x–y
m|>δ
K(x,y, . . . ,ym)f(y)· · ·fm(ym)dy· · ·dym.
The vector-valued multilinear Calderón-Zygmund operatorTqand vector-valued
max-imal multilinear operatorTq∗associated withT are defined and studied in [, ].
Tq(f)(x) =T(f, . . . ,fm)(x)q=T(f·, . . . ,fm·)(x)lq
= ∞
k=
T(fk, . . . ,fmk)(x) q
/q
,
Tq∗(f)(x) =T∗(f, . . . ,fm)(x)q=T∗(f·, . . . ,fm·)(x)lq
= ∞
k=
T∗(fk, . . . ,fmk)(x) q
/q
,
wherefi={fik}∞k=fori= , . . . ,m.
Theorem A [] Assume that T is a multilinear Calderón-Zygmund operator. Let <
p, . . . ,pm<∞, <q, . . . ,qm<∞and/m<p,q<∞such that p = p +· · ·+pm, q =
q+· · ·+
qm.If(ω
p , . . . ,ω
pm
m )∈(Ap, . . . ,Apm),then there exists a constant C> such that
Tq(f)Lp(ωp
···ω
p m)≤C
m
j=
|fj|qj
Lpj(ωjpj).
Theorem B [] Assume that T is a multilinear Calderón-Zygmund operator. Let ≤
p, . . . ,pm <∞, <q, . . . ,qm <∞ and <p,q<∞ such that p = p +· · ·+ pm, q =
q+· · ·+
qm.
(i) If <p, . . . ,pm<∞andω∈Ap∩ · · · ∩Apm,then there exists a constantC> such
that
Tq(f)Lp(ω)≤C
m
j=
|fj|qj
Lpj(ω).
(ii) If at least onepj= andω∈A,then there exists a constantC> such that
Tq(f)Lp,∞(ω)≤C
m
j=
|fj|qjLpj(ω).
We will replace (.) by a weaker regularity condition on the kernel K. Assume that operatorsAtare associated with kernelsat(x,y) in the sense that
Atf(x) =
Rnat(x,y)f(y)dy
for every functionf∈Lp(Rn), ≤p≤ ∞, anda
t(x,y) satisfy the following size condition:
at(x,y)≤ht(x,y) =t–n/sh
|x–y|s
t
, (.)
wheresis a positive fixed constant andhis a positive, bounded, decreasing function sat-isfying that for someη> ,
lim
r→∞r
n+ηhrs= . (.)
Thejth transposeT∗,jofTis defined via
T∗,j(f, . . . ,fm),g
=T(f, . . . ,fj–,g,fj+, . . . ,fm),fj
for allf, . . . ,fm,ginS(Rn). It is easy to check that the kernelK∗,jofT∗,jis related to the
kernelKofT via the identity
K∗,j(x,y, . . . ,yj–,yj,yj+, . . . ,ym) =K(yj,y, . . . ,yj–,x,yj+, . . . ,ym).
To maintain uniform notation, we may occasionally denoteT=T∗,andK=K∗,.
Assumption (H) We always assume that there exist some ≤q, . . . ,qm<∞and some
<q<∞with
q=
q+· · ·+
qm such that bothT
∗andTmapLq× · · · ×LqmtoLq,∞.
Assumption (H) Assume that there exist operators{A(ti)}t>with kernelsa(ti)(x,y) that
satisfy the conditions (.) and (.) with constantssandηfor eachi= , . . . ,mand that for everyj= , , , . . . ,m, there exists kernelKt∗,j,(i)(x,y, . . . ,ym) such that
T∗,jf
, . . . ,A(ti)fi, . . . ,fm
,g
=
Rn
(Rn)m
Kt∗,j,(i)(x,y, . . . ,ym)f(y)· · ·fm(ym)g(x)dy· · ·dymdx,
for allf, . . . ,fminS(Rn) with
m
k=suppfk∩suppg=φ. Also assume that there exist a
non-negative functionφ∈C(R) withsuppφ∈[–, ] and a constant> so that for every
j∈ {, , . . . ,m}and everyi∈ {, , . . . ,m}, allt> and allx,y, . . . ,yn∈Rn, we have
K∗,j(x,y, . . . ,ym) –Kt∗,j,(i)(x,y, . . . ,ym)
≤ A
(|x–y|+· · ·+|x–ym|)mn m
k=,k=i
φ
|yi–yk|
t/s
+ At
/s
(|x–y|+· · ·+|x–ym|)mn+
,
KernelsKthat satisfy the size estimate (.) and Assumption (H) with parametersm,
A,s,η,εare called generalized Calderón-Zygmund kernels, and their collection is denoted bym-GCZK(A,s,η,ε). We say thatTis of classm-GCZO(A,s,η,ε) ifT has an associated kernelKinm-GCZK(A,s,η,ε).
Assumption (H) Assume that there exist operators{At}t>with kernelsat(x,y) that
sat-isfy conditions (.) and (.) with constants sandη, and there exist kernelsKt()(x,y, . . . ,ym) such that the representation is valid
Kt()(x,y, . . . ,ym) =
RnK(z,y, . . . ,ym)at(x,z)dz
and that there exist a non-negative functionφ∈C(R) andsuppφ⊂[–, ] and a positive constantεsuch that
K(x,y, . . . ,ym) –Kt()(x,y, . . . ,ym)
≤ A
(mk=|x–yk|)mn m
k=
k=j
φ
|x
–yk|
t/s
+ At
ε/s
(mk=|x–yk|)mn+ε
(.)
for some A> , whenever t/s ≤ max
≤j≤m|x –yj|. Moreover, assume that for all
x,y, . . . ,ym∈Rn,
Kt()(x,y, . . . ,ym)≤
A
(mk=|x–yk|)mn
,
whenever t/s≤min
≤j≤m|x–yj|, and for allx,x,y, . . . ,ym∈Rn,
K(x,y, . . . ,ym) –Kt()
x,y, . . . ,ym≤
Atε/s
(mk=|x–yk|)mn+ε
,
whenever t/s≤min
≤j≤m|x–yj|and |x–x| ≤t/s.
The commutators associated withTandT∗are defined respectively by
Tb(f)(x) =
b,
b, . . .
bl–, [bl,T]l
l–· · ·
(f)(x)
=
(Rn)m
l
j=
bj(x) –bj(yj)
K(x,y, . . . ,ym) m
i=
fi(yi)dy,
and
T∗b(f)(x) =sup δ>
b,
b, . . .
bl–, [bl,Tδ]l
l–· · ·
(f)(x)
=sup δ>
|x–y|+···+|x–y
m|>δ
l
j=
bj(x) –bj(yj)
K(x,y, . . . ,ym) m
i=
fi(yi)dy
.
Here and subsequently, we often writey= (y, . . . ,ym) anddy=dy· · ·dym.
vector-valuedTqandTq∗can be defined by
Tb,q(f)(x) =Tb(f)(x)q=Tb(f·, . . . ,fm·)(x)lq=
∞
k=
Tb(fk)(x) q
/q
,
T∗
b,q(f)(x) =T
∗
b(f)(x)q=T
∗
b(f·, . . . ,fm·)(x)lq=
∞
k=
T∗
b(fk)(x) q
/q
.
From now on, we always assume thatT is a multilinear operator inm-GCZO(A,s,η,ε) and its kernel satisfies Assumption (H). Recently, ifl=m, Penget al.[] obtained the following weighted strong type estimates forTbandT∗bwith multiple weights (see
Def-inition .).
Theorem C[] Letb∈BMOm, p =p +· · ·+
pm with <pj<∞,j= , . . . ,m.Then we have
(i) There exists a constantCsuch that
T∗
b(f)Lp(ν ω)≤C
m
i=
biBMO m
i=
fiLpi(Mωi).
(ii) If eachωi∈Apj,then there exists a constantCsuch that
T∗
b(f)Lp(ν ω)≤C
m
i=
biBMO m
i=
fiLpi(ωi),
whereνω=
m
i=ω
p/pj
i .Similar results still hold forTb,which extend the results in
[]significantly.
In this work, we first pursue results parallel to Theorems A and B, then extend Theo-rem C to a vector-valued version. The main results can be stated as follows.
Theorem . Let <p, . . . ,pm<∞, <q, . . . ,qm<∞and/m<p,q<∞such that p=
p+· · ·+
pm,
q=
q+· · ·+
qm.If(ω
p , . . . ,ω
pm
m )∈(Ap, . . . ,Apm),then there exists a constant C> such that
Tq(f)Lp(ωp
···ω
p m)≤C
m
j=
|fj|qj
Lpj(ωjpj).
Moreover,similar estimates hold for T∗.
Theorem . Let≤p, . . . ,pm<∞, <q, . . . ,qm<∞and <p,q<∞such that p =
p+· · ·+
pm,
q=
q +· · ·+
qm.
(i) If <p, . . . ,pm<∞andω∈Ap∩ · · · ∩Apm,then there exists a constantC> such
that
Tq(f)Lp(ω)≤C
m
j=
(ii) If at least onepj= andω∈A,then there exists a constantC> such that
Tq(f)Lp,∞(ω)≤C
m
j=
|fj|qjLpj(ω).
Moreover,similar estimates hold for T∗.
Theorem . Let/m<p<∞, p = p +· · ·+
pm with <p, . . . ,pm<∞, /m<q<∞, and q
+· · ·+
qm =
q with <q, . . . ,qm <∞.Suppose that ω∈Ap,νω=
m
i=ω
p pi
i and
b∈(BMO)l.Then we have
(i) There then exists a constantC> such that
Tb,q(f)Lp(ν ω)≤
l
j=
bjBMO m
j=
|fj|qj
Lpj(Mwj).
(ii) Ifωj∈Apj,then there exists a constantC> such that
Tb,q(f)Lp(ν ω)≤
l
j=
bjBMO m
j=
|fj|qj
Lpj(wj).
Moreover,similar estimates hold for T∗.
Remark . Ifl= andl=m, Theorem . can be seen as the vector-valued extension of Theorem . in [] and Theorem C, respectively.
2 Proofs of Theorem 1.1 and Theorem 1.2
Let us begin with the definition of Hardy-Littlewood maximal operator, that is
Mf(x) =sup
Qx
|Q|
Q
f(y)dy.
The sharp maximal function is defined by
Mf(x) =sup
Qx
inf
c
|Q|
Q
f(y) –cdy≈sup
Qx
|Q|
Q
f(y) –fQdy.
Forδ> , we also need the maximal functionMδf =M(|f|δ)
δ andMδf=M(|f|δ)
δ.
The new maximal functionMcan be defined by
M(f)(x) =sup
Qx m
j=
|Q|
Q
fj(yj)dyj.
For ≤l≤m, as in [], a modified maximal functionMlis given by
Ml(f)(x) =sup Qx
∞
k= –knl
l
j=
|Q|
Q
fj(yj)dyj m
j=l+
|kQ|
kQ
fj(yj)dyj
For exponentsp, . . . ,pm, we will often writepfor the number given by /p= /p+· · ·+ /pm, andpfor the vectorp= (p, . . . ,pm). Let us recall the definition ofApweights.
Definition . Let ≤p, . . . ,pm<∞. Givenω= (ω, . . . ,ωm), setνω=
m
i=ω
p/pi
i . We say
thatωsatisfies theApcondition if
sup
Q
|Q|
Q m
i=
ω p pi
i
pm i=
|Q|
Q
ω–p i
i
p i
<∞,
whenpi= , (|Q|
Qω
–pi i )
pi
is understood as (infQωi)–.
We will use the following lemmas in the proof of Theorem . and Theorem ..
Lemma .[] LetT be an m-linear operator,and let <q, . . . ,qm<∞and m <q<∞
be fixed indices such thatq =q +· · ·+
qm.For(ω
q , . . . ,ω
qm
m )∈(Aq, . . . ,Aqm),the following estimate holds:
T(f)Lq(ωq
···ω
q m)≤C
m
j=
fjLqj(ωqj j )
.
Then, for all indices, < p, . . . ,pm < ∞ and m < p< ∞ satisfy p = p +· · ·+ pm,
<s, . . . ,sm <∞ and m <s<∞ such that s = s +· · ·+ sm, and all (ωp, . . . ,ω
pm
m )∈
(Ap, . . . ,Apm).Then the following inequality holds:
k
T(fk, . . . ,fmk)s
s
Lp(ωp···ωpm) ≤C
m
j=
k
|fjk|sj
sj
Lpj(ωpjj ) .
Lemma .[] Suppose that for some≤q,q, . . . ,qm–≤ ∞,qm∈(,∞)and q∈(,∞)
satisfying q = q +· · ·+
qm, T maps L
q× · · · ×Lqm to Lq.Let ≤p
, . . . ,pm<∞, p =
p+· · ·+
pm,ω= (ω, . . . ,ωm)∈Apandp= (p, . . . ,pm).Then
(i) T∗can be extended to a bounded operator fromLp(ω
)× · · · ×Lpm(ωm)toLp(νω) if all the exponentspjare strictly greater than.
(ii) T∗can be extended to a bounded operator fromLp(ω
)× · · · ×Lpm(ωm)toLp,∞(νω)
if some exponentspjequal.
Similar results hold for T.
Note that if eachωj∈Apj, then
m
j=Apj ⊂Ap and this inclusion is strict (see [] for
details). This fact together with Lemma . yields the following weighted estimates.
Lemma . Consider an m-tuple(ωp , . . . ,ω
pm
m )∈(Ap, . . . ,Apm),where <p, . . . ,pm<∞
andm <p<∞satisfy p=p +· · ·+
pm.Then there exists a constant C such that
T(f)Lp(ωp
···ω
p m)≤C
m
j=
fjLpj(ωpj j )
and
T∗(f)Lp(ωp
···ω
p m)≤C
m
j=
fjLpj(ωpj j )
.
Proof of Theorem.and Theorem. As a consequence of Lemma . and Lemma ., we obtain Theorem . (see the proof of Corollary in []). From [] we know that for all ≤l≤m,f = (f, . . . ,fm) andx∈Rn, the following two inequalities hold:
M(f)(x)≤Ml(f)(x)≤ m
j=
M(fj)(x),
TfLp(ν ω)≤C
m
l= Ml(f)
Lp(ν ω)
.
So, we getTfLp(ν ω)≤C
m
j=M(fj)Lp(ν
ω). A similar inequality still holds forT
∗.
The-orem . follows by repeating the same steps as in Corollary . in []. In fact, we apply Theorem . in [] to the families
F T(f, . . . ,fm), m
j=
Mfj
, F T∗(f, . . . ,fm), m
j=
Mfj
.
Hölder’s inequality and the normal inequalities for the maximal operator yield the desired
results.
3 Proof of Theorem 1.3
We begin with some lemmas which will be used in the proof of Theorem ..
Lemma .[] Let <p,δ<∞and letω be a weight in A∞.Then there exists C > (depending upon the A∞condition ofω)such that
Rn
Mδf(x) p
ω(x)dx≤C
Rn
Mδf(x)pω(x)dx (.)
for every function such that the left-hand side is finite.
Lemma . Let <δ< /m, /m<q<∞and/q= /q+· · ·+/qmwith <q, . . . ,qm<∞.
Then there exists a constant C> such that
Mδ
Tq(f)
(x)≤C
m
j=
M|fj|qj
(x)
for any smooth vector function{fk}∞k=for any x∈Rn.
Proof Fix a pointx∈Rnand a cubeQcentered atx. Setf
j=fj+fj∞, wherefj=fjχQ∗. Let
fα=fα
· · ·fmαm andQ∗= (
√
n+ )Q. It is easy to see
Tq(f)(z) –C≤Tq
f(z)+ α,...,αm
whereC=α,...,αm|T(f
α
· · ·fmαm)(x)|q and in the last sum eachαj= or∞and in each
term there is at least oneαj=∞. Since <δ< /m< , it follows that
|Q|
Q
Tq(f)(z)δ–|C|δ
dz
/δ
≤C
|Q|
Q
T(f)(z) –cδqdz
/δ
≤C
|Q|
Q
Tf(z)δqdz
/δ
+C
α,...,αm
|Q|
Q
Tfα(z) –cδqdz
/δ
P+P,
whereC=|c|q= (
k≥|ck|q)/q.
Applying Kolmogorov’s inequality and Theorem . toP, we have
|Q|
Q
Tq
f(z)δdz
/δ
≤CTq
fL/m,∞(Q,dz |Q|)
≤C
m
j=
|Q|
Q
fj(z)qjdz
≤C
m
j=
M|fj|qj
(x).
We proceed to the estimate forP. We can taket= [√nl(Q)]s. Ifα=· · ·=αm=∞, we
have
|Q|
Q
T(f)(z) –cδqdz
/δ
≤|Q|C
Q
Tf∞(z) –cqdz
≤|Q|C
Q
∞
k=
Tfk∞(z) –Tfk∞(x)q /q
dz
≤|Q|C
Q
∞
k=
(Rn\Q∗)m
K(z,y) –K(x,y)|fk· · ·fmk|dy
q
/q
dz
≤ C |Q|
Q
∞
k=
(Rn\Q∗)m
K(z,y) –Kt()(z,y)|fk· · ·fmk|dy
q
/q
dz
+ C
|Q|
Q
∞
k=
(Rn\Q∗)m
Kt()(z,y) –K(x,y)|fk· · ·fmk|dy
q
/q
dz
Sincez∈Qandyj∈Rn\(√n+ )Q, we get|yj–z|> (√n+ )l(Q) > t/sfor allj=
, . . . ,m. Applying Assumption (H), we obtain
P≤
C |Q|
Q
(Rn\Q∗)m
Atε/s
(|z–y|+· · ·+|z–ym|)mn+ε|
f|q· · · |fm|qmdy dz
≤
∞
k= kε
m
j= (k+)n|Q∗|
k+Q∗|fj|qjdyj ≤C
m
j=
M|fj|qj
(x).
Sincex,z∈Q,|z–x| ≤√nl(Q)≤t/s. Note that|y
j–z|> (
√
n+ )l(Q) > t/s, for all
j= , . . . ,m, henceφ(|yj–z|
t/s ) = . Similarly, we getP≤
m
j=M(|fj|qj)(x).
Now we estimate the typical representative ofP, that is,α=· · ·=αl=∞andαl+=
· · ·=αm= .
Tf∞, . . . ,fl∞,fl+, . . . ,fm(z) –Tf∞, . . . ,fl∞,fl+, . . . ,fm(x)q
≤
(Rn)m
K(z,y) –Kt()(z,y)+Kt()(z,y) –K(x,y)|f|q· · · |fm|qmdy
=
(Rn)m
K(z,y) –Kt()(z,y)|f|q· · · |fm|qmdy
+
(Rn)m
K()
t (z,y) –K(x,y)|f|q· · · |fm|qmdy.
Thus, we get
|Q|
Q
T
f∞, . . . ,fl∞,fl+, . . . ,fm(z) –cδqdz
/δ
≤ C |Q|
Q
Tf∞, . . . ,fl∞,fl+, . . . ,fm(z) –Tf∞, . . . ,fl∞,fl+, . . . ,fm(x)qdz
≤|Q|C
Q
(Rn)m
K(z,y) –Kt()(z,y)|f|q· · · |fm|qmdy dz
+ C
|Q|
Q
(Rn)m
Kt()(z,y) –K(x,y)|f|q· · · |fm|qmdy dz
=P+P.
ForP, by Assumption (H), we have
P≤
C |Q|
Q
(Rn\Q∗)l
tε/sl
j=|fj(yj)|qjdyj
(j∈{,,...,l}|z–yj|)mn+ε
+
(Rn\Q∗)l
l
j=|fj(yj)|qjdyj
(j∈{,,...,l}|z–yj|)mn
m
j=l+
Q∗
fj(yj)q
jdyjdz
≤
∞
k=
|Q∗|ε/n
(k|Q∗|/n)mn+ε
(Rn\Q∗)l
l
j=
fj(yj)q
+ ∞
k= (k|Q∗|/n)mn
(k+Q∗\kQ∗)l
l
j=
fj(yj)q
jdyj
m
j=l+
Q∗
fj(yj)q
jdyj ≤C
m
j=
M|fj|qj
(x).
By a similar argument, we deduce thatP≤C
m
j=M(|fj|qj)(x). In other case, we can
also deduce the same estimate with minor modifications on the above arguments. We
have thus proved Lemma ..
Lemma . Let <δ <ε< /m, /m <q< ∞ and /q= /q +· · ·+ /qm with <
q, . . . ,qm<∞. Suppose thatb∈(BMO)l.There then exists a constant C> depending
only onδandεsuch that
Mδ(Tb,qf)(x)≤C l
j=
bjBMO m
j=
ML(logL)
|fj|qj
(x) +Mε(Tqf)(x)
+C
l–
j=
σ∈Cjl
i∈σ
biBMOMε(Tbσ,qf)(x)
for any smooth vector function{fk}∞k=for any x∈Rn,whereσ={, . . . ,l} \σ.
Proof For simplicity of notation, we write F(y) instead of the product ofm functions
f(y)· · ·fm(ym) and letλj=|Q|
Qbj(z)dz, forj= , . . . ,l. Letx∈RnandQbe a cube
cen-tered atx. Then we have
Tb(f)(x) =
(Rn)m
b(x) –b(y)
· · ·bl(x) –bl(yl)
K(x,y)F(y)dy
=
(Rn)m
b(x) –λ
–b(y) –λ
· · ·bl(x) –λl
–bl(y) –λl
×K(x,y)F(y)dy
=
l
i=
σ∈Cl i
(–)l–j
j∈σ
bj(x) –λj
(Rn)m
j∈σ
bj(yj) –λj
K(x,y)F(y)dy
=b(x) –λ
· · ·bl(x) –λl
T(f)(x) +Tb(·) –λ
· · ·bl(·l) –λl
f(x)
+
l–
i=
σ∈Cli
(–)l–j
j∈σ
bj(x) –λj
(Rn)m
j∈σ
bj(yj) –λj
K(x,y)F(y)dy.
Noting thatj∈σ(bj(yj) –λj) =
j∈σ[(bj(yj) –bj(x)) – (bj(x) –λj)]. Then we get
Tb,qf(x)≤b(x) –λ
· · ·bl(x) –λlTq(f)(x)
+Tb(·) –λ
· · ·bl(·l) –λl
f(x)q
+C
l–
i=
σ∈Cli
j∈σ
Since <δ< /m< , it follows that
|Q|
Q
Tb,q(f)(z)
δ
–|C|δdz
/δ
≤C
|Q|
Q
Tb(f)(z) –c
δ
qdz
/δ
≤C
|Q|
Q
b(z) –λ
· · ·bl(z) –λlT(f)(z)
δ
qdz
/δ
+C
l–
i=
σ∈Cli
|Q|
Q
j∈σ
bj(z) –λjTbσ,qf(z)
δ
dz
/δ
+C
|Q|
Q
Tb(·) –λ
· · ·bl(·l) –λl
f(z) –cδqdz
/δ
I+II+III,
whereC=|c|q= (
k≥|ck|q)/q. We can choose <p, . . . ,pl<∞with p+· · ·+p
l+
ε=
δ. Since <δ<ε< /m, Hölder’s inequality gives
I≤C
l
j=
bjBMOMε(Tqf)(x),
II≤C
l–
i=
σ∈Cli
j∈σ
bjBMOMε(Tbσ,qf)(x).
Let us estimate termIII. Setfj=fj+fj∞, wherefj=fjχQ∗. Letfα=fα· · ·fmαm andQ∗=
(√n+ )Q. TakingC=α,...,αm|T((b(·) –λ)· · ·(bl(·l) –λl)f
α
· · ·fmαm)(x)|q, we have
Tb,q(f)(z) –C
≤Tq
b(·) –λ
· · ·bl(·l) –λl
f(z)
+C
α,...,αm
Tb(·) –λ
· · ·bl(·l) –λl
fα(z)
–Tb(·) –λ
· · ·bl(·l) –λl
fα(x)q,
where in the last sum eachαj= or∞and in each term there is at least oneαj=∞.
Ifα=· · ·=αm= , applying Kolmogorov’s inequality and Theorem ., we get
|Q|
Q
Tq
b(·) –λ
· · ·bl(·l) –λl
f(z)δdz
/δ
≤CTq
b(·) –λ
· · ·bl(·l) –λl
fL/m,∞(Q,dz |Q|)
≤C
l
j=
|Q|
Q
bj(yj) –λjfj(z)qjdz m
j=l+
|Q|
Q
≤C
l
j=
bjBMO|fj|qjL(logL),Q m
j=l+
|Q|
Q
fj(z)q
jdz
≤C
l
j=
bjBMO m
j=
ML(logL)
|fj|qj
(x).
Ifα=· · ·=αm=∞, we have
|Q|
Q
Tb
f∞(z) –cδqdz
/δ
≤ C |Q|
Q
Tb
f∞(z) –cqdz
≤ C |Q|
Q
∞
k=
Tb
fk∞(z) –Tb
fk∞(x)q /q
dz
≤|Q|C
Q
∞
k=
(Rn\Q∗)m
K(z,y) –K(x,y)
×b(y) –λ
· · ·bl(yl) –λl
fk· · ·fmkdy
q
/q
dz
≤|Q|C
Q
∞
k=
(Rn\Q∗)m
K(z,y) –Kt()(z,y)
×b(y) –λ
· · ·bl(yl) –λl
fk· · ·fmkdy
q
/q
dz
+ C
|Q|
Q
∞
k=
(Rn\Q∗)m
Kt()(z,y) –K(x,y)
×b(y) –λ
· · ·bl(yl) –λl
fk· · ·fmkdy
q
/q
dz
=III+III.
Consider now the termIII. Takingt= [√nl(Q)]s, we have by Assumption (H)
III≤
C |Q|
Q
(Rn\Q∗)m
∞
k=
|Q∗|ε/n
(k|Q∗|/n)mn+ε
×b(y) –λ
· · ·bl(yl) –λl|f|q· · · |fm|qmdy dz
≤C
l
j= ∞
k= kε
(k+)n|Q∗|
×
k+Q∗bj(yj) –λj|fj|qjdyj
m
j=l+ (k+)n|Q∗|
k+Q∗|fj|qjdyj ≤C
l
j= ∞
k=
k
kεbjBMO|fj|qjL(logL),k+Q∗
m
j=l+ (k+)n|Q∗|
k+Q∗|
≤C
l
j=
bjBMO|fj|qjL(logL),k+Q∗
m
j=l+ (k+)n|Q∗|
k+Q∗|fj|qjdyj ≤C
l
j=
bjBMO m
j=
ML(logL)
|fj|qj
(x).
Similarly, we getIII≤
l
j=bjBMO
m
j=ML(logL)(|fj|qj)(x). Now we consider only the
typ-ical representative ofIII. Similar to the estimates ofPin Lemma ., we have
|Q|
Q
Tb(·) –λ
· · ·bl(·l) –λl
f∞, . . . ,fl∞,fl+, . . . ,fm(z) –cδqdz
/δ
≤|Q|C
Q
T
b(·) –λ
· · ·bl(·l) –λl
f∞, . . . ,fl∞,fl+, . . . ,fm(z) –Tb(·) –λ
· · ·bl(·l) –λl
f∞, . . . ,fl∞,fl+, . . . ,fm(x)qdz ≤
l
j=
(Rn\Q∗)l tε/s|(b
(y) –λ)· · ·(bl(yl) –λl)|
l
j=|fj(yj)|qjdyj
(j∈{,,...,l}|z–yj|)mn+ε
+
(Rn\Q∗)l
|(b(y) –λ)· · ·(bl(yl) –λl)|
l
j=|fj(yj)|qjdyj
(j∈{,,...,l}|z–yj|)mn
m
j=l+
Q∗
fj(yj)q
jdyj
≤C
l
j=
bjBMO|fj|qjL(logL),k+Q∗
m
j=l+
|Q∗|
Q∗
fj(z)q
jdz
≤C
l
j=
bjBMO m
j=
ML(logL)
|fj|qj
(x).
Then Lemma . is proved.
Lemma . Let <p<∞, /m<q<∞,andq
+· · ·+
qm =
qwith <q, . . . ,qm<∞and
let w∈A∞.Suppose thatb∈(BMO)l.Then there exists a constant C> such that
Rn|Tb,q
f|pw(x)dx≤C
l
j=
bjpBMO
Rn
m
j=
ML(logL)
|fj|qj
(x)
p
w(x)dx (.)
for any smooth functionf with compact support.
Proof We assume that the right-hand side of (.) is finite, since otherwise there is nothing to be proved. Forl= , by using Lemma . and Lemma ., we obtain
Tb,qfLp(ω)≤ MδT
b,qfLp(ω)≤CMδT
b,qfLp(ω) ≤CbBMO
Mε(Tqf)Lp(ω)+
m
j=
ML(logL)
|fj|qj
Lp(ω)
≤CbBMO
TqfLp(ω)+
m
j=
ML(logL)
|fj|qj
Lp(ω)
≤CbBMO
m
j=
M|fj|qj
Lp(ω)
+
m
j=
ML(logL)
|fj|qj
Lp(ω)
≤CbBMO
m
j=
ML(logL)
|fj|qj
Lp(ω)
. (.)
For the general casel≥, similarly to the case forl= , we have
Tb,qfLp(ω)≤C
l
j=
bjBMO
m
j=
ML(logL)
|fj|qj
Lp(ω)
.
To apply the Fefferman-Stein inequality in (.), one needs to verify now that
Tb,qfLp(ω) <∞andTqfLp(ω)<∞. We will only show the first one since the proof
of another one is very similar but easier.
Suppose that the symbolsbjand the weightωare bounded functions. Sincef has
com-pact support, we may assumesuppfj⊂B(,R). Then we have
Tb,qfLp(ω)=
|x|≤R
Tb,qf(x) p
ω(x)dx+
|x|>R
Tb,qf(x) p
ω(x)dx
=I+I.
We chooses>p ands, . . . ,sm > such that /s= /s+· · ·+ /sm. Theorem . and
Hölder’s inequality imply
I≤C
|x|≤R
|Tb,qf|pdx≤C
Rn|Tq f|sdx
p/s
Rn(s–p)/s
≤C
m
j=
|fj|qj
Lsj
p
Rn(s–p)/s<∞.
Consider now the termI. Since|x|> R, we have by Assumption (H)
Tb,qf(x)≤C
(B(,R))m
K(x,y)b(x) –b(y)· · ·bl(x) –bl(yl)
×f(y)· · ·fm(ym)dy
q
≤C
(B(,R))m
K(x,y) –Kt()(x,y)
m
j=
fj(yj)qjdy
+
(B(,R))m
Kt()(x,y)
m
j=
fj(yj)qjdy
≤C
m
j=
|x|n
B(,|x|)
fj(yj)q
jdyj≤
m
j=
ML(logL)
|fj|qj
(x).
Thus,I≤C
Rn(
m
j=ML(logL)(|fj|qj)(x))pw(x)dx<∞.
For the general case, we can check the limit (see the proof of [], Theorem ., we omit
Proof of Theorem. (i) Sinceνω∈Amp⊂A∞, Lemma . gives
Rn
Tb,q(f)(x) p
νωdx≤C
Rn
m
j=
MpL(logL)|fj|qj
(x)νωdx.
It follows from [] that there existsr> such thatνω∈A(p
r,...,pmr ). On the other hand, since (t) =t( +log+t)≤trfor allt> , the generalized Jensen inequality yields that
fjL(logL),Q≤C
|Q|
Q
fj(y) r
dy
/r
for allj. One sees immediately that
Tb,q(f)Lp(ν ω)≤C
m
j=
Mr
|fj|qj
Lp(ν ω)
.
Sinceνω∈A(p
r,...,pmr ), it follows from Hölder’s inequality and the well-known inequality of
Fefferman-Stein [] that
m
j=
M|fj|qj
Lp/r(ν ω)
≤C
m
j=
|fj|qj
Lpj/r(Mωj).
We then have
Tb,q(f)Lp(ν ω)≤C
l
j=
bjBMO m
j=
|fj|qj
Lpj(Mωj).
This is the desired conclusion.
(ii) Sinceωj∈Apj, there existsr> such thatωj∈Apj/r. Analysis similar to that in the
proof of Theorem . (i) shows that
Tb,q(f)Lp(ν ω)≤C
m
j=
Mr
|fj|qj
Lp(ν ω)
.
The Hölder inequality implies
m
j=
Mr
|fj|qj
Lp(ν ω)
=
m
j=
M|fj|rqj
Lp/r(ν ω)
/r
≤
m
j=
M
|fj|rqjLpj/r(ω)
/r
≤
m
j=
|fj|qjLpj(ωj). (.)
above lemmas. The key for tackling the new complexities is a very careful deal with the supremum, we refer the reader to []. This concludes the proof of the theorem.
Competing interests
The author declares that he has no competing interests.
Acknowledgements
This work was supported by the Mathematical Tianyuan Foundation of China (No. 11226102). The author thanks the referees for carefully reading the manuscript and providing many valuable suggestions, which have improved this article.
Received: 11 March 2013 Accepted: 30 April 2013 Published: 17 May 2013
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