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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 373050, 17 pages
doi:10.1155/2008/373050
Research Article
A Class of Commutators for Multilinear Fractional
Integrals in Nonhomogeneous Spaces
Jiali Lian and Huoxiong Wu
School of Mathematical Sciences, Xiamen University, Xiamen Fujian, 361005, China
Correspondence should be addressed to Huoxiong Wu,
Received 3 March 2008; Accepted 16 July 2008
Recommended by Nikolaos Papageorgiou
Let μ be a nondoubling measure on R
d
. A class of commutators associated with multilinear
fractional integrals and RBMOμ functions are introduced and shown to be bounded on product
of Lebesgue spaces with μ.
Copyright q 2008 J. Lian and H. Wu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
In recent years, the study of multilinear operators and their commutator has been attracting
many researchers. Many results which parallel to the linear theory of classical integral
operators are obtained. For details, one can see 1–4, and so forth. Meanwhile, as a further
development, harmonic analysis on R
d
with nondoubling measures has been developed
rapidly. Many results of singular integrals and the related operators on Euclidean spaces with
Lebesgue measure have been generalized to the Lebesgue spaces with nondoubling measures
see 5–10,etc.. Motivated by 5, 8, we will consider the commutators generated by a class
of multilinear fractional integrals and RBMO functions with nondoubling measure, which
were introduced by Tolsa in 11 .
Before stating our results, we recall some definitions and notations. Let μ be a Radon
measure on R
d
satisfying the following growth condition; there exist constants C>0and
n ∈ 0,d, such that
μQ ≤ ClQ
n
, 1.1
for any cube Q ⊂ R
d
with sides parallel to the coordinate axes, where lQ stands for the
side length of Q. For r>0, rQ will denote the cube with the same center as Q and with
lrQrlQ.
2 Journal of Inequalities and Applications
Let 0 ≤ β<n, given two cubes Q ⊂ R in R
d
,weset
K
β
Q,R
1
N
Q,R
k1
μ
2
k
Q
l
2
k
Q
n
1−β/n
, 1.2
where N
Q,R
is the first integer k such that l2
k
Q ≥ lR.Ifβ 0, then K
0
Q,R
K
Q,R
.Thelater
quantity was introduced by Tolsa in 11.
Given β
d
depending on d large enough e.g., β
d
> 2
n
, we say that a cube Q ⊂ R
d
is
doubling if μ2Q ≤ β
d
μQ.
Given a cube Q ⊂ R
d
,letN be the smallest nonnegative integer such that 2
N
Q is
doubling. We denote this cube by
Q.
Let η>1 be a fixed constant. We say that b ∈ L
1
loc
μ is in RBMOμ if there exists a
constant C
1
such that for any cube Q
1
μηQ
Q
by − m
Q
b
dμy ≤ C
1
,
m
Q
b − m
R
b
≤ C
1
K
Q,R
, for any two doubling cubes Q ⊂ R,
1.3
where m
Q
b μQ
−1
Q
bydμy. The minimal constant C
1
is the RBMOμ norm of b,andit
will be denoted by b
∗
.In11, Tolsa obtained equivalent norm in the space RBMOμ with
different parameters η>1andβ
d
> 2
n
.
We consider the following multilinear fractional integral operator
I
α,m
f
1
, ,f
m
x
R
d
m
f
1
x − y
1
f
2
x − y
2
···f
m
x − y
m
y
1
,y
2
, ,y
m
mn−α
dμ
y
1
···dμ
y
m
. 1.4
For m 1, we denote I
α,1
by I
α
, which is the Riesz potential operator related to μ.
Given m ∈ N, for all 1 ≤ j ≤ m, we denote by C
m
j
the family of all finite subsets
σ {σ1, ,σj} of {1, 2, ,m} of j different elements. For any σ ∈ C
m
j
, we denote σ
{1, 2, ,m}\σ {σ
j 1, ,σ
m}. Moreover, for b
j
∈ RBMOμ, j 1, 2, ,m,let
b b
1
,b
2
, ,b
m
and denote by
b
σ
b
σ1
, ,b
σj
and by b
σ
xb
σ1
x ···b
σj
x.
Also, we denote
f f
1
, ,f
m
,
f
σ
f
σ1
, ,f
σj
,
b
σ
f
σ
b
σ
j1
f
σj1
, ,b
σ
m
f
σ
m
.
We define a kind of commutator of I
α,m
as follows:
b, I
α,m
f
x
m
j0
σ∈C
m
j
−1
m−j
b
σ
xI
α,m
f
σ
,
b
σ
f
σ
x. 1.5
In particular, for m 2, we define
b
1
,b
2
,I
α,2
f
1
,f
2
xb
1
xb
2
xI
α,2
f
1
,f
2
x − b
1
xI
α,2
f
1
,b
2
f
2
x
− b
2
xI
α,2
b
1
f
1
,f
2
xI
α,2
b
1
f
1
,b
2
f
2
x.
1.6
Obviously, for m 1, the operator defined in 1.5 is the Coifman-Rochberg-Weiss
type commutator of fractional integral, b, I
α
. Under the assumption that μ is a nondoubling
measure, Chen and Sawyer 5 established the L
p
,L
q
-boundedness of b, I
α
also see 9
for the more general case. In this paper, we will extend the result of 5 as follows.
J. Lian and H. Wu 3
Theorem 1.1. Let μ be defined as above and μ ∞, b
j
∈ RBMOR
d
, j 1, 2, 0 <α<2n.Then
b
1
,b
2
,I
α,2
is a bounded operator from L
q
1
× L
q
2
to L
q
with 1/q 1/q
1
1/q
2
− α/2n>0 and
1 <q
1
, q
2
< ∞.
Remark 1.2. By Lemma 2.2 in Section 2, Theorem 1.1 for the case μ < ∞ also holds provided
I
α,2
, b
1
,b
2
,I
α,2
, b
1
,I
α,2
,andb
2
,I
α,2
satisfy certain T1 type conditions. For instance, if I
α,2
satisfies the T1 condition, that is, I
∗1
α,2
0, then we can easily obtain
I
α,2
f
1
,f
2
xdμx0
see 3 for the notation I
∗1
α,2
.
More generally, we have the following theorem.
Theorem 1.3. Let m ∈ N, μ be defined as above, and μ ∞, b
j
∈ RBMOR
d
, j 1, 2, ,m,
0 <α<mn.Then
b, I
α,m
f
L
q
μ
≤ C
m
j1
b
j
∗
f
j
L
q
j
μ
, 1.7
where 1/q 1/q
1
1/q
2
··· 1/q
m
− α/mn > 0 and 1 <q
j
< ∞, j 1, 2, ,m.
Clearly, 5, Theorem 1 is the special case of our Theorem 1.3 for m 1. Throughout
this paper, we always use the letter C to denote a positive constant that may vary at each
occurrence but is independent of the essential variable.
2. Proofs of theorems
We only prove Theorem 1.1 since Theorem 1.3 can follow from the same arguments and an
analogous version of the following Lemma 2.5, which can be deduced by induction on m.
Before proving our results, we need to recall some notation and establish some lemmas which
play important roles in the proofs.
Let f be a function in L
1
loc
R
d
, we define the noncentered maximal operator
M
β
p,η
fxsup
Qx
1
μηQ
1−βp/n
Q
fy
p
dμy
1/p
, 2.1
and the sharp maximal function
M
#,β
fxsup
Qx
1
μ
3/2Q
Q
fy − m
Q
f
dμy sup
R⊃Qx
Q,R doubling
m
Q
f − m
R
f
K
β
Q,R
, 2.2
where the supremum is taken over all cubes Q with sides parallel to the coordinate axes,
m
Q
f is the mean value of f on the cube Q. When β 0, we denote M
0
p,η
f by M
p,η
f and
M
#,0
f by M
#
f.
We also consider the noncentered doubling maximal operator N, defined by
Nfx sup
Qx
Q doubling
1
μQ
Q
fy
dμy. 2.3
4 Journal of Inequalities and Applications
Lemma 2.1 see 11. Let 1 ≤ p<∞ and 1 <ρ<∞.Thenb ∈ RBMOμ, if and only if for any
cube Q ⊂ R
d
,
1
μρQ
Q
bx − m
Q
b
p
dμx ≤ Cb
p
∗
, 2.4
and for any doubling cubes Q ⊂ R,
m
Q
b − m
R
b
≤ CK
Q,R
b
∗
. 2.5
Lemma 2.2 see 5. Let f ∈ L
1
loc
μ with
fdμ 0 if μ < ∞. For 1 <p<∞,ifinf1, Nf ∈
L
p
μ, then for 0 ≤ β<nwe have
Nf
L
p
μ
≤ C
M
#,β
f
L
p
μ
. 2.6
Lemma 2.3 see 5. Let p<r<n/αand 1/q 1/r − α/n.Then
M
α
p,η
f
L
q
μ
≤ Cf
L
r
μ
, 2.7
where η>1 and 0 ≤ α<n/p.
Lemma 2.4. Suppose μ is a Radon measure satisfying 1.1.Letm ∈ N and 1/s 1/r
1
···1/r
m
−
α/n > 0 with 0 <α<mn, 1 ≤ r
j
≤∞. Then,
a if each r
j
> 1,
I
α,m
f
1
, ,f
m
L
s
μ
≤ C
m
j1
f
j
L
r
j
μ
; 2.8
b if r
j
1 for some j,
I
α,m
f
1
, ,f
m
L
s,∞
μ
≤ C
m
j1
f
j
L
r
j
μ
. 2.9
Proof. The proof follows the idea that, for the classical setting, can be found in 4. For the
sake of completeness, we will show it again.
Since α>0, some r
i
< ∞.Ifsay,r
l1
··· r
m
∞,1 ≤ l<m, because α/n <
1/r
1
··· 1/r
l
≤ l,sothatmn − α>m − ln, integration in y
l1
, ,y
m
reduces matters to
the case when all r
i
are finite and m l. Thus, we may assume that all r
i
< ∞. Now, observe
that if 0 <c
i
, i 1, ,m,and0<α<
m
i1
c
i
, we can find 0 <α
i
<c
i
such that α
m
i1
α
i
.
Apply this observation to c
i
n/r
i
,and1/s
i
1/r
i
−α
i
/n. Since
m
i1
1/s
i
1/s,0<α
i
/n ≤ 1,
1 <s
i
< ∞,and
y
1
n−α
1
y
2
n−α
2
···
y
m
n−α
m
≤
y
1
, ,y
m
nm−α
, 2.10
where α
m
i1
α
i
. It follows that
I
α,m
f
1
, ,f
m
x ≤
m
i1
I
α
i
f
i
x. 2.11
Then, by 5, Lemma 1, page 1289or see 6, page 1269 and H
¨
older’s inequality see 12,
page 15 for weak spaces when some r
i
1, we can get Lemma 2.4.
J. Lian and H. Wu 5
Lemma 2.5. Let b
1
,b
2
,I
α,2
be as in 1.6, 0 <α<2n, τ>1, b
1
,b
2
∈ RBMOμ. Then there exists
a constant C>0 such that for all f
1
∈ L
q
1
μ, f
2
∈ L
q
2
μ, and x ∈ R
d
,
M
#,α
b
1
,b
2
,I
α,2
f
1
,f
2
x ≤ C
b
1
∗
b
2
∗
M
τ,3/2
I
α,2
f
1
,f
2
x
b
1
∗
M
τ,3/2
b
2
,I
α,2
f
1
,f
2
x
b
2
∗
M
τ,3/2
b
1
,I
α,2
f
1
,f
2
x
b
1
∗
b
2
∗
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x
,
2.12
M
#,α
b
1
,I
α,2
f
1
,f
2
x ≤ C
b
1
∗
M
τ,3/2
I
α,2
f
1
,f
2
x
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x
,
2.13
M
#,α
b
2
,I
α,2
f
1
,f
2
x ≤ C
b
2
∗
M
τ,3/2
I
α,2
f
1
,f
2
x
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x
,
2.14
where
b
1
,I
α,2
f
1
,f
2
xb
1
xI
α,2
f
1
,f
2
x − I
α,2
b
1
f
1
,f
2
x,
b
2
,I
α,2
f
1
,f
2
xb
2
xI
α,2
f
1
,f
2
x − I
α,2
f
1
,b
2
f
2
x.
2.15
Proof. By the definition, to obtain 2.12,itsuffices to prove that for any x ∈ R
d
and a cube
Q x,
1
μ
3/2Q
Q
b
1
,b
2
,I
α,2
f
1
,f
2
z − h
Q
dμz≤C
b
1
∗
b
2
∗
M
τ,3/2
I
α,2
f
1
,f
2
x
b
1
∗
M
τ,3/2
b
2
,I
α,2
f
1
,f
2
x
b
2
∗
M
τ,3/2
b
1
,I
α,2
f
1
,f
2
x
b
1
∗
b
2
∗
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x
,
2.16
and for any cubes Q ⊂ R, where Q is an arbitrary cube and R is doubling,
h
Q
− h
R
≤ CK
2
Q,R
K
α
Q,R
b
1
∗
b
2
∗
M
τ,3/2
I
α,2
f
1
,f
2
x
b
1
∗
M
τ,3/2
b
2
,I
α,2
f
1
,f
2
x
b
2
∗
M
τ,3/2
b
1
,I
α,2
f
1
,f
2
x
b
1
∗
b
2
∗
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x
,
2.17
where
h
Q
m
Q
I
α,2
m
Q
b
1
− b
1
f
1
χ
R
d
\4/3Q
,
m
Q
b
2
− b
2
f
2
χ
R
d
\4/3Q
,
h
R
m
R
I
α,2
m
R
b
1
− b
1
f
1
χ
R
d
\4/3R
,
m
R
b
2
− b
2
f
2
χ
R
d
\4/3R
.
2.18
6 Journal of Inequalities and Applications
First of all, it is easy to see that
b
1
,b
2
,I
α,2
f
1
,f
2
z − h
Q
≤
b
1
z − m
Q
b
1
b
2
z − m
Q
b
2
I
α,2
f
1
,f
2
z
b
1
z − m
Q
b
1
I
α,2
f
1
,
b
2
z − b
2
f
2
z
b
2
z − m
Q
b
2
I
α,2
b
1
z − b
2
f
1
,f
2
z
I
α,2
b
1
− m
Q
b
1
f
1
,
b
1
− m
Q
b
1
f
2
z − h
Q
: IzIIzIIIzIVz.
2.19
Consequently,
1
μ
3/2Q
Q
b
1
,b
2
,I
α,2
f
1
,f
2
z − h
Q
dμz ≤ CI II III IV, 2.20
where I μ3/2Q
−1
Q
Izdμz, and II, III, IV are defined in the same way.
In what follows, we estimate I–IV, respectively. For I, by H
¨
older’s inequality and
Lemma 2.1, we have
I
1
μ
3/2Q
Q
Izdμz
≤ C
1
μ
3/2Q
Q
b
1
z − m
Q
b
1
τ
1
dμz
1/τ
1
×
1
μ
3/2Q
Q
b
2
z − m
Q
b
2
τ
2
dμz
1/τ
2
×
1
μ
3/2Q
Q
I
α,2
f
1
,f
2
τ
dμz
1/τ
≤ C
b
1
∗
b
2
∗
M
τ,3/2
I
α,2
f
1
,f
2
x,
2.21
where τ
1
> 1, τ
2
> 1and1/τ 1/τ
1
1/τ
2
1.
For II, we have
II
1
μ
3/2Q
Q
IIzdμz
≤ C
1
μ
3/2Q
Q
b
1
z − m
Q
b
1
s
dμz
1/s
×
1
μ
3/2Q
Q
b
2
,I
α,2
f
1
,f
2
z
τ
dμz
1/τ
≤ C
b
1
∗
M
τ,3/2
b
2
,I
α,2
f
1
,f
2
x,
2.22
where s>1and1/s 1/τ 1.
J. Lian and H. Wu 7
Similarly, we have
III ≤ C
b
2
∗
M
τ,3/2
b
2
,I
α,2
f
1
,f
2
x. 2.23
It remains to estimate IV. For convenience, we set f
0
j
f
j
χ
4/3Q
, f
j
f
0
j
f
∞
j
, j 1, 2.
Then,
IVz
≤
I
α,2
b
1
− m
Q
b
1
f
0
1
,
b
2
− m
Q
b
2
f
0
2
z
I
α,2
b
1
− m
Q
b
1
f
0
1
,
b
2
− m
Q
b
2
f
∞
2
z
I
α,2
b
1
− m
Q
b
1
f
∞
1
,
b
2
− m
Q
b
2
f
0
2
z
I
α,2
b
1
− m
Q
b
1
f
∞
1
,
b
2
− m
Q
b
2
f
∞
2
z − h
Q
IV
1
zIV
2
zIV
3
zIV
4
z,
2.24
and so we have
1
μ
3/2Q
Q
IVz
dμz ≤
4
j1
1
μ
3/2Q
Q
IV
j
zdμz :
4
j1
IV
j
. 2.25
To estimate IV
1
,sets
1
√
p
1
, s
2
√
p
2
,and1/v 1/s
1
1/s
2
−α/n. It follows from H
¨
older’s
inequality and Lemma 2.4 that
IV
1
≤
μQ
1−1/v
μ
3/2Q
I
α,2
b
1
− m
Q
b
1
f
0
1
,
b
2
− m
Q
b
2
f
0
2
L
v
μ
≤ Cμ
3/2Q
−1/v
b
1
− m
Q
b
1
f
0
1
L
s
1
μ
b
2
− m
Q
b
2
f
0
2
L
s
2
μ
≤
C
μ
3/2Q
1/v
4/3Q
f
1
y
1
p
1
dμy
1
1/p
1
×
4/3Q
b
1
y
1
− m
Q
b
1
p
1
/
√
p
1
−1
dμy
1
√
p
1
−1/p
1
×
4/3Q
f
2
y
2
p
2
dμy
2
1/p
2
4/3Q
b
2
y
1
− m
Q
b
i
p
2
/
√
p
2
−1
dμy
i
√
p
2
−1/p
2
≤ C
2
i1
1
μ
3/2Q
1−αp
i
/2n
4/3Q
f
i
y
i
p
i
dμy
i
1/p
i
×
1
μ
3/2Q
4/3Q
b
i
y
i
− m
Q
b
i
p
i
/
√
p
i
−1
dμy
i
√
p
i
−1/p
i
≤ C
b
1
∗
b
2
∗
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x.
2.26
8 Journal of Inequalities and Applications
For term IV
2
,byLemma 2.1, we have
IV
2
1
μ
3/2Q
Q
IV
2
zdμz
≤ C
1
μ
3/2Q
Q
R
d
\4/3Q
4/3Q
×
b
1
y
1
− m
Q
b
1
f
0
1
y
1
b
2
y
2
− m
Q
b
2
f
∞
2
y
2
z − y
1
,z− y
2
2n−α
dμ
y
1
dμ
y
2
dμz
≤
C
μ
3/2Q
Q
4/3Q
b
1
y − m
Q
b
1
f
0
1
y
1
dμ
y
1
×
R
d
\4/3Q
b
2
y
2
− m
Q
b
2
f
∞
2
y
2
z − y
2
2n−α
dμ
y
2
dμz
≤ C
1
μ
3/2Q
1−αp
1
/2n
4/3Q
f
1
y
1
p
1
dμ
y
1
1/p
1
×
1
μ
3/2Q
4/3Q
b
1
− m
Q
b
1
p
1
dμ
y
1
1/p
1
× μ
3
2
Q
−α/2n
μQ
∞
k1
2
k
4/3Q\2
k−1
4/3Q
b
2
y
2
− m
Q
b
2
f
2
y
2
2
k2n−α
lQ
2n−α
dμ
y
2
≤ C
b
1
∗
M
α
p
1
,9/8
f
1
x
∞
k1
2
−kn−α/2
l
2
k
3
2
Q
−nα/2
×
2
k
4/3Q
b
2
y
2
− m
Q
b
2
f
2
y
2
dμ
y
2
≤ C
b
1
∗
M
α
p
1
,9/8
f
1
x
∞
k1
2
−kn−α/2
l
2
k
3
2
Q
−nα/2
×
2
k
4/3Q
b
2
y
2
− m
2
k
4/3Q
b
2
f
2
y
2
dμ
y
2
m
2
k
4/3Q
b
2
− m
Q
b
2
2
k
3/2Q
f
2
y
2
dμ
y
2
≤ C
b
1
∗
M
α
p
1
,9/8
f
1
x
×
∞
k1
2
−kn−α/2
1
l
2
k
3/2Q
n
2
k
4/3Q
b
2
y
2
− m
2
k
4/3Q
b
2
p
2
dμ
y
2
1/p
2
×
1
l
2
k
3/2Q
n−αp
2
/2
2
k
4/3Q
f
2
y
2
p
2
dμ
y
2
1/p
2
∞
k1
k2
−kn−α/2
b
2
∗
1
l
2
k
3/2Q
n−α/2
2
k
4/3Q
f
2
y
2
dμ
y
2
≤ C
b
1
∗
b
2
∗
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x,
2.27
J. Lian and H. Wu 9
where the last inequality follows from the following two facts:
1
l
2
k
3/2Q
n−α/2
2
k
4/3Q
f
2
y
2
dμ
y
2
≤
μ
2
k
4/3Q
1−1/p
2
l
2
k
3/2Q
n−α/2
2
k
4/3Q
f
2
y
2
p
2
dμ
y
2
1/p
2
≤ C
μ
2
k1
4/3Q
1−1/p
2
1/p
2
−α/2n
l
2
k1
4/3Q
n−α/2
1
μ
2
k1
4/3Q
1−αp
2
/2n
2
k1
4/3Q
f
2
y
2
p
2
dμ
y
2
1/p
2
≤ CM
α
p
2
,9/8
f
2
x,
2.28
and see11
m
2
k
4/3Q
b
j
− m
Q
b
j
≤ C
b
j
∗
K
Q,
2
k
4/3Q
≤ C
b
j
∗
K
Q,2
k
4/3Q
≤ Ck
b
j
∗
,j 1, 2.
2.29
Similarly,
IV
3
≤ C
b
1
∗
b
2
∗
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x. 2.30
For term IV
4
, we have
I
α,2
b
1
− m
Q
b
1
f
∞
1
,
b
2
− m
Q
b
2
f
∞
2
z − I
α,2
b
1
− m
Q
b
1
f
∞
1
,
b
2
− m
Q
b
2
f
∞
2
y
≤
R
d
\4/3Q
R
d
\4/3Q
1
z − y
1
,z− y
2
2n−α
−
1
y − y
1
,y− y
2
2n−α
×
2
i1
b
i
y
i
− m
Q
b
i
f
∞
i
y
i
dμ
y
1
dμ
y
2
≤
R
d
\4/3Q
R
d
\4/3Q
|z − y|
y − y
1
,y− y
2
2n−α1
×
2
i1
b
i
y
i
− m
Q
b
i
f
∞
i
y
i
dμ
y
1
dμ
y
2
≤ C
2
i1
R
d
\4/3Q
|z − y|
1/2
y − y
i
n−α/21/2
b
i
y
i
− m
Q
b
i
f
∞
i
y
i
dμ
y
i
≤ C
2
i1
∞
k1
2
k
4/3Q\2
k−1
4/3Q
2
−k/2
1
l
2
k
Q
n−α/2
b
i
y
i
− m
Q
b
i
f
∞
i
y
i
dμ
y
i
≤ C
2
i1
∞
k1
2
−k/2
1
l
2
k
3/2Q
n
2
k
4/3Q
b
i
y
i
− m
Q
b
i
p
i
dμ
y
i
1/p
i
×
1
l
2
k
3/2Q
n−αp
i
/2
2
k
4/3Q
f
i
y
i
p
i
dμ
y
i
1/p
i
10 Journal of Inequalities and Applications
≤ C
2
i1
∞
k1
2
−k/2
M
α
p
i
,9/8
f
i
x
×
1
l
2
k
3/2Q
n
2
k
4/3Q
b
i
y
i
−m
2
k
4/3Q
b
i
m
2
k
4/3Q
b
i
−m
Q
b
i
p
i
dμ
y
i
1/p
i
≤ C
2
i1
∞
k1
2
−k/2
k
b
i
∗
M
α
p
i
,9/8
f
i
x
≤ C
b
1
∗
b
2
∗
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x.
2.31
Taking the mean over y ∈ Q,weobtain
I
α,2
b
1
−m
Q
b1
f
∞
1
,
b
1
−m
Q
b
1
f
∞
2
z−h
Q
≤ C
b
1
∗
b
2
∗
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x.
2.32
Thus,
IV
4
1
μ
3/2Q
Q
IV
4
zdμz ≤ C
b
1
∗
b
2
∗
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x. 2.33
Combing 2.20–2.33,weobtain2.16.
Now we turn to estimate 2.17. For any cubes, Q ⊂ R with x ∈ Q, where Q is arbitrary
and R is doubling. We denote N
Q,R
1simplybyN, write
h
Q
− h
R
m
Q
I
α,2
b
1
− m
Q
b
1
f
∞
1
,
b
2
− m
Q
b
2
f
∞
2
− m
R
I
α,2
b
1
− m
R
b
1
f
∞
1
,
b
2
− m
R
b
2
f
∞
2
≤
m
R
I
α,2
b
1
− m
Q
b
1
f
1
χ
R
d
\2
N
Q
,
b
2
− m
Q
b
2
f
2
χ
R
d
\2
N
Q
− m
Q
I
α,2
b
1
− m
Q
b
1
f
1
χ
R
d
\2
N
Q
,
b
2
− m
Q
b
2
f
2
χ
R
d
\2
N
Q
m
R
I
α,2
b
1
− m
R
b
1
f
1
χ
R
d
\2
N
Q
,
b
2
− m
R
b
2
f
2
χ
R
d
\2
N
Q
− m
R
I
α,2
b
1
− m
Q
b
1
f
1
χ
R
d
\2
N
Q
,
b
2
− m
Q
b
2
f
2
χ
R
d
\2
N
Q
m
Q
I
α,2
b
1
− m
Q
b
1
f
1
χ
2
N
Q\4/3Q
,
b
2
− m
Q
b
2
f
2
χ
R
d
\4/3Q
m
Q
I
α,2
b
1
− m
Q
b
1
f
1
χ
R
d
\2
N
Q
,
b
2
− m
Q
b
2
f
2
χ
2
N
Q\4/3Q
m
R
I
α,2
b
1
− m
R
b
1
f
1
χ
R
d
\4/3R
,
b
2
− m
R
b
2
f
2
χ
2
N
Q\4/3R
m
R
I
α,2
b
1
− m
R
b
1
f
1
χ
2
N
Q\4/3R
,
b
2
− m
R
b
2
f
2
χ
R
d
\2
N
Q
6
i1
A
i
.
2.34
By the similar arguments used in proving 2.33,weobtainthat
A
1
≤ C
K
Q,R
2
b
1
∗
b
2
∗
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x. 2.35
J. Lian and H. Wu 11
To estimate A
2
, we write
I
α,2
b
1
− m
R
b
1
f
1
χ
R
d
\2
N
Q
,
b
2
− m
R
b
2
f
2
χ
R
d
\2
N
Q
z
− I
α,2
b
1
− m
Q
b
1
f
1
χ
R
d
\2
N
Q
,
b
2
− m
Q
b
2
f
2
χ
R
d
\2
N
Q
z
m
R
b
2
− m
Q
b
2
I
α,2
b
1
− m
R
b
1
f
1
χ
R
d
\2
N
Q
,f
2
χ
R
d
\2
N
Q
z
m
R
b
1
− m
Q
b
1
I
α,2
f
1
χ
R
d
\2
N
Q
,
b
2
− m
R
b
2
f
2
χ
R
d
\2
N
Q
z
m
R
b
1
− m
Q
b
1
m
R
b
2
− m
Q
b
2
I
α,2
f
1
χ
R
d
\2
N
Q
,f
2
χ
R
d
\2
N
Q
z.
2.36
Then,
A
2
≤
m
R
b
2
− m
Q
b
2
1
μR
R
I
α,2
b
1
− m
R
b
1
f
1
χ
R
d
\2
N
Q
,f
2
χ
R
d
\2
N
Q
zdμz
m
R
b
1
− m
Q
b
1
1
μR
R
I
α,2
f
1
χ
R
d
\2
N
Q
,
b
2
− m
R
b
2
f
2
χ
R
d
\2
N
Q
zdμz
m
R
b
1
− m
Q
b
1
m
R
b
2
− m
Q
b
2
1
μR
R
I
α,2
f
1
χ
R
d
\2
N
Q
,f
2
χ
R
d
\2
N
Q
zdμz
A
21
A
22
A
23
.
2.37
It is obvious that
A
23
≤ CK
2
Q,R
b
1
∗
b
2
∗
M
τ,3/2
I
α,2
f
1
,f
2
x. 2.38
In order to estimate term A
21
, we write
I
α,2
b
1
− m
R
b
1
f
1
χ
R
d
\2
N
Q
,f
2
χ
R
d
\2
N
Q
z
I
α,2
b
1
− m
R
b
1
f
1
,f
2
z − I
α,2
b
1
− m
R
b
1
f
1
χ
2
N
Q
χ
4/3R
,f
2
χ
4/3R
z
− I
α,2
b
1
− m
R
b
1
f
1
χ
4/3R
,f
2
χ
2
N
Q
χ
4/3R
z
I
α,2
b
1
− m
R
b
1
f
1
χ
2
N
Q
χ
4/3R
,f
2
χ
2
N
Q
χ
4/3R
z
− I
α,2
b
1
− m
R
b
1
f
1
χ
R
d
\4/3R
,f
2
χ
2
N
Q
z
− I
α,2
b
1
− m
R
b
1
f
1
χ
2
N
Q
,f
2
χ
R
d
\4/3R
z
I
α,2
b
1
− m
R
b
1
f
1
χ
2
N
Q\4/3R
,f
2
χ
2
N
Q\4/3R
z
7
j1
B
j
z.
2.39
For B
1
z, we write
I
α,2
b
1
− m
R
b
1
f
1
,f
2
z
≤
I
α,2
b
1
− b
1
z
f
1
,f
2
z
I
α,2
b
1
z − m
R
b
1
f
1
,f
2
z
.
2.40
12 Journal of Inequalities and Applications
By H
¨
older’s inequality and the fact that R is doubling, we have
1
μR
R
I
α,2
b
1
− m
R
b
1
f
1
,f
2
zdμz ≤ C
b
1
∗
M
τ,3/2
I
α,2
f
1
,f
2
x,
1
μR
R
I
α,2
b
1
− b
1
z
f
1
,f
2
zdμz ≤ CM
τ,3/2
b
1
,I
α,2
f
1
,f
2
x,
2.41
which imply
m
R
B
1
≤ C
b
1
∗
M
τ,3/2
I
α,2
f
1
,f
2
xM
τ,3/2
b
1
,I
α,2
f
1
,f
2
x
. 2.42
For B
2
z,sets
1
√
p
1
, s
2
p
2
,and1/v 1/s
1
1/s
2
− α/n.UsingH
¨
older’s inequality and
Lemma 2.4, we have
1
μR
R
B
2
zdμz ≤ C
μR
1−1/v
μR
I
α,2
b
1
− m
R
b
1
f
1
χ
2
N
Q
χ
4/3R
,f
2
χ
4/3R
L
v
μ
≤ Cμ
3
2
R
−1/v
b
1
− m
R
b
1
f
1
χ
2
N
Q
χ
4/3R
L
s
1
μ
f
2
χ
4/3R
L
s
2
μ
≤ Cμ
3
2
R
−1/v
4/3R
f
2
y
p
2
dμy
1/p
2
4/3R
f
1
x
p
1
dμy
1/p
1
×
4/3R
b
1
y − m
R
b
1
p
1
/
√
p
1
−1
dμy
√
p
1
−1/p
1
≤ C
1
μ
3/2R
1−αp
1
/2n
4/3R
f
1
y
p
1
dμy
1/p
1
×
1
μ
3/2R
4/3R
b
1
y − m
R
b
1
p
1
/
√
p
1
−1
dμy
√
p
1
−1/p
1
×
1
μ
3/2R
1−αp
2
/2n
4/3Q
f
2
y
p
2
dμy
1/p
2
≤ C
b
1
∗
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x,
2.43
which implies
m
R
B
2
≤ C
b
1
∗
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x. 2.44
Similarly,
m
R
B
3
≤ C
b
1
∗
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x,
m
R
B
4
≤ C
b
1
∗
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x.
2.45
J. Lian and H. Wu 13
For B
5
z,sincez ∈ R, we have
B
5
z
≤
2
N
Q
R
d
\4/3Q
b
1
− m
R
b
1
f
1
f
2
y
2
z − y
1
,z− y
2
2n−α
dμ
y
1
dμ
y
2
≤
2
N
Q
f
2
y
2
dμ
y
2
R
d
\4/3Q
b
1
y
1
− m
R
b
1
f
1
z − y
1
2n−α
dμ
y
1
≤
C
l2R
n−α/2
2
N
Q
f
2
y
2
dμ
y
2
×
∞
k1
2
−kn
l
2
k
4/3
n−α/2
2
k
4/3R\2
k−1
4/3R
b
1
y
1
− m
R
b
1
f
1
dμ
y
1
≤ C
1
l2R
n−α/2
2
N
Q
f
2
y
2
dμ
y
2
×
∞
k1
2
−kn
l
2
k
4/3R
n−α/2
×
2
k
4/3R
b
1
y
1
− m
2
k
4/3R
b
1
f
1
dμ
y
1
2
k
4/3R
m
2
k
4/3R
b
1
− m
R
b
1
y
1
f
1
y
1
dμ
y
1
≤ C
1
l2R
n−α/2
2
N
Q
f
2
y
2
dμ
y
2
×
∞
k1
2
−kn
1
l
2
k
4/3R
n
2
k
4/3R
b
1
y
1
− m
2
k
4/3R
b
1
p
1
dμ
y
1
1/p
1
×
1
l
2
k
4/3R
n−αp
1
/2
2
k
4/3R
f
1
p
1
dμ
y
1
1/p
1
C
b
1
∗
1
l
2
k
4/3R
n−α/2
2
k
4/3R
f
1
dμ
y
1
≤ C
1
l2R
n−α/2
2
N
Q
f
2
y
2
dμ
y
2
∞
k1
2
−kn
b
1
∗
M
α
p
1
,9/8
f
1
xM
p
1
,9/8
f
1
x
≤ C
N
k1
1
l2R
n−α/2
2
k
Q\2
k−1
Q
f
2
y
2
dμ
y
2
b
1
∗
M
α
p
1
,9/8
f
1
x
1
l2R
n−α/2
Q
f
2
y
2
dμ
y
2
b
1
∗
M
α
p
1
,9/8
f
1
x
≤ C
N
k0
1
l2R
n−α/2
2
k
Q
f
2
y
2
dμ
y
2
b
1
∗
M
α
p
1
,9/8
f
1
x
14 Journal of Inequalities and Applications
≤ C
N
k0
μ
2
k1
Q
1−α/2n
l
2
k1
Q
n−α/2
l
2
k1
Q
n−α/2
l2R
n−α/2
1
μ
2
k1
Q
1−α/2n
×
2
k
Q
f
2
y
2
dμ
y
2
b
1
∗
M
α
p
1
,9/8
f
1
x
≤ C
N
k0
μ
2
k1
Q
1−α/2n
l
2
k1
Q
n−α/2
1
μ
9/82
k
Q
1−α/2n
2
k
Q
f
2
y
2
dμ
y
2
b
1
∗
M
α
p
1
,9/8
f
1
x
≤ CK
α
Q,R
b
1
∗
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x.
2.46
Taking the mean on z over R,weobtain
m
R
B
5
≤ CK
α
Q,R
b
1
∗
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x. 2.47
Similarly, we have
m
R
B
6
≤ CK
α
Q,R
b
1
∗
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x,
m
R
B
7
≤ CK
α
Q,R
b
1
∗
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x.
2.48
Summing up the estimates 2.39–2.48,weobtain
A
21
≤ CK
Q,R
K
α
Q,R
b
1
∗
b
2
∗
M
τ,3/2
I
α,2
f
1
,f
2
x
b
2
∗
M
τ,3/2
b
1
,I
α,2
f
1
,f
2
x
b
1
∗
b
2
∗
M
p
1
,9/8
f
1
xM
p
2
,9/8
f
2
x
.
2.49
By the same arguments, we can get
A
22
≤ CK
Q,R
K
α
Q,R
b
1
∗
b
2
∗
M
τ,3/2
I
α,2
f
1
,f
2
x
b
1
∗
M
τ,3/2
b
2
,I
α,2
f
1
,f
2
x
b
1
∗
b
2
∗
M
p
1
,9/8
f
1
xM
p
2
,9/8
f
2
x
.
2.50
Consequently,
A
2
≤ CK
2
Q,R
K
α
Q,R
b
1
∗
b
2
∗
M
τ,3/2
I
α,2
f
1
,f
2
x
b
1
∗
M
τ,3/2
b
2
,I
α,2
f
1
,f
2
x
b
2
∗
M
τ,3/2
b
1
,I
α,2
f
1
,f
2
x
b
1
∗
b
2
∗
M
p
1
,9/8
f
1
xM
p
2
,9/8
f
2
x
.
2.51
Using the similar arguments to those used in proving B
5
z, we can conclude that
A
3
A
4
A
5
A
6
≤ CK
2
Q,R
K
α
Q,R
b
1
∗
b
2
∗
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
x. 2.52
J. Lian and H. Wu 15
Therefore, we obtain
h
Q
− h
R
≤ CK
2
Q,R
K
α
Q,R
b
1
∗
b
2
∗
M
τ,3/2
I
α,2
f
1
,f
2
x
b
1
∗
M
τ,3/2
b
2
,I
α,2
f
1
,f
2
x
b
2
∗
M
τ,3/2
b
1
,I
α,2
f
1
,f
2
x
b
1
∗
b
2
∗
M
p
1
,9/8
f
1
xM
p
2
,9/8
f
2
x
,
2.53
which implies 2.17.
Finally, we show how to derive 2.12 from 2.16 and 2.17.From2.16,ifQ is
doubling and x ∈ Q, we have
m
Q
b
1
,b
2
,I
α,2
f
1
,f
2
− h
Q
≤
1
μQ
Q
b
1
,b
2
,I
α,2
f
1
,f
2
z − h
Q
dμz
≤ C
b
1
∗
b
2
∗
M
τ,3/2
I
α,2
f
1
,f
2
x
b
1
∗
M
τ,3/2
b
2
,I
α,2
f
1
,f
2
x
b
2
∗
M
τ,3/2
b
1
,I
α,2
f
1
,f
2
x
b
1
∗
b
2
∗
M
p
1
,9/8
f
1
xM
p
2
,9/8
f
2
x
.
2.54
Also, for any cube Q with x ∈ Q, K
Q,
Q
≤ C and K
α
Q,
Q
≤ C,by2.16, 2.17,and2.54,we
have
1
μ
3/2Q
Q
b
1
,b
2
,I
α,2
f
1
,f
2
z − m
Q
b
1
,b
2
,I
α,2
f
1
,f
2
dμz
≤
1
μ
3/2Q
b
1
,b
2
,I
α,2
f
1
,f
2
z − h
Q
dμz
1
μ
3/2Q
Q
m
Q
b
1
,b
2
,I
α,2
f
1
,f
2
− h
Q
dμz
1
μ
3/2Q
Q
h
Q
− h
Q
dμz
≤ C
b
1
∗
b
2
∗
M
τ,3/2
I
α,2
f
1
,f
2
x
b
1
∗
M
τ,3/2
b
2
,I
α,2
f
1
,f
2
x
b
2
∗
M
τ,3/2
b
1
,I
α,2
f
1
,f
2
x
b
1
∗
b
2
∗
M
p
1
,9/8
f
1
xM
p
2
,9/8
f
2
x
.
2.55
On the other hand, for all doubling cubes Q ⊂ R with x ∈ Q, such that K
α
Q,R
≤ P
α
, where P
α
is
the constant in 5, Lemma 6,by2.17, we have
h
Q
− h
R
≤ CK
2
Q,R
P
α
b
1
∗
b
2
∗
M
τ,3/2
I
α,2
f
1
,f
2
x
b
1
∗
M
τ,3/2
b
2
,I
α,2
f
1
,f
2
x
b
2
∗
M
τ,3/2
b
1
,I
α,2
f
1
,f
2
x
b
1
∗
b
2
∗
M
p
1
,9/8
f
1
xM
p
2
,9/8
f
2
x
.
2.56
16 Journal of Inequalities and Applications
Hence, by 5, Lemma 6,weget
h
Q
− h
R
≤ CK
α
Q,R
b
1
∗
b
2
∗
M
τ,3/2
I
α,2
f
1
,f
2
x
b
1
∗
M
τ,3/2
b
2
,I
α,2
f
1
,f
2
x
b
2
∗
M
τ,3/2
b
1
,I
α,2
f
1
,f
2
x
b
1
∗
b
2
∗
M
p
1
,9/8
f
1
xM
p
2
,9/8
f
2
x
,
2.57
for all doubling cubes Q ⊂ R with x ∈ Q. Invoking 2.55 yields that
m
Q
b
1
,b
2
,I
α,2
f
1
,f
2
− m
R
b
1
,b
2
,I
α,2
f
1
,f
2
≤
m
Q
b
1
,b
2
,I
α,2
f
1
,f
2
− h
Q
h
R
− m
R
b
1
,b
2
,I
α,2
f
1
,f
2
h
Q
− h
R
≤ CK
α
Q,R
b
1
∗
b
2
∗
M
τ,3/2
I
α,2
f
1
,f
2
x
b
1
∗
M
τ,3/2
b
2
,I
α,2
f
1
,f
2
x
b
2
∗
M
τ,3/2
b
1
,I
α,2
f
1
,f
2
x
b
1
∗
b
2
∗
M
p
1
,9/8
f
1
xM
p
2
,9/8
f
2
x
.
2.58
From this estimate and the definition of the sharp maximal function, we complete the proof
of 2.12. Similarly, we can deduce 2.13 and 2.14. The details are omitted.
Now we are in the position to prove Theorem 1.1.
Proof of Theorem 1.1. By the Lebesgue differentiation theorem, it is easy to see that for any
f ∈ L
1
loc
R
d
,
fx
≤Nfx, 2.59
for μ − a.e. x ∈ R
d
,see11 for details. By Lemmas 2.2–2.5, we have
b
1
,b
2
,I
α,2
f
1
,f
2
L
q
≤
N
b
1
,b
2
,I
α,2
f
1
,f
2
L
q
≤
M
#,β
b
1
,b
2
,I
α,2
f
1
,f
2
L
q
≤ C
b
1
∗
b
2
∗
M
τ,3/2
I
α,2
f
1
,f
2
L
q
b
1
∗
M
τ,3/2
b
2
,I
α,2
f
1
,f
2
L
q
b
2
∗
M
τ,3/2
b
1
,I
α,2
f
1
,f
2
L
q
b
1
∗
b
2
∗
M
α
p
1
,9/8
f
1
xM
α
p
2
,9/8
f
2
L
q
≤ C
b
1
∗
b
2
∗
f
1
L
q
1
f
2
L
q
2
b
1
∗
b
2
,I
α,2
f
1
,f
2
L
q
b
2
∗
b
1
,I
α,2
f
1
,f
2
L
q
J. Lian and H. Wu 17
≤ C
b
1
∗
b
2
∗
f
1
L
q
1
f
2
L
q
2
b
1
∗
M
#,β
b
2
,I
α,2
f
1
,f
2
L
p
1
b
2
∗
M
#,β
b
1
,I
α,2
f
1
,f
2
L
p
2
≤ C
b
1
∗
b
2
∗
f
1
L
q
1
f
2
L
q
2
.
2.60
This proves Theorem 1.1.
Acknowledgments
The authors would like to express their deep thanks to the referees for their valuable remarks
and suggestions. Huoxiong Wu was partially supported by the NSF of China G10571122.
References
1 C. P
´
erez and R. H. Torres, “Sharp maximal function estimates for multilinear singular integrals,”
in Harmonic analysis at Mount Holyoke, vol. 320 of Contemporary Mathematics, pp. 323–331, American
Mathematical Society, Providence, RI, USA, 2003.
2 C. P
´
erez and R. Trujillo-Gonz
´
alez, “Sharp weighted estimates for multilinear commutators,” Journal
of the London Mathematical Society, vol. 65, no. 3, pp. 672–692, 2002.
3 L. Grafakos and R. H. Torres, “Multilinear Calder
´
on-Zygmund theory,” Advances in Mathematics, vol.
165, no. 1, pp. 124–164, 2002.
4 C. E. Kenig and E. M. Stein, “Multilinear estimates and fractional integration,” Mathematical Research
Letters, vol. 6, no. 1, pp. 1–15, 1999.
5 W. Chen and E. Sawyer, “A note on commutators of fractional integrals with RBMO μ functions,”
Illinois Journal of Mathematics, vol. 46, no. 4, pp. 1287–1298, 2002.
6 J. Garc
´
ıa-Cuerva and J. Mar
´
ıa-Martell, “Two-weight norm inequalities for maximal operators and
fractional integrals on non-homogeneous spaces,” Indiana University Mathematics Journal, vol. 50, no.
3, pp. 1241–1280, 2001.
7 J. Xu, “Boundedness of multilinear singular integrals for non-doubling measures,” Journal of
Mathematical Analysis and Applications, vol. 327, no. 1, pp. 471–480, 2007.
8 J. Xu, “Boundedness in Lebesgue spaces for commutators of multilinear singular integrals and RBMO
functions with non-doubling measures,” Science in China. Series A, vol. 50, no. 3, pp. 361–376, 2007.
9 G. Hu, Y. Meng, and D. Yang, “Multilinear commutators for fractional integrals in non-homogeneous
spaces,” Publicacions Matem
`
atiques, vol. 48, no. 2, pp. 335–367, 2004.
10 G. Hu, Y. Meng, and D. Yang, “Multilinear commutators of singular integrals with non doubling
measures,” Integral Equations and Operator Theory, vol. 51, no. 2, pp. 235–255, 2005.
11 X. Tolsa, “BMO, H
1
, and Calder
´
on-Zygmund operators for non doubling measures,” Mathematische
Annalen, vol. 319, no. 1, pp. 89–149, 2001.
12 L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education/Prentice Hall, Upper Saddle
River, NJ, USA, 2004.
