Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
28 (2012), 37–46
www.emis.de/journals
ISSN 1786-0091
SOME HARDY SPACE ESTIMATES FOR MULTILINEAR
SINGULAR INTEGRAL OPERATOR
ZHOU XIAOSHA AND LIU LANZHE
Abstract. In this paper, we establish the boundedness for some multilinear singular integral operators on Hardy and Herz type Hardy spaces. The
operators include Calderon-Zygmund singular integral operators.
1. Introduction
Let b ∈ BM O(Rn ) and T be the Calderon-Zygmund operator. The commutator [b, T ] generated by b and T is defined by [b, T ]f (x) = b(x)T f (x) −
T (bf )(x). By a classical result of Coifman, Rochberg and Weiss(see [6]), we
know that the commutator [b, T ] is bounded on Lp (Rn ) for 1 < p < ∞. However, it was observed that [b, T ] is not bounded, in general, from H p (Rn ) to
Lp (Rn ) and from L1 (Rn ) to L1,∞ (Rn ) for p ≤ 1. But, if H p (Rn ) is replaced
by a suitable atomic space Hbp (Rn )(see [1, 14]), then [b, T ] is bounded from
Hbp (Rn ) to Lp (Rn ) for p ∈ (n/(n + 1), 1]. In recent years, the theory of Herz
space and Herz type Hardy space, as a local version of Lebesgue space and
Hardy space, have been developed (see [8, 9, 11, 12]). The main purpose of
this paper is to establish the boundedness properties of some multilinear operators related to certain non-convolution type singular integral operators on
Hardy and Herz type Hardy spaces. The operators include Calderón-Zygmund
singular integral operators.
2. Notations and Theorems
In this paper, we study the singular integral operators as following. Let
T : S → S 0 be a linear operator and there exists a locally integrable function
2010 Mathematics Subject Classification. 42B20, 42B25.
Key words and phrases. Multilinear operator; Singular integral operator; BMO; Hardy
space, Herz-Hardy space.
Supported by the Scientific Research Fund of Hunan Provincial Education Departments(09C057) and (10C0365) and Scientific Research Fund of Hunan Provincial Science
and Technology Departments (2010SK3026) and (2011FJ6056).
37
38
ZHOU XIAOSHA AND LIU LANZHE
K(x, y) on Rn × Rn \ {(x, y) ∈ Rn × Rn : x = y} such that
Z
T f (x) =
K(x, y)f (y)dy
Rn
for every bounded and compactly supported function f , where K satisfies: for
fixed ε > 0 and n > δ ≥ 0,
|K(x, y)| ≤ C|x − y|−n+δ
and
|K(y, x) − K(z, x)| + |K(x, y) − K(x, z)| ≤ C|y − z|ε |x − z|−n−ε+δ
if 2|y −z| ≤ |x−z|. Let mi be positive integers(i = 1, . . . , l), m1 +· · ·+ml = m
and Ai be some functions on Rn (i = 1, . . . , l). The multilinear operator related
to T is defined by
Z Ql
A
i=1 Rmi +1 (Ai ; x, y)
T (f )(x) =
K(x, y)f (y)dy,
|x − y|m
Rn
where
Rmi +1 (Ai ; x, y) = Ai (x) −
X 1
Dβ Ai (y)(x − y)β .
β!
|β|≤mi
A
Note that when m = 0, T is just the multilinear commutator of T and A (see
[16]). While when m > 0, T A is non-trivial generalizations of the commutator.
It is well known that multilinear operators are of great interest in harmonic
analysis and have been widely studied by many authors (see [3, 5, 4, 13]). In
[7], the weighted Lp (p > 1)-boundedness of the multilinear operator related to
some singular integral operator are obtained. The main purpose of this paper
is to study the boundedness of the multilinear singular integral operator T A
on some Hardy and Herz-Hardy spaces.
First, let us introduce some notations. Throughout this paper, Q will denote
a cube of Rn with sides parallel to the axes. For any locally integrable function
f , the sharp function of f is defined by
Z
1
#
f (x) = sup
|f (y) − fQ |dy,
x∈Q |Q| Q
R
where, and in what follows, fQ = |Q|−1 Q f (x)dx. It is well-known that (see
[10])
Z
1
#
f (x) ≈ sup inf
|f (y) − c|dy.
x∈Q c∈C |Q| Q
We say that f belongs to BM O(Rn ) if f # belongs to L∞ (Rn ) and ||f ||BM O =
||f # ||L∞ .
Definition 1. Let Ai be some function on Rn and mi be positive integers(i =
1, . . . , l), m1 + · · · + ml = m and 0 < p ≤ 1. A bounded measurable function
a on Rn is said to be a (p, Dm A) atom if
SOME HARDY SPACE ESTIMATES
39
i) supp a ⊂ Q = Q(x0 , r),
−1/p
∞
ii) ||a||
R L ≤ |Q| R ,
Q
iii) Rn a(y)dy = Rn a(y) kν=1 Dβ Aν (y)dy = 0 for |β| = mi , i = 1, . . . , l
and k = 1, . . . , l;
p
n
A tempered distribution f is said to belong to HD
m A (R ), if, in the Schwartz
distributional sense, it can be written as
∞
X
f (x) =
λj aj (x),
j=1
where aj ’s are (p, D A) atoms, λj ∈ C and
P
1/p
∞
p
||f ||HDp m A ≈
|λ
|
.
j
j=1
m
P∞
j=1
|λj |p < ∞. Moreover,
Definition 2. Let 0 < p, q < ∞, α ∈ R. For k ∈ Z, define Bk = {x ∈ Rn :
|x| ≤ 2k } and Ck = Bk \ Bk−1 . Denote by χk the characteristic function of Ck
and χ0 the characteristic function of B0 .
The homogeneous Herz space is defined by
(1)
K̇qα,p (Rn ) = {f ∈ Lqloc (Rn \ {0}) : ||f ||K̇qα,p < ∞},
where
"
||f ||K̇qα,p =
∞
X
#1/p
2kαp ||f χk ||pLq
.
k=−∞
The nonhomogeneous Herz space is defined by
(2)
Kqα,p (Rn ) = {f ∈ Lqloc (Rn ) : ||f ||Kqα,p < ∞},
where
"
||f ||Kqα,p =
∞
X
#1/p
kαp
2
||f χk ||pLq
+
||f χB0 ||pLq
.
k=1
Definition 3. Let Ai be a function on Rn and mi be positive integers (i =
1, . . . , l), m1 + · · · + ml = m, α ∈ R, 0 < p < ∞, 1 < q ≤ ∞. A function a(x)
on Rn is called a central (α, q, Dm A)-atom (or a central (a, q, Dm A)-atom of
restrict type), if
1) supp a ⊂ B(0, r) for some r > 0 (or for some r ≥ 1),
−α/n
q
2) ||a||
,
R L ≤ |B(0, Rr)|
Q
3) Rn a(x)dx = Rn a(y) ki=1 Dβ Ai (y)dy = 0 for |β| = mi , i = 1, . . . , l
and k = 1, . . . , l;
α,p
α,p
n
A tempered distribution f is said to belong to H K̇q,D
(Rn ) (or HKq,D
m A (R )),
P∞ m A
P∞
if it can be written as f = j=−∞ λj aj (or f = j=1 λj aj ) in the S 0 (Rn ) sense,
m
where aj is a central (α, q, Dm A)-atomP(or a central (α, q, DP
A)-atom of re∞
p
p
j
strict type) supported on B(0, 2 ) and j=−∞ |λj | < ∞(or ∞
j=1 |λj | < ∞),
1/p
P
p
α,p
.
moreover, ||f ||H K̇ α,p m (or ||f ||HKq,D
)≈
|λ
|
j
j
mA
q,D
A
40
ZHOU XIAOSHA AND LIU LANZHE
Now, we can state our results as following.
Theorem 1. Let max(n/(n + 1), n/(n + ε − δ)) < q ≤ 1, 1/q = 1/p − δ/n,
Dβ Ai ∈ BM O(Rn ) for all β with |β| = mi and i = 1, . . . , l. Suppose that T A
is bounded from Ls (Rn ) to Lr (Rn ) for any 1 < s < n/δ and 1/r = 1/s − δ/n.
p
n
q
n
Then T A is bounded from HD
m A (R ) to L (R ).
Theorem 2. Let 0 < p < ∞, 1 < q1 , q2 < ∞, 1/q1 −1/q2 = δ/n, n(1−1/q1 ) ≤
α < min(n(1 − 1/q1 ) + 1, n(1 − 1/q1 ) + ε) and Dβ Ai ∈ BM O(Rn ) for all β
with |β| = mi and i = 1, . . . , l. Suppose that T A is bounded from Ls (Rn ) to
Lr (Rn ) for any 1 < s < n/δ and 1/r = 1/s − δ/n. Then T A is bounded from
n
α,p
H K̇qα,p
(Rn ).
m (R ) to K̇q
2
1 ,D A
Remark 1. Theorem 2 is also hold for nonhomogeneous Herz and Herz type
Hardy space.
3. Proofs of Theorems
To prove the theorems, we need the following lemma.
Lemma 1 (see [4]). Let A be a function on Rn and Dβ A ∈ Lq (Rn ) for |β| = m
and some q > n. Then
1/q
Z
X 1
m
β
q
|Rm (A; x, y)| ≤ C|x − y|
|D A(z)| dz
,
|
Q̃(x,
y)|
Q̃(x,y)
|β|=m
√
where Q̃(x, y) is the cube centered at x and having side length 5 n|x − y|.
Proof of Theorem 1: It suffices to prove that there exists a constant C > 0
such that for every (p, Dm A) atom a,
||T A (a)||Lq ≤ C.
Let a be a (p, Dm A) atom supported on a cube Q = Q(x0 , d). We write
Z
Z
Z
A
q
A
q
|T (a)(x)| dx =
|T (a)(x)| dx +
|T A (a)(x)|q dx = I + II.
Rn
2Q
(2Q)c
For I, taking r, s > 1 with q < s < n/δ and 1/r = 1/s − δ/n, by Holder’s
inequality and the (Ls , Lr )-boundedness of T A , we get
I ≤ C||T A (a)||qLr |Q(x0 , 2d)|1−q/r ≤ C||a||qLs |Q|1−q/r ≤ C|Q|−q/p+q/s+1−q/r ≤ C.
To obtain the estimate of II, we need to estimate T A (a)(x) for x ∈ (2Q)c .
Without loss of generality, we may assume l = 2. Let Ãi (x) = Ai (x) −
P
1
β
β
β
β
|β|=mi β! (D Ai )Q x . Then Rmi (Ai ; x, y) = Rmi (Ãi ; x, y) and D Ãi = D Ai −
SOME HARDY SPACE ESTIMATES
41
(Dβ Ai )Q for |β| = mi . We write, by the vanishing moment of a,
Z K(x, y)
K(x, x0 )
A
T (a)(x) =
−
Rm1 (Ã1 ; x, y)Rm2 (Ã2 ; x, y)a(y)dy
m
|x − x0 |m
Rn |x − y|
Z
K(x, x0 )
+
[Rm1 (Ã1 ; x, y) − Rm1 (Ã1 ; x, x0 )]Rm2 (Ã2 ; x, y)a(y)dy
m
Rn |x − x0 |
Z
K(x, x0 )
+
[Rm2 (Ã2 ; x, y) − Rm2 (Ã2 ; x, x0 )]Rm1 (Ã1 ; x, x0 )a(y)dy
m
Rn |x − x0 |
X 1 Z K(x, y)(x − y)β2
K(x, x0 )(x − x0 )β2
−
−
×
β2 ! Rn
|x − y|m
|x − x0 |m
|β2 |=m2
× Rm1 (Ã1 ; x, y)Dβ2 Ã2 (y)a(y)dy
X 1 Z K(x, x0 )(x − x0 )β2
[Rm1 (Ã1 ; x, y) − Rm1 (Ã1 ; x, x0 )]×
−
β2 ! Rn
|x − x0 |m
|β2 |=m2
−
× Dβ2 Ã2 (y)a(y)dy
X 1 Z K(x, y)(x − y)β1
|β1 |=m1
β1 !
Rn
|x − y|m
K(x, x0 )(x − x0 )β1
−
×
|x − x0 |m
× Rm2 (Ã2 ; x, y)Dβ1 Ã1 (y)a(y)dy
X 1 Z K(x, x0 )(x − x0 )β1
[Rm1 (Ã2 ; x, y) − Rm2 (Ã2 ; x, x0 )]×
−
β2 ! Rn
|x − x0 |m
|β1 |=m1
+
× Dβ1 Ã1 (y)a(y)dy
Z
X
1
|β1 |=m1 ,|β2 |=m2
β1 !β2 !
Rn
K(x, y)(x − y)β1 +β2
K(x, x0 )(x − x0 )β1 +β2
−
×
|x − y|m
|x − x0 |m
× Dβ1 Ã1 (y)Dβ2 Ã2 (y)a(y)dy
= II1 (x) + II2 (x) + II3 (x) + II4 (x) + II5 (x) + II6 (x) + II7 (x) + II8 (x).
By Lemma and the following inequality (see [17])
|bQ1 − bQ2 | ≤ C log(|Q2 |/|Q1 |)||b||BM O
for Q1 ⊂ Q2 ,
we know that, for y ∈ Q and x ∈ 2 Q \ 2 Q,
X
|Rmi (Ãi ; x, y)| ≤ C|x − y|mi
(||Dβ Ai ||BM O + |(Dβ Ai )Q(x,y) − (Dβ Ai )Q |)
k+1
k
|β|=mi
≤ Ck|x − y|
mi
X
||Dβ Ai ||BM O .
|β|=mi
Note that |x − y| ∼ |x − x0 | for y ∈ Q and x ∈ Rn \ 2Q, we obtain, by the
condition on K,
|II1 (x)| ≤
42
ZHOU XIAOSHA AND LIU LANZHE
Y
2
|y − x0 |ε
|y − x0 |
|Rmi (Ãi ; x, y)||a(y)|dy
≤C
+
|x − x0 |m+n+1−δ |x − x0 |m+n+ε−δ i=1
Rn


Z 2
Y
X
|y − x0 |
|y − x0 |ε
2
βi


k
≤C
||D Ai ||BM O
+
|a(y)|dy
n+1−δ
n+ε−δ
|x
−
x
|
|x
−
x
|
0
0
Q
i=1
|βi |=mi


2
Y
X
|Q|ε/n+1−1/p
|Q|1/n+1−1/p
βi
2


≤C
+
.
||D Ai ||BM O k
|x − x0 |n+1−δ |x − x0 |n+ε−δ
i=1
Z
|βi |=mi
For II2 (x), by the formula (see [5]):
Rmi (Ãi ; x, y) − Rmi (Ãi ; x, x0 ) =
X 1
Rm −|γ| (Dγ Ãi ; x, x0 )(x − y)γ
γ! i
|γ|<mi
and Lemma, we have
|Rmi (Ãi ; x, y)−Rmi (Ãi ; x, x0 )| ≤ C
X X
|x−x0 |mi −|γ| |x−y||γ| ||Dβ Ai ||BM O ,
|γ|<mi |β|=mi
thus
|II2 (x)| ≤ C
2
Y
i=1
≤C
2
Y



X
||D Ai ||BM O 
|βi |=mi


X

||Dβi Ai ||BM O  k
|βi |=mi
i=1
Z
βi
k
Q
|y − x0 |
|a(y)|dy
|x − x0 |n+1−δ
|Q|1/n+1−1/p
.
|x − x0 |n+1−δ
Similarly,
|II3 (x)| ≤ C
2
Y
i=1


X
|βi |=mi

||Dβi Ai ||BM O  k
|Q|1/n+1−1/p
.
|x − x0 |n+1−δ
For II4 (x), similar to the proof of II1 (x) and II2 (x), we get
X
X |Q|1/n−1/p
|Q|ε/n−1/p
β1
+
×
|II4 (x)| ≤ C
||D A1 ||BM O
k
|x − x0 |n+1−δ |x − x0 |n+ε−δ
|β1 |=m1
|β2 |=m2
Z
× |Dβ2 A2 (y) − (Dβ2 A2 )Q |dy
Q


2
ε/n+1−1/p
1/n+1−1/p
Y
X
|Q|
|Q|
β

+
.
≤C
||D i Ai ||BM O  k
|x − x0 |n+1−δ |x − x0 |n+ε−δ
i=1
|βi |=mi
SOME HARDY SPACE ESTIMATES
43
Similarly,
2
Y
|II5 (x)| ≤ C
i=1
2
Y
|II6 (x)| ≤ C
i=1
2
Y
|II7 (x)| ≤ C

X


||Dβi Ai ||BM O  k
|βi |=mi

X


||D Ai ||BM O  k
βi
|βi |=mi

X

|βi |=mi
i=1

|Q|1/n+1−1/p
.
|x − x0 |n+1−δ
|Q|1/n+1−1/p
|Q|ε/n+1−1/p
+
|x − x0 |n+1−δ |x − x0 |n+ε−δ
.
|Q|1/n+1−1/p

||D Ai ||BM O k
.
|x − x0 |n+1−δ
βi
For II8 (x), taking 1 < r1 , r2 < ∞ such that 1/r1 + 1/r2 = 1, then, by Holder’s
inequality,
|II8 (x)| ≤
X
≤C
K(x, y)(x − y)β1 +β2
K(x, x0 )(x − x0 )β1 +β2 −
×
|x − y|m
|x − x0 |m
Rn
Z
|β1 |=m1 ,|β2 |=m2
×|Dβ1 Ã1 (y)||Dβ2 Ã2 (y)||a(y)|dy
1/r1
X Z
α1
β1
r1
|D A1 (y) − (D A1 )Q | dy
≤C
Q
|β1 |=m1
×
|β2 |=m2
≤C
1/r
2
X Z
2
Y


|D A2 (y) − (D A2 )Q | dy
α2
β2
Q
X

||D Ai ||BM O 
βi
|βi |=mi
i=1
r2
|Q|1/n−1/p
|Q|ε/n−1/p
+
|x − x0 |n+1−δ |x − x0 |n+ε−δ
|Q|1/n+1−1/p
|Q|ε/n+1−1/p
+
|x − x0 |n+1−δ |x − x0 |n+ε−δ
.
Thus, recall that max(n/(n + 1), n/(n + ε − δ)) < q ≤ 1,
∞ Z
X
II ≤
|T A (a)(x)|q dx
2k+1 Q\2k Q
k=1
≤C
2
Y


i=1
X

||D Ai ||BM O 
|βi |=mi
βi
∞ Z
X
k=1
k 2q ×
2k+1 Q\2k Q
q
|Q|ε/n+1−1/p
|Q|1/n+1−1/p
+
dx
×
|x − x0 |n+1−δ |x − x0 |n+ε−δ
 
q
2
∞
Y
X
X
βi




≤C
||D Ai ||BM O
k 2q [2knq(1/p−1/n−1) + 2knq(1/p−ε/n−1) ]
i=1
|βi |=mi
k=1
44
ZHOU XIAOSHA AND LIU LANZHE
 
q
2
Y
X
≤C 
||Dβi Ai ||BM O  .
i=1
|βi |=mi
This completes the proof of Theorem 1.
Proof of Theorem 2: Without loss
of
generality,
we
may
assume
l
=
2.
Let
P∞
n
f ∈ H K̇qα,p
m (R ) and f (x) =
j=−∞ λj aj (x) be the atomic decomposition
1 ,D A
for f as in Definition 3. We write
!p
∞
k−3
X
X
||T A (f )||pK̇ α,p ≤
2kαp
|λj |||T A (aj )χk ||Lq2
q2
+
k=−∞
j=−∞
∞
X
∞
X
2kαp
k=−∞
!p
|λj |||T A (aj )χk ||Lq2
j=k−2
= J + JJ.
For JJ, by the (Lq1 , Lq2 )-boundedness of T A , we get
!p
∞
∞
X
X
kαp
|λj |||aj ||Lq1
2
JJ ≤ C
≤C
k=−∞
j=k−2
∞
X
∞
X
k=−∞
2kαp
!p
|λj |2−jα
j=k−2
P
 P
∞
kαp
p −jαp
C ∞
2
|λ
|
2
, 0<p≤1
j
k=−∞
j=k−2
P
p/p0
≤
P∞
∞
 C P∞
kαp
p −jαp/2
−jαp0 /2
2
|λ
|
2
2
, p>1
j
k=−∞
j=k−2
j=k−2
 P
Pj+2
p
(k−j)αp
C ∞
|λ
|
, 0<p≤1
j
j=−∞
k=−∞ 2
P
p/p0
≤
Pj+2
j+2
 C P∞
p
(k−j)αp/2
(k−j)αp0 /2
, p>1
j=−∞ |λj |
k=−∞ 2
k=−∞ 2
≤C
∞
X
j=−∞
|λj |p ≤ C||f ||pH K̇ α,p
q1 ,D m A
.
For J, similar to the proof of Theorem 1, we get, for x ∈ Ck , j ≤ k − 3,
|T A (aj )(x)| ≤


Z
2
ε/n
1/n
Y
X
|B
|
|B
|
j
j
β
i

+ n+ε−δ
|aj (y)|dy
≤C
||D Ai ||BM O 
n+1−δ
|x|
|x|
n
R
i=1
|βi |=mi
X Z
X
|Bj |ε/n
|Bj |1/n
β1
+ n+ε−δ
+C
||D A1 ||BM O
|aj (y)||Dβ2 Ã2 (y)|dy
n+1−δ
|x|
|x|
Rn
|β1 |=m1
|β2 |=m2
SOME HARDY SPACE ESTIMATES
45
X Z
|Bj |1/n
|Bj |ε/n
+C
||D A2 ||BM O
+
|aj (y)||Dβ1 Ã1 (y)|dy
|x|n+1−δ |x|n+ε−δ
n
|β2 |=m2
|β1 |=m1 R
Z
X
|Bj |1/n
|Bj |ε/n
+C
+
|aj (y)||Dβ1 Ã1 (y)||Dβ2 Ã2 (y)|dy
n+1−δ
n+ε−δ
|x|
|x|
n
|β1 |=m1 ,|β2 |=m2 R
j(1+n(1−1/q1 )−α)
2j(ε+n(1−1/q1 )−α)
2
+
≤C
.
|x|n+1−δ
|x|n+ε−δ
X
β2
To be simply, denote W (j, k) = 2(j−k)(1+n(1−1/q1 )−α) + 2(j−k)(ε+n(1−1/q1 )−α) , then
p !
j(1+n(1−1/q1 )−α)
∞
k−3
X
X
2j(ε+n(1−1/q1 )−α)
kαp
p 2
J ≤C
+
2knp/q2
2
|λj |
k(n+1−δ)
k(n+ε−δ)
2
2
j=−∞
k=−∞
P
( P∞
p
C j=−∞ |λj |p ∞
k=j+3 W (j, k) , 0 < p ≤ 1
h
i hP
ip/p0
P∞
P∞
≤
∞
p
p/2
p0 /2
C j=−∞ |λj |
, p>1
k=j+3 W (j, k)
k=j+3 W (j, k)
≤C
∞
X
|λj |p ≤ C||f ||pH K̇ α,p
q1 ,D m A
j=−∞
.
These yield the desired result and finish the proof of Theorem 2.
4. Examples
In this section we shall apply Theorem 1 and 2 of the paper to the CalderónZygmund singular integral operator.
Let T be the Calderón-Zygmund operator (see [2, 10, 15, 18], the multilinear
operator related to T is defined by
Z Ql
A
i=1 Rmi +1 (Ai ; x, y)
T (f )(x) =
K(x, y)f (y)dy.
|x − y|m
Rn
In particular, the multilinear commutator related to T is (see [11])
#
Z "Y
l
T A (f )(x) =
(Ai (x) − Ai (y)) K(x, y)f (y)dy.
Rn
i=1
Then it is easily to see that T satisfies the conditions in Theorem 1 and 2.
References
[1] J. Alvarez. Continuity properties for linear commutators of Calderón-Zygmund operators. Collect. Math., 49(1):17–31, 1998.
[2] J. Alvarez and M. Milman. H p continuity properties of Calderón-Zygmund-type operators. J. Math. Anal. Appl., 118(1):63–79, 1986.
[3] J. Cohen. A sharp estimate for a multilinear singular integral in Rn . Indiana Univ.
Math. J., 30(5):693–702, 1981.
[4] J. Cohen and J. Gosselin. A BMO estimate for multilinear singular integrals. Illinois
J. Math., 30(3):445–464, 1986.
46
ZHOU XIAOSHA AND LIU LANZHE
[5] J. Cohen and J. A. Gosselin. On multilinear singular integrals on Rn . Studia Math.,
72(3):199–223, 1982.
[6] R. R. Coifman, R. Rochberg, and G. Weiss. Factorization theorems for Hardy spaces
in several variables. Ann. of Math. (2), 103(3):611–635, 1976.
[7] Y. Ding and S. Z. Lu. Weighted boundedness for a class of rough multilinear operators.
Acta Math. Sin. (Engl. Ser.), 17(3):517–526, 2001.
[8] J. Garcı́a-Cuerva. Hardy spaces and Beurling algebras. J. London Math. Soc. (2),
39(3):499–513, 1989.
[9] J. Garcı́a-Cuerva and M.-J. L. Herrero. A theory of Hardy spaces associated to the Herz
spaces. Proc. London Math. Soc. (3), 69(3):605–628, 1994.
[10] J. Garcı́a-Cuerva and J. L. Rubio de Francia. Weighted norm inequalities and related
topics, volume 116 of North-Holland Mathematics Studies. North-Holland Publishing
Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104.
[11] S. Z. Lu and D. C. Yang. The decomposition of weighted Herz space on Rn and its
applications. Sci. China Ser. A, 38(2):147–158, 1995.
[12] S. Z. Lu and D. C. Yang. The weighted Herz-type Hardy space and its applications.
Sci. China Ser. A, 38(6):662–673, 1995.
[13] Y. Meyer and R. Coifman. Wavelets, volume 48 of Cambridge Studies in Advanced
Mathematics. Cambridge University Press, Cambridge, 1997. Calderón-Zygmund and
multilinear operators, Translated from the 1990 and 1991 French originals by David
Salinger.
[14] C. Pérez. Endpoint estimates for commutators of singular integral operators. J. Funct.
Anal., 128(1):163–185, 1995.
[15] C. Pérez and G. Pradolini. Sharp weighted endpoint estimates for commutators of
singular integrals. Michigan Math. J., 49(1):23–37, 2001.
[16] C. Pérez and R. Trujillo-González. Sharp weighted estimates for multilinear commutators. J. London Math. Soc. (2), 65(3):672–692, 2002.
[17] E. M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory
integrals, volume 43 of Princeton Mathematical Series. Princeton University Press,
Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III.
[18] A. Torchinsky. Real-variable methods in harmonic analysis, volume 123 of Pure and
Applied Mathematics. Academic Press Inc., Orlando, FL, 1986.
Received June 3, 2010.
College of Mathematics,
Changsha University of Science and Technology,
Changsha 410077,
P. R. of China
E-mail address: lanzheliu@163.com