R E S E A R C H
Open Access
Weighted boundedness of multilinear
singular integral operator with general
kernels for the extreme cases
Mingjun Zhang and Yanfeng Guo
**Correspondence:
[email protected] College of Science, Guangxi University of Science and Technology, Liuzhou, 545006, P.R. China
Abstract
We prove the weighted boundedness properties for the multilinear operator
associated to the singular integral operator with general kernels for the extreme cases.
MSC: 42B20; 42B25
Keywords: multilinear operator; singular integral operator; BMO space
1 Introduction and preliminaries
As for the development of singular integral operators, their commutators and multilinear operators have been well studied (see [–]). LetT be the Calderón-Zygmund singular integral operator andb∈BMO(Rn), a classical result of Coifmanet al.(see []) stated that the commutator [b,T](f) =T(bf) –bT(f) is bounded onLp(Rn) for <p<∞. In [], the authors obtain the boundedness properties of the commutators for the extreme values ofp(that is,p= andp=∞). Note that [b,T] is not bounded for the end point bound-edness. The purpose of this paper is to introduce some multilinear operator associated to the singular integral operator with general kernels (see []) and prove the weighted boundedness properties of the multilinear operators for the extreme cases.
First, let us introduce some preliminaries (see [, ]). Throughout this paper, Qwill denote a cube of Rnwith sides parallel to the axes. For a locally integrable functionsb and a weight functionw(that is, a non-negative locally integrable function), letw(Q) =
Qw(x)dx,wQ=|Q|–
Qw(x)dx, the weighted sharp function ofbis defined by
b#(x) =sup
Qx w(Q)
Q
b(y) –bQw(y)dy.
We say thatb belongs toBMO(w) ifb# belongs to L∞(w), and we defineb
BMO(w)=
b#
L∞(w). Ifw= , we denoteBMO(w) =BMO(Rn). It has been known that (see [])
b–bkQBMO≤CkbBMO.
We also define the centralBMOspace byCMO(Rn), which is the space of those functions f ∈Lloc(Rn) such that
fCMO=sup r>
Q(,r)–
Q
f(x) –fQdx<∞.
It is well known that (see [, ])
fCMO≈sup r>
inf
c∈C
Q(,r)–
Q
f(x) –cdx.
Definition Let <p<∞andwbe a non-negative weight functions onRn. We shall call Bp(w) the space of those functionsf onRnsuch that
fBp(w)=sup r>
wQ(,r)–/pfχQ(,r)Lp(w)<∞.
TheApweight is defined by (see [])
Ap= <w∈Lloc
Rn:sup
Q
|Q|
Q w(x)dx
|Q|
Q
w(x)–/(p–)dx
p– <∞
,
<p<∞,
and
A= <w∈Lloc
Rn:sup
Qx
|Q|
Q
w(y)dy≤Cw(x), a.e.
.
2 Theorems
In this paper, we will study the following multilinear singular integral operator (see []).
Definition LetT:S→S be a linear operator such thatTis bounded onL(Rn) and has a kernelK, that is, there exists a locally integrable functionK(x,y) onRn×Rn\ {(x,y)∈ Rn×Rn:x=y}such that
T(f)(x) =
Rn
K(x,y)f(y)dy
for every bounded and compactly supported functionf, whereKsatisfies
K(x,y)≤C|x–y|–n,
|y–z|<|x–y|
K(x,y) –K(x,z)+K(y,x) –K(z,x)dx≤C,
and there is a sequence of positive constant numbers{Ck}such that for anyk≥,
k|z–y|≤|x–y|<k+|z–y|
K(x,y) –K(x,z)+K(y,x) –K(z,x)qdy
/q
≤Ck
Letmjbe the positive integers (j= , . . . ,l),m+· · ·+ml=mandbjbe the functions on Rn(j= , . . . ,l). Set, for ≤j≤l,
Rmj+(bj;x,y) =bj(x) –
|α|≤mj
α!D
α
bj(y)(x–y)α.
The multilinear operator associated toTis defined by
Tb(f)(x) =
Rn
l
j=Rmj+(bj;x,y)
|x–y|m K(x,y)f(y)dy.
Note that the classical Calderón-Zygmund singular integral operator satisfies Defini-tion (see [, ]). Also note that whenm= ,Tbis just a multilinear commutator ofTand b(see [–]). It is well known that a multilinear operator, as a non-trivial extension of a commutator, is of great interest in harmonic analysis and has been widely studied by many authors (see [–]). In this paper, we will study the weighted boundedness properties of the multilinear operatorsTbfor the extreme cases (see [–]).
We shall prove the following theorems in Section .
Theorem Let T be the singular integral operator as Definition,and we have the
se-quence{kmC
k} ∈l,w∈A and Dαbj∈BMO(Rn)for allα with|α|=mjand j= , . . . ,l.
Then Tbis bounded from L∞(w)to BMO(w).
Theorem Let T be the singular integral operator as Definition,and we have the
se-quence{kmC
k} ∈l, <p<∞,w∈A,and Dαbj∈BMO(Rn)for allαwith|α|=mjand j= , . . . ,l.Then Tbis bounded from Bp(w)to CMO(w).
3 Proofs of theorems
We begin with two preliminaries lemmas.
Lemma (see []) Let b be a function on Rnand Dαb∈Lq(Rn)for|α|=m and some q>n. Then
Rm(b;x,y)≤C|x–y|m
|α|=m
| ˜Q(x,y)|
˜
Q(x,y)
Dαb(z)qdz
/q ,
whereQ˜(x,y)is the cube centered at x and having side length√n|x–y|.
Lemma (see []) Let T be the singular integral operator as Definition,and we have
the sequence{Ck} ∈l.Then T is bounded on Lp(Rn,w)for w∈Apwith <p<∞.
Proof of Theorem It is only for us to prove that there exists a constantCQsuch that
w(Q)
Q
Tb(f)(x) –CQw(x)dx≤CfL∞(w)
holds for any cube Q. Without loss of generality, we may assume l = . Fix a cube Q=Q(x,d). LetQ˜ =
√
nQand b˜j(x) =bj(x) –|α|=mα!(D
αbj) ˜
Rm(b˜j;x,y) andDαb˜
j=Dαbj– (Dαbj)Q˜ for|α|=mj. We write, forf=fχQ˜ andf=fχRn\ ˜Q,
Tb(f)(x) =
Rn
j=Rmj+(b˜j;x,y)
|x–y|m K(x,y)f(y)dy
=
Rn
j=Rmj(b˜j;x,y)
|x–y|m K(x,y)f(y)dy
–
|α|=m
α!
Rn
Rm(b˜;x,y)(x–y) αDαb˜
(y)
|x–y|m K(x,y)f(y)dy
–
|α|=m
α!
Rn
Rm(b˜;x,y)(x–y) αDαb˜
(y)
|x–y|m K(x,y)f(y)dy
+
|α|=m,|α|=m
α!α!
Rn
(x–y)α+αDαb˜
(y)Dαb˜(y)
|x–y|m K(x,y)f(y)dy
+
Rn
j=Rmj+(b˜j;x,y)
|x–y|m K(x,y)f(y)dy
=T
j=Rmj(b˜j;x,·) |x–·|m f
–T
|α|=m
α!
Rm(b˜;x,·)(x–·) αDαb˜
|x–·|m f
–T
|α|=m
α!
Rm(b˜;x,·)(x–·)αDαb˜
|x–·|m f
+T
|α|=m,|α|=m
α!α!
(x–·)α+αDαb˜
Dαb˜
|x–·|m f
+T
j=Rmj+(b˜j;x,·) |x–·|m f
,
then
Tb(f)(x) –Tb˜(f)(x)≤
T
j=Rmj(b˜j;x,·) |x–·|m f
+T
|α|=m
α!
Rm(b˜;x,·)(x–·)αDαb˜
|x–·|m f
+T
|α|=m
α!
Rm(b˜;x,·)(x–·) αDαb˜
|x–·|m f
+T
|α|=m,|α|=m
α!α!
(x–·)α+αDαb˜
Dαb˜
|x–·|m f
and
w(Q)
Q
Tb(f)(x) –Tb˜(f)(x)w(x)dx
≤
w(Q)
Q
I(x)w(x)dx+ w(Q)
Q
I(x)w(x)dx+ w(Q)
Q
I(x)w(x)dx
+
w(Q)
Q
I(x)w(x)dx+ w(Q)
Q
I(x)w(x)dx
=I+I+I+I+I.
Now, let us estimateI,I,I,I, andI, respectively. First, forx∈Qandy∈ ˜Q, by Lemma , we get
Rm(b˜j;x,y)≤C|x–y|m
|αj|=m
Dαjb
jBMO,
thus, by theLp(w)-boundedness ofTfor <p<∞(Lemma ) and Hölder’s inequality, we obtain
I≤C
j= |αj|=mj
Dαjb
jBMO
w(Q)
Q
T(f)(x)w(x)dx
≤C
j= |αj|=mj
Dαjb
jBMO
w(Q)
Rn
T(f)(x) p
w(x)dx
/p
≤C
j= |αj|=mj
Dαjb
jBMO
w(Q)
Rn
f(x) p
w(x)dx
/p
≤C
j= |αj|=mj
Dαjb
jBMO
w(Q˜) w(Q)
/p
fL∞(w)
≤C
j= |αj|=mj
Dαjb
jBMO
fL∞(w).
ForI, sincew∈A,wsatisfies the reverse of Hölder’s inequality:
|Q|
Q
w(x)pdx
/p
≤ C
|Q|
Q w(x)dx
for all cubeQand some <p<∞(see []), thus, by theLp-boundedness ofTforp> , we get
I≤C
|α|=m
Dα
bBMO
|α|=m
w(Q)
Q
T
Dαb˜f
(x)w(x)dx
≤C
|α|=m
Dαb
BMO
|α|=m
w(Q)
Rn
TDαb˜
f
(x)pw(x)dx
≤C
|α|=m
Dαb
BMO
|α|=m
w(Q)
Rn
Dαb˜
(x)f(x)pw(x)dx
/p
≤C
|α|=m
Dαb
BMO
|α|=m
|Q|
˜
Q
Dαb
(x) –
Dαb
˜
Q pp
dx
/pp
×w(Q)–/p|Q|/p
| ˜Q|
˜
Q
w(x)pdx
/pp
fL∞(w)
≤C
j= |αj|=mj
Dαjb
jBMO
w(Q)–/p|Q|/p
| ˜Q|
˜
Q w(x)dx
/p
fL∞(w)
≤C
j= |αj|=mj
Dαjb
jBMO
fL∞(w).
ForI, similar to the proof ofI, we get
I≤C
j= |αj|=mj
Dαjb
jBMO
fL∞(w).
Similarly, forI, choose <r,r<∞such that /r+ /r+ /p= , we obtain, by Hölder’s inequality and the reverse of Hölder’s inequality,
I≤C
|α|=m,|α|=m
w(Q)
Q
TDαb˜
Dαb˜f
(x)w(x)dx
≤C
|α|=m,|α|=m
w(Q)
Rn
TDαb˜
Dαb˜f
(x)pw(x)dx
/p
≤C
|α|=m,|α|=m
w(Q)–/p
Rn
Dαb˜
(x)Dαb˜(x)f(x)pw(x)dx
/p
≤C
|α|=m,|α|=m
| ˜Q|
˜
Q
Dαb˜
(x) pr
dx
/pr | ˜Q|
˜
Q
Dαb˜
(x) pr
dx
/pr
×w(Q)–/p|Q|/p
| ˜Q|
˜
Q
w(x)pdx
/pp
fL∞(w)
≤C
j= |αj|=mj
Dαjb
jBMO
fL∞(w).
ForI, we write
Tb˜(f)(x) –Tb˜(f)(x)
=
Rn
K(x,y)
|x–y|m –
K(x,y)
|x–y|m
j=
Rmj(b˜j;x,y)f(y)dy
+
Rn
Rm(b˜;x,y) –Rm(b˜;x,y)
Rm(b˜;x,y) |x–y|m
+
Rn
Rm(b˜;x,y) –Rm(b˜;x,y)
Rm(b˜;x,y) |x–y|m
K(x,y)f(y)dy
–
|α|=m
α!
Rn
Rm(b˜;x,y)(x–y) α
|x–y|m K(x,y) –
Rm(b˜;x,y)(x–y) α
|x–y|m
K(x,y)
×Dαb˜
(y)f(y)dy
–
|α|=m
α!
Rn
Rm(b˜;x,y)(x–y) α
|x–y|m K(x,y) –
Rm(b˜;x,y)(x–y) α
|x–y|m
K(x,y)
×Dαb˜
(y)f(y)dy
+
|α|=m,|α|=m
α!α!
Rn
(x–y)α+α
|x–y|m K(x,y) –
(x–y)α+α
|x–y|m
K(x,y)
×Dαb˜
(y)Dαb˜(y)f(y)dy
=I()(x) +I()(x) +I()(x) +I()(x) +I() (x) +I()(x).
By Lemma and the inequality (see [])
|bQ–bQ| ≤Clog
|Q|/|Q|
bBMO forQ⊂Q,
we know that, forx∈Qandy∈k+Q˜ \kQ˜,
Rmj(b˜j;x,y)≤C|x–y|
mj
|α|=mj
Dα
bjBMO+Dαbj
˜
Q(x,y)–
Dαbj
˜
Q
≤Ck|x–y|mj |α|=mj
DαbjBMO.
Note that|x–y| ∼ |x–y|forx∈Qandy∈Rn\ ˜Q, we obtain, by the conditions onK,
I()(x)≤
Rn
|x–y|m –
|x–y|m
K(x,y)
j=
Rmj(b˜j;x,y)f(y)dy
+
Rn
K(x,y) –K(x,y)|x–y|–m
j=
Rmj(b˜j;x,y)f(y)dy
≤ ∞
k=
k+Q˜\kQ˜
|x–y|m –
|x–y|m
K(x,y)
j=
Rmj(b˜j;x,y)f(y)dy
+
∞
k=
k+Q˜\kQ˜
K(x,y) –K(x,y)|x–y|–m
j=
Rmj(b˜j;x,y)f(y)dy
≤C
j= |αj|=mj
Dαjb
jBMO
∞
k=
k
k+Q˜\kQ˜
|x–x|
|x–y|n+
f(y)dy
+C
j= |αj|=mj
Dαjb
jBMO
∞
k=
k
k+Q˜\kQ˜
f(y)q dy
×
k+Q˜\kQ˜
K(x,y) –K(x,y)qdy
/q
≤C
j= |αj|=mj
Dαjb
jBMO
∞
k=
k–k+Ck
fL∞(w)
≤C
j= |αj|=mj
Dαjb
jBMO
fL∞(w).
ForI()(x), by the formula (see []):
Rmj(b˜j;x,y) –Rmj(b˜j;x,y) =
|β|<m
β!Rm–|β|
Dβb˜j;x,x
(x–y)β
and Lemma , we have
Rmj(b˜j;x,y) –Rmj(b˜j;x,y)≤C
|β|<mj
|α|=mj
|x–x|mj–|β||x–y||β|DαbjBMO,
thus
I()(x)≤C
j= |αj|=mj
Dαjb
jBMO
∞
k=
k+Q˜\kQ˜k
|x–x|
|x–y|n+
f(y)dy
≤C
j= |αj|=mj
Dαjb
jBMO
∞
k=
k–kfL∞(w)
≤C
j= |αj|=mj
Dαjb
jBMO
fL∞(w).
Similarly,
I()(x)≤C
j= |αj|=mj
Dαjb
jBMO
fL∞(w).
ForI()(x), similar to the proof ofI()(x) andI() (x), we get
I()(x)≤C
|α|=m
Rn
Rm(b˜;x,y) –Rm(b˜;x,y)
|(x–y)αK(x,y)|
|x–y|m
×Dαb˜
(y)f(y)dy
+C
|α|=m
Rn
(|xx––yy)|αm –
(x–y)α
|x–y|m
K(x,y)Rm(b˜;x,y)
×Dαb˜
(y)f(y)dy
+C
|α|=m
Rn
K(x,y) –K(x,y)
(x–y)α
|x–y|m
×Dαb˜
(y)f(y)dy
≤C
|α|=m
Dαb
BMO
∞
k=
k–k
|α|=m
|kQ˜|
kQ˜
Dαb˜
(y)dy
fL∞(w)
+C
|α|=m
Dαb
BMO
|α|=m ∞
k=
k
k+Q˜\kQ˜
K(x,y) –K(x,y) q
dy
/q
×
k+Q˜\kQ˜
Dαb˜
(y)q dy
/q
fL∞(w)
≤C
j= |αj|=mj
Dαjb
jBMO
∞
k=
k–k+Ck
fL∞(w)
≤C
j= |αj|=mj
Dαjb
jBMO
fL∞(w).
Similarly,
I()(x)≤C
j= |αj|=mj
Dαjb
jBMO
fL∞(w).
ForI()(x), taking <r,r<∞such that /r+ /r= , then
I()(x)≤C
|α|=m,|α|=m
Rn
(x–y)α+αK(x,y) |x–y|m –
(x–y)α+αK(x,y)
|x–y|m
×Dαb˜
(y)Dαb˜(y)f(y)dy
≤C
|α|=m,|α|=m ∞
k=
–k+Ck
fL∞(w)
×
|kQ˜|
kQ˜
Dαb˜
(y)rdy
/r |kQ˜|
kQ˜
Dαb˜
(y)rdy
/r
≤C
j= |αj|=mj
Dαjb
jBMO
∞
k=
k–k+Ck
fL∞(w)
≤C
j= |αj|=mj
Dαjb
jBMO
fL∞(w).
Thus
I≤C
j= |αj|=mj
Dαjb
jBMO
fL∞(w).
Proof of Theorem It is only for us to prove that there exists a constantCQsuch that
w(Q)
Q
Tb(f)(x) –CQw(x)dx≤CfBp(w)
holds for any cubeQ=Q(,d) withd> . Without loss of generality, we may assumel= . Fix a cubeQ=Q(,d) withd> . LetQ˜ = √nQandb˜j(x) =bj(x) –|α|=m
j
α!(D
αbj) ˜
Qxα, thenRmj(bj;x,y) =Rmj(b˜j;x,y) andD
αb˜
j=Dαbj– (Dαbj)Q˜ for|α|=mj. Similar to the proof
of Theorem , we write, forf=fχQ˜ andf=fχRn\ ˜Q,
w(Q)
Q
Tb(f)(x) –Tb˜(f)()w(x)dx
≤
w(Q)
Q
T
j=Rmj(b˜j;x,·) |x–·|m f
w(x)dx
+
w(Q)
Q
T
|α|=m
α!
Rm(b˜;x,·)(x–·) αDαb˜
|x–·|m f
w(x)dx
+
w(Q)
Q
T
|α|=m
α!
Rm(b˜;x,·)(x–·) αDαb˜
|x–·|m f
w(x)dx
+
w(Q)
Q
T
|α|=m,|α|=m
α!α!
(x–·)α+αDαb˜
Dαb˜
|x–·|m f
w(x)dx
+
w(Q)
Q
Tb˜(f)(x) –Tb˜(f)()w(x)dx
=L+L+L+L+L.
Similar to the proof of Theorem , we get
L≤C
j= |αj|=mj
Dαjb
jBMO
w(Q)
Rn
T(f)(x) p
w(x)dx
/p
≤C
j= |αj|=mj
Dαjb
jBMO
w(Q˜)–/pfχQ˜Lp(w)
≤C
j= |αj|=mj
Dαjb
jBMO
fBp(w).
ForL, takingr,s,t> such thatr<p,t=pp/(p–r), and /s+ /(p/r) + /t= , then, by the reverse of Hölder’s inequality,
L≤C
|α|=m
Dα
bBMO
|α|=m
w(Q)
Rn
T
Dαb˜f
(x)rw(x)dx
/r
≤C
|α|=m
Dαb
BMOw(Q)–/r
|α|=m
Rn
Dαb˜
(x)f(x)rw(x)dx
≤C
|α|=m
Dαb
BMOw(Q)–/r
|α|=m
˜
Q
Dαb˜(x)rsdx
/rs
×
˜
Q
f(x)pw(x)dx
/p
˜
Q
w(x)(–r/p)tdx
/rt
≤C
j= |αj|=mj
Dαjb
jBMO
w(Q)–/r|Q|/rsfχQ˜Lp(w)|Q|/rt
×
| ˜Q|
˜
Q
w(x)pdx
/rt
≤C
j= |αj|=mj
Dαjb
jBMO
w(Q)–/r|Q|/rsfχQ˜Lp(w)
| ˜Q|
˜
Q w(x)dx
p/rt
≤C
j= |αj|=mj
Dαjb
jBMO
w(Q˜)–/pfχQ˜Lp(w)
≤C
j= |αj|=mj
Dαjb
jBMO
fBp(w),
L≤C
j= |αj|=mj
Dαjb
jBMO
fBp(w).
ForL, takingr,s,s,t> such thatr<p,t=pp/(p–r), and /s+ /s+ /(p/r) + /t= , then, by the reverse of Hölder’s inequality,
L≤C
|α|=m,|α|=m
w(Q)
Rn
TDαb˜
Dαb˜f
(x)rw(x)dx
/r
≤Cw(Q)–/r
|α|=m,|α|=m
Rn
Dαb˜
(x)Dαb˜(x)f(x)rw(x)dx
/r
≤Cw(Q)–/r
|α|=m,|α|=m
˜
Q
Dαb˜(x) rs
dx
/rs ˜
Q
Dαb˜(x) rs
dx
/rs
×
˜
Q
f(x)pw(x)dx
/p
˜
Q
w(x)(–r/p)tdx
/rt
≤C
j= |αj|=mj
Dαjb
jBMO
w(Q)–/r|Q|/rs+/rs+/rtfχ
˜
QLp(w)
×
| ˜Q|
˜
Q w(x)dx
p/rt
≤C
j= |αj|=mj
Dαjb
jBMO
w(Q˜)–/pfχQ˜Lp(w)
≤C
j= |αj|=mj
Dαjb
jBMO
ForL, similar to the proof of the proof ofIin Theorem , we have
Tb˜(f)(x) –Tb˜(f)()
=
Rn
K(x,y)
|x–y|m – K(,y)
|y|m
j=
Rmj(b˜j;x,y)f(y)dy
+
Rn
Rm(b˜;x,y) –Rm(b˜; ,y)
Rm(b˜;x,y)
|y|m K(,y)f(y)dy
+
Rn
Rm(b˜;x,y) –Rm(b˜; ,y)
Rm(b˜;x,y)
|y|m K(,y)f(y)dy
–
|α|=m
α!
Rn
Rm(b˜;x,y)(x–y) α
|x–y|m K(x,y) –
Rm(b˜; ,y)(–y) α
|y|m K(,y)
×Dαb˜
(y)f(y)dy
–
|α|=m
α!
Rn
Rm(b˜;x,y)(x–y) α
|x–y|m K(x,y) –
Rm(b˜; ,y)(–y) α
|y|m K(,y)
×Dαb˜
(y)f(y)dy
+
|α|=m,|α|=m
α!α!
Rn
(x–y)α+α
|x–y|m K(x,y) –
(–y)α+α |y|m K(,y)
×Dαb˜
(y)Dαb˜(y)f(y)dy
=L() (x) +L()(x) +L() (x) +L() (x) +L() (x) +L() (x).
ForL() (x), taking <r<∞such that /p+ /q+ /r= , byw∈A⊂Ap/r+, we get
L() (x)≤C
∞
k=
k+Q˜\kQ˜
|x–y|m –
|y|m
K(x,y)
j=
Rmj(b˜j;x,y)f(y)dy
+
∞
k=
k+Q˜\kQ˜
K(x,y) –K(,y)|y|–m
×
j=
Rmj(b˜j;x,y)f(y)w(y)
/pw(y)–/pdy
≤C
j= |α|=mj
Dα
bjBMO
∞
k=
k+Q˜\kQ˜k
d
(kd)n+f(y)dy
+C
j= |α|=mj
DαbjBMO
∞
k=
k
k+Q˜\kQ˜
K(x,y) –K(,y)qdy
/q
×
k+Q˜
f(y)pw(y)dy
/p
k+Q˜w(y)
–r/pdy
/r
≤C
j= |αj|=mj
Dαjb
jBMO
∞
k=
k–kwkQ˜–/p
kQ˜
f(y)pw(y)dy
×
|kQ˜|
kQ˜w
(y)dy
/p
|kQ˜|
kQ˜ w
(y)–/(p–)dy
(p–)/p
+C
j= |αj|=mj
Dαjb
jBMO
∞
k=
kCkw
kQ˜–/p
kQ˜
f(y)pw(y)dy
/p
×
|kQ˜|
kQ˜w
(y)dy
/p
|kQ˜|
kQ˜ w
(y)–r/pdy
/r
≤C
j= |αj|=mj
Dαjb
jBMO
fBp(w).
Similarly, we get, for <r,r,r,r,s<∞with /p+ /r+ /s= , /p+ /q+ /r+ /s= , and /p+ /q+ /r+ /r+ /s= ,
L() (x) +L() (x) +L() (x) +L() (x) +L() (x)
≤C
|α|=m
DαbBMO
|α|=m ∞
k=
k+Q˜\kQ˜k
d (kd)n+D
αb˜
(y)f(y)dy
+C
|α|=m
Dαb
BMO
|α|=m ∞
k=
k+Q˜\kQ˜
k d
(kd)n+D
αb˜
(y)f(y)dy
+C
|α|=m
DαbBMO
|α|=m ∞
k=
k
k+Q˜\kQ˜
K(x,y) –K(,y)
×Dαb˜
(y)f(y)dy
+C
|α|=m
DαbBMO
|α|=m ∞
k=
k
k+Q˜\kQ˜
K(x,y) –K(,y)
×Dαb˜
(y)f(y)dy
+C
|α|=m,|α|=m ∞
k=
k+Q˜\kQ˜
K(x,y) –K(,y)Dαb˜
(y)Dαb˜(y)f(y)dy
≤C
|α|=m
Dαb
BMO
|α|=m ∞
k=
k
k+Q˜
Dαb˜
(y) r
dy
/r
×
k+Q˜
f(y)pw(y)dy
/p
k+Q˜w(y)
–s/pdy
/s
+C
|α|=m
Dα
bBMO
|α|=m ∞
k=
k
k+Q˜
Dα˜
b(y) r
dy
/r
×
k+Q˜
f(y)pw(y)dy
/p
k+Q˜w(y)
–s/pdy
/s
+C
|α|=m
Dα
bBMO
|α|=m ∞
k=
k
k+Q˜\kQ˜
K(x,y) –K(,y)q dy
/q
×
k+Q˜
Dαb˜
(y) r
dy
/r
k+Q˜
f(y)pw(y)dy
/p k+Q˜
w(y)–s/pdy
+C
|α|=m
DαbBMO
|α|=m ∞
k=
k
k+Q˜\kQ˜
K(x,y) –K(,y)qdy
/q
×
k+Q˜
Dα˜
b(y) r
dy
/r
k+Q˜
f(y)pw(y)dy
/p k+Q˜w
(y)–s/pdy
/s
+C
|α|=m,|α|=m ∞
k=
k+Q˜\kQ˜
K(x,y) –K(,y)qdy
/q
k+Q˜ w(y)
–s/pdy
/s
×
k+Q˜
Dα˜
b(y) r
dy
/r
k+Q˜
Dα˜
b(y) r
dy
/r
×
k+Q˜
f(y)pw(y)dy
/p
≤C
j= |αj|=mj
Dαjb
jBMO
∞
k=
k–k+Ck
wkQ˜–/p
kQ˜
f(y)pw(y)dy
/p
≤C
j= |αj|=mj
Dαjb
jBMO
fBp(w).
Thus
L≤C
j= αj|=mj
Dαjb
jBMO
fBp(w).
This finishes the proof of Theorem .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors completed the paper. MZ carried out the ideas and methods of studies, YG participated in the design of the study and performed the statistical analysis, the sequence alignment and drafted the manuscript. The authors read and approved the final manuscript.
Acknowledgements
The work is supported by NNSFC (No. 11301097), Master Foundation of Guangxi University of Science and Technology (No. 0816208), Guangxi Education Institution Scientific Research Item (No. 2013YB170), GXNSF Grant
(No. 2013GXNSFAA019001).
Received: 1 June 2014 Accepted: 21 August 2014 Published:03 Sep 2014
References
1. Cohen, J: A sharp estimate for a multilinear singular integral onRn. Indiana Univ. Math. J.30, 693-702 (1981)
2. Cohen, J, Gosselin, J: On multilinear singular integral operators onRn. Stud. Math.72, 199-223 (1982)
3. Cohen, J, Gosselin, J: A BMO estimate for multilinear singular integral operators. Ill. J. Math.30, 445-465 (1986) 4. Coifman, R, Meyer, Y: Wavelets, Calderón-Zygmund and Multilinear Operators. Cambridge Studies in Advanced
Math., vol. 48. Cambridge University Press, Cambridge (1997)
5. Coifman, RR, Rochberg, R, Weiss, G: Factorization theorems for Hardy spaces in several variables. Ann. Math.103, 611-635 (1976)
6. Garcia-Cuerva, J, Rubio de Francia, JL: Weighted Norm Inequalities and Related Topics. North-Holland Mathematics Studies, vol. 116. North-Holland, Amsterdam (1985)
7. Pérez, C, Pradolini, G: Sharp weighted endpoint estimates for commutators of singular integral operators. Mich. Math. J.49, 23-37 (2001)
8. Pérez, C, Trujillo-Gonzalez, R: Sharp weighted estimates for multilinear commutators. J. Lond. Math. Soc.65, 672-692 (2002)
10. Harboure, E, Segovia, C, Torrea, JL: Boundedness of commutators of fractional and singular integrals for the extreme values ofp. Ill. J. Math.41, 676-700 (1997)
11. Chang, DC, Li, JF, Xiao, J: Weighted scale estimates for Calderón-Zygmund type operators. Contemp. Math.446, 61-70 (2007)
12. Lin, Y: Sharp maximal function estimates for Calderón-Zygmund type operators and commutators. Acta Math. Sci. Ser. A Chin. Ed.31, 206-215 (2011)
13. Liu, LZ: Weighted boundedness of multilinear operators for the extreme cases. Taiwan. J. Math.10, 669-690 (2006) 14. Liu, LZ: Weighted Herz type spaces estimates of multilinear singular integral operators for the extreme cases. Rev. R.
Acad. Cienc. Exactas Fís. Nat., Ser. A Mat.101, 87-98 (2007)
15. Liu, LZ: Endpoint estimates for multilinear fractional singular integral operators on some Hardy spaces. Math. Notes
88, 735-752 (2010)
10.1186/1029-242X-2014-341
