Báo cáo toán học: "Weighted Estimates of Multilinear Singular Integral Operators with Variable Calder´n-Zygmund Kernel for the Extreme Cases" ppt

Vietnam Journal of Mathematics 34:1 (2006) 51–61
Weighted Estimates of Multilinear
Singular Integral Operators with Variable
Calder´on-Zygmund Kernel for the Extreme Cases
*
Liu Lanzhe
College of Math. and Compt., Changsha Univ. o f Sci. and Tech.
Changsha 410077, China
Received February 28, 2005
Abstract. The weighted endpoint estimates for the multilinear singular integral
operators with variable Calder´on-Zygmund kernel on some Hardy and Herz type Hardy
spaces are obtained.
1. Introduction
Let b ∈ BMO(R
n
)andT be the Calder´on-Zygmund operator. The commutator
[b, T] generated by b and T is defined by [b, T]f (x)=b(x)Tf(x) −T(bf)(x). By
a classical result of Coifman, Rochberg and Weiss(see [9]), we know that the
commutator [b, T] is bounded on L
p
(R
n
)for1<p<∞. In [13], the bound-
edness properties of the commutators for the extreme values of p are proved,
and in [3], the weak (H
1
, L
1
)-boundedness of the multilinear operator related
to some singular integral operator are obtained. In [2], Calder´on and Zygmund
introduce some singular integral operators with variable kernel and discuss their

boundedness. In [10], the authors obtain the boundedness for the commuta-
tors generated by the singular integral operators with variable kernel and BMO
functions. In [16], the authors prove the boundedness for the multilinear oscilla-
tory singular integral operators generated by the operators and BMO functions.
In recent years, the theory of Herz space and Herz type Hardy space, as a lo-
cal version of Lebesgue space and Hardy space, have been developed (see[11,
∗
Supported by the NNSF (Grant: 10271071).
52 Liu Lanzhe
14, 15]). The main purpose of this paper is to establish the weighted endpoint
continuity properties of the multilinear singular integral operators with variable
Calder´on-Zygmund kernel on Hardy and Herz type Hardy spaces.
2. Notations and Theorems
Throughout this paper, we denote the Muckenhoupt weights by A
p
for 1 
p<∞ (see [12]). Q will denote a cube of R
n
with sides parallel to the axes.
For a cube Q and a locally integrable function f ,letf(Q)=

Q
f(x)dx, f
Q
=
|Q|
−1

Q
f(x)dx and f

#
(x)=sup
x∈Q
|Q|
−1

Q
|f(y) − f
Q
|dy.Moreover,f is said to
belong to BMO(R
n
)iff
#
∈ L
∞
(R
n
) and define that ||f ||
BMO
= ||f
#
||
L
∞
.
Also, we give the concepts of the atom and weighted H
1
space. A function a
is called a H

1
(w) atom if there exists a cube Q such that a is supported on Q,
||a||
L
∞
(w)
 w(Q)
−1
and

R
n
a(x)dx = 0. It is well known that the weighted
Hardy space H
1
(w) has the atomic decomposition characterization (see [1, 12]).
For k ∈ Z, define B
k
= {x ∈ R
n
: |x|  2
k
} and C
k
= B
k
\B
k−1
.Denoteby
χ

k
the characteristic function of C
k
and ˜χ
k
the characteristic function of C
k
for
k ≥ 1and˜χ
0
the characteristic function of B
0
.
Definition 1. Let 1 <p<∞ and w
1
, w
2
be two non-negative weight functions
on R
n
.
(1) The homogeneous weighted Herz space is define d by
˙
K
p
(w
1
,w
2
; R

n
)={f ∈ L
p
loc
(R
n
\{0}):f
˙
K
p
(w
1
,w
2
)
< ∞} ,
where
f
˙
K
p
(w
1
,w
2
)
=
∞

k=−∞

[w
1
(B
k
)]
1−1/p
fχ
k

L
p
(w
2
)
;
(2) The nonhomogeneous weighted Herz space is defined by
K
p
(w
1
,w
2
; R
n
)={f ∈ L
p
loc
(R
n
):f

K
p
(w
1
,w
2
)
< ∞} ,
where
f
K
p
=
∞

k=0
[w
1
(B
k
)]
1−1/p
f ˜χ
k

L
p
(w
2
)

;
(3) The homogeneous weighted Herz type Hardy sp a ce is define d by
H
˙
K
p
(w
1
,w
2
; R
n
)={f ∈ S

(R
n
):G(f) ∈
˙
K
p
(w
1
,w
2
; R
n
)},
where
||f||
H

˙
K
p
(w
1
,w
2
)
= ||G(f)||
˙
K
p
(w
1
,w
2
)
;
(4) The nonhomogeneous weighted Herz type Hardy space is defined by
Weighted Estimates of Multilinear Singular Integral Operators 53
HK
p
(w
1
,w
2
; R
n
)={f ∈ S


(R
n
):G(f) ∈ K
p
(w
1
,w
2
; R
n
)},
where
f
HK
p
(w
1
,w
2
)
= G(f )
K
p
(w
1
,w
2
)
and G(f) is the grand maximal function of f .
TheHerztypeHardyspaceshavetheatomic decomposition characteriza-

tion.
Definition 2. Let 1 <p<∞ and w
1
,w
2
∈ A
1
.Afunctiona(x) on R
n
is called
acentral(n(1 −1/p),p; w
1
,w
2
)-atom (or a central (n(1 −1/p),p; w
1
,w
2
)-atom
of restrict type), if
(1) Supp a ⊂ B(0,r)forsomer>0 (or for some r ≥ 1);
(2) ||a||
L
p
(w
2
)
 [w
1
(B(0,r))]

1/p−1
,
(3)

R
n
a(x)dx =0.
Lemma 1. (see [11, 15]) Let w
1
,w
2
∈ A
1
and 1 <p<∞. A temperate distri-
bution f belongs to H
˙
K
p
(w
1
,w
2
; R
n
)(or HK
p
(w
1
,w
2

; R
n
))ifandonlyifthere
exist centr al (n(1−1/p),p; w
1
,w
2
)-atoms (or central (n(1−1/p),p; w
1
,w
2
)-atoms
of restrict typ e) a
j
supported on B
j
= B(0, 2
j
) and constants λ
j
,

j
|λ
j
| < ∞
such that f =

∞
j=−∞

λ
j
a
j
(or f =

∞
j=0
λ
j
a
j
)intheS

(R
n
) sense, and
f
H
˙
K
p
(w
1
,w
2
)
( or f 
HK
p

(w
1
,w
2
)
) ≈

j
|λ
j
|.
In this paper, we will study a class of multilinear operators related to the
singular integral operators with variable kernel, whose definitions are following.
Definition 3. Let k(x)=Ω(x)/|x|
n
: R
n
\{0}−→R. k is said to b e a
Calder´on-Zygmund kernel if
(a) Ω ∈ C
∞
(R
n
\{0});
(b) Ω is homogeneous of degree zero;
(c)

Σ
Ω(x)x
α

dσ(x)=0for all multi-indices α ∈ (N

{0})
n
with |α| = N,
where Σ={x ∈ R
n
: |x| =1} is the unit spher e of R
n
.
Definition 4. Let k(x, y)=Ω(x, y)/|y|
n
: R
n
× (R
n
\{0}) −→ R. k is said to
be a variable Calder´on-Zygmund kernel if
(d) k(x, ·) is a Calder´on-Zygmund kernel for a.e. x ∈ R
n
;
(e) max
|γ|2n






∂

|γ|
∂
γ
y
Ω(x, y)






L
∞
(R
n
×Σ)
= M<∞.
Let m be a positive integer and A be a function on R
n
.Set
R
m+1
(A; x, y)=A(x) −

|α|m
1
α!
D
α
A(y)(x −y)

α
and
54 Liu Lanzhe
Q
m+1
(A; x, y)=R
m
(A; x, y) −

|α|=m
1
α!
D
α
A(x)(x −y)
α
.
The multiline ar singular integral operators with variable Calder´on-Zygmund
kernel are defined by
˜
T
A
(f)(x)=

R
n
Ω(x, x − y)
|x − y|
n+m
Q

m+1
(A; x, y)f(y)dy
and
T
A
(f)(x)=

R
n
Ω(x, x − y)
|x − y|
n+m
R
m+1
(A; x, y)f(y)dy,
where Ω(x, y)/|y|
n
is a variable Calder´on-Zygmund kernel. We also define
T (f)(x)=

R
n
Ω(x, x − y)
|x − y|
n
f(y)dy,
which is the singular integral operator with variable Calder´on-Zygmund kernel
(see [2]).
Note that when m =0,T
A

is just the commutator of T and A (see [10]).
While when m>0, T
A
is the non-trivial generalizations of the commutator.
From [16], we know that T
A
is bounded on L
p
(w)for1<p ∞ and w ∈ A
1
.
In this paper, we will study the weighted endpoint continuity properties of the
multilinear operators
˜
T
A
on Hardy and Herz type Hardy spaces.
We shall prove the following theorems in Sec. 3.
Theorem 1. Let w ∈ A
1
and D
α
A ∈ BMO(R
n
) for all α with |α| = m.Then
˜
T
A
is bounded from H
1

(w) to L
1
(w).
Theorem 2. Let 1 <p<∞, w
1
,w
2
∈ A
1
and D
α
A ∈ BMO(R
n
) for all α with
|α| = m.Then
˜
T
A
is bounded fr om
˙
HK
p
(w
1
,w
2
; R
n
)(resp. HK
p

(w
1
,w
2
; R
n
))
to
˙
K
p
(w
1
,w
2
; R
n
)(resp. HK
p
(w
1
,w
2
; R
n
)).
3. Proofs of Theorems
To prove the theorems, we need the following lemma.
Lemma 2. (see [7]) Let A be a function on R
n

and D
α
A ∈ L
q
(R
n
) for |α| = m
and some q>n.Then
|R
m
(A; x, y)|  C|x − y|
m

|α|=m

1
|
˜
Q(x, y)|

˜
Q(x,y)
|D
α
A(z)|
q
dz

1/q
,

Weighted Estimates of Multilinear Singular Integral Operators 55
where
˜
Q(x, y) isthecubecenteredatx and having side length 5
√
n|x − y|.
Proof of The orem 1. It suffices to show that there exists a constant C>0such
that for every H
1
(w)-atom a (that is that a satisfies: suppa ⊂ Q = Q(x
0
,r),
||a||
L
∞
(w)
 w(Q)
−1
and

a(y)dy = 0 (see [1])), the following holds:
||
˜
T
A
(a)||
L
1
(w)
 C.

Without loss of generality, we may assume l =2. Write

R
n
˜
T
A
(a)(x)w(x)dx =


2Q
+

(2Q)
c

˜
T
A
(a)(x)w(x)dx := I
1
+ I
2
.
For I
1
, by the following equality
Q
m+1
(A; x, y)=R

m+1
(A; x, y)+

|α|=m
1
α!
(x − y)
α
(D
α
A(x) − D
α
A(y)),
we get
|
˜
T
A
(a)(x)|  |T
A
(a)(x)| + C

|α|=m
|[D
α
A, T ]a(x)|,
thus,
˜
T
A

is L
p
(w)-bounded for 1 <p ∞ (see[10, 16]), we see that
I
1
 C||
˜
T
A
(a)||
L
∞
(w)
w(2Q)  C||a||
L
∞
(w)
w(Q)  C.
To obtain the estimate of I
2
, we need to estimate
˜
T
A
(a)(x)forx ∈ (2Q)
c
.
Denote
˜
A(x)=A(x) −


|α|=m
1
α!
(D
α
A)
2Q
x
α
,thenD
α
˜
A = D
α
A − (D
α
A)
2Q
for |α| = m, Q
m
(A; x, y)=Q
m
(
˜
A; x, y)andQ
m+1
(A; x, y)=R
m
(A; x, y) −


|α|=m
1
α!
D
α
A(x)(x − y)
α
. By [4, 10], we know that
˜
T
A
(f)(x)=
∞

k=1
g
k

h=1
a
hk
(x)

R
n
Y
hk
(x −y)
|x − y|

n+m
Q
m+1
(A; x, y)f(y)dy
:=
∞

k=1
g
k

h=1
a
hk
(x)S
A
hk
(f)(x),
where g
k
 Ck
n−2
, ||a
hk
||
L
∞
 Ck
−2n
, |Y

hk
(x − y)|  Ck
n/2−1
and



Y
hk
(x −y)
|x − y|
n
−
Y
hk
(x − x
0
)
|x − x
0
|
n



 Ck
n/2
|x
0
− y|/|x − x

0
|
n+1
for |x −x
0
| > 2|x
0
− y| > 0. we write, by the vanishing moment of a and for
x ∈ (2Q)
c
,
56 Liu Lanzhe
S
A
hk
(a)(x)=

R
n

Y
hk
(x − y)
|x − y|
m+n
−
Y
hk
(x −x
0

)
|x − x
0
|
m+n

R
m
(
˜
A; x, y)a(y)dy
+

R
n
Y
hk
(x − x
0
)
|x − x
0
|
m+n
[R
m
(
˜
A; x, y) −R
m

(
˜
A; x, x
0
)]a(y)dy
− C

|α|=m

R
n

Y
hk
(x − y)(x −y)
α
|x − y|
m+n
−
Y
hk
(x − x
0
)(x − x
0
)
α
|x − x
0
|

m+n

D
α
˜
A(x)a(y)dy
= I
(1)
2
(x)+I
(2)
2
(x)+I
(3)
2
(x);
For I
(1)
2
(x), by Lemma 1 and the following inequality (see [17])
|b
Q
1
− b
Q
2
|  C log(|Q
2
|/|Q
1

|)||b||
BMO
for Q
1
⊂ Q
2
,
we know that, for x ∈ Q and y ∈ 2
j+1
Q \ 2
j
Q(j ≥ 1),
|R
m
(
˜
A; x, y)|  C|x − y|
m

|α|=m
(||D
α
A||
BMO
+ |(D
α
A)
2Q(x,y)
− (D
α

A)
2Q
|)
 Cj|x −y|
m

|α|=m
||D
α
A||
BMO
,
note that |x −y|∼|x −x
0
| for y ∈ Q and x ∈ R
n
\ 2Q,then
|I
(1)
2
(x)|  Ck
n/2

R
n
|y − x
0
|
|x − x
0

|
m+n+1
|R
m
(
˜
A; x, y)||a(y)|dy
 Ck
n/2

|α|=m
||D
α
A||
BMO
j

Q
|y − x
0
|
|x − x
0
|
n+1
|a(y)|dy
 Ck
n/2

|α|=m

||D
α
A||
BMO
j
|Q|
1/n+1
|x − x
0
|
n+1
w(Q)
−1
.
For I
(2)
2
(x), by the formula (see [7]):
R
m
(
˜
A; x, y) −R
m
(
˜
A; x, x
0
)=


|β|<m
1
β!
R
m−|β|
(D
β
˜
A; x, x
0
)(x −y)
β
and Lemma 1, we have
|R
m
(
˜
A; x, y) −R
m
(
˜
A; x, x
0
)|  C

|β|<m

|α|=m
|x − x
0

|
m−|β|
|x −y|
|β|
D
α
A
BMO
,
then
Weighted Estimates of Multilinear Singular Integral Operators 57
|I
(2)
2
(x)|  Ck
n/2

|α|=m
D
α
A
BMO
j

Q
|y − x
0
|
|x − x
0

|
n+1
|a(y)|dy
 Ck
n/2

|α|=m
D
α
A
BMO
j
|Q|
1/n+1
|x − x
0
|
n+1
w(Q)
−1
.
Similarly,
|I
(3)
2
(x)|  Ck
n/2

|α|=m
|Q|

1/n+1
|x − x
0
|
n+1
w(Q)
−1
|D
α
˜
A(x)|;
Thus, for x ∈ 2
j+1
Q \ 2
j
Q(j ≥ 1),
|
˜
T
A
(a)(x)|  C
∞

k=1
g
k

h=1
|a
hk

(x)|k
n/2

|α|=m
D
α
A
BMO
j
|Q|
1/n+1
|x − x
0
|
n+1
w(Q)
−1
+ C
∞

k=1
g
k

h=1
|a
hk
(x)|k
n/2


|α|=m
|Q|
1/n+1
|x − x
0
|
n+1
w(Q)
−1
|D
α
˜
A(x)|
 C
∞

k=1
k
−2n+n/2+n−2

|α|=m
D
α
A
BMO
j
|Q|
1/n+1
|x − x
0

|
n+1
w(Q)
−1
+ C
∞

k=1
k
−2n+n/2+n−2

|α|=m
|Q|
1/n+1
|x − x
0
|
n+1
w(Q)
−1
|D
α
˜
A(x)|
 C

|α|=m
||D
α
A||

BMO
j
|Q|
1/n+1
|x − x
0
|
n+1
w(Q)
−1
+ C

|α|=m
|Q|
1/n+1
|x − x
0
|
n+1
w(Q)
−1
|D
α
˜
A(x)|;
Notice that if w ∈ A
1
,then
w(Q
2

)
|Q
2
|
|Q
1
|
w(Q
1
)
 C
for all cubes Q
1
,Q
2
with Q
1
⊂ Q
2
,andw satisfies the reverse of H¨older’
inequality:

1
|Q|

Q
w(x)
q
dx


1/q

C
|Q|

Q
w(x)dx
for all cube Q and some 1 <q<∞(see[12]). Thus, by H¨older’s inequality, we
obtain
58 Liu Lanzhe
I
2

∞

j=1

2
j+1
Q\2
j
Q
|
˜
T
A
(a)(x)|dx
 C

|α|=m

D
α
A
BMO
∞

j=1
j2
−j
|Q|
w(Q)
w(2
j+1
Q)
|2
j+1
Q|
+ C

|α|=m
∞

j=1
2
−j
|Q|
w(Q)

1
|2

j+1
Q|

2
j+1
Q
|D
α
˜
A(x)|
q

dx

1/q

×

1
|2
j+1
Q|

2
j+1
Q
w(x)
q
dx


1/q
 C

|α|=m
D
α
A
BMO
∞

j=1
j2
−j
w(2
j+1
Q)
|2
j+1
Q|
|Q|
w(Q)
 C

|α|=m
D
α
A
BMO
.
This completes the proof of Theorem 1.


Proof of Theorem 2. We only give the proof for case of homogeneous Herz
type Hardy space. Without loss of generality, we may assume l =2. Let
f ∈ H
˙
K
p
(w
1
,w
2
; R
n
), by Lemma 1, f =

∞
j=−∞
λ
j
a
j
,wherea

j
s are the central
(n(1 −1/p),p; w
1
,w
2
)-atom with suppa

j
⊂ B
j
= B(0, 2
j
)and||f||
H
˙
K
p
(w
1
,w
2
)
≈

j
|λ
j
|.Write
||
˜
T
A
(f)||
˙
K
p
(w

1
,w
2
)
=
∞

k=−∞
[w
1
(B
k
)]
1−1/p
||χ
k
˜
T
A
(f)||
L
p
(w
2
)

∞

k=−∞
[w

1
(B
k
)]
1−1/p
k−1

j=−∞
|λ
j
|||χ
k
˜
T
A
(a
j
)||
L
p
(w
2
)
+
∞

k=−∞
[w
1
(B

k
)]
1−1/p
∞

j=k
|λ
j
|||χ
k
˜
T
A
(a
j
)||
L
p
(w
2
)
= J
1
+ J
2
.
For J
2
,bytheL
p

(w)-boundedness of
˜
T
A
for 1 <p<∞ and w ∈ A
1
,weget
J
2
 C
∞

k=−∞
[w
1
(B
k
)]
1−1/p
∞

j=k
|λ
j
|||a
j
||
L
p
(w

2
)
 C
∞

k=−∞
[w
1
(B
k
)]
1−1/p
∞

j=k
|λ
j
|[w
1
(B
j
)]
−(1−1/p)
 C
∞

j=−∞
|λ
j
|

j

k=−∞

w
1
(B
k
)
w
1
(B
j
)

1−1/p
 C
∞

j=−∞
|λ
j
|  C||f||
H
˙
K
p
(w
1
,w

2
)
.
Weighted Estimates of Multilinear Singular Integral Operators 59
To estimate J
1
,wedenotethat
˜
A(x)=A(x) −

|α|=m
1
α!
(D
α
A)
2Q
x
α
.Then
Q
m
(A; x, y)=Q
m
(
˜
A; x, y). Similar to the proof of Theorem 1, we know
˜
T
A

(a
j
)(x)=
∞

k=1
g
k

h=1
a
hk
(x)

R
n
Y
hk
(x −y)
|x − y|
n+m
Q
m+1
(A; x, y)a
j
(y)dy
=
∞

k=1

g
k

h=1
a
hk
(x)S
A
hk
(a
j
)(x),
and write, by the vanishing moment of a
j
,forx ∈ (2Q)
c
,
S
A
hk
(a
j
)(x)=

R
n

Y
hk
(x −y)

|x − y|
m+n
−
Y
hk
(x)
|x|
m+n

R
m
(
˜
A; x, y)a
j
(y)dy
+

R
n
Y
hk
(x)
|x|
m+n
[R
m
(
˜
A; x, y) − R

m
(
˜
A; x, 0)]a(y)dy
− C

|α|=m

R
n

Y
hk
(x − y)(x −y)
α
|x − y|
m+n
−
Y
hk
(x)x
α
|x|
m+n

D
α
˜
A(x)a
j

(y)dy;
Similar to the proof of Theorem 1, we obtain
|
˜
T
A
(a
j
)(x)|  C

|α|=m
||D
α
A||
BMO
2
j
2
k(n+1)

B
j
|a
j
(y)|dy
+ C

|α|=m
|D
α

˜
A(x)|
2
j
2
k(n+1)

B
j
|a
j
(y)|dy
 C

|α|=m
||D
α
A||
BMO
2
j
2
k(n+1)
||a
j
||
L
p
(w
2

)


B
j
w
2
(y)
−1/(p−1)
dy

(p−1)/p
+ C

|α|=m
|D
α
˜
A(x)|
2
j
2
k(n+1)
||a
j
||
L
p
(w
2

)


B
j
w
2
(y)
−1/(p−1)
dy

(p−1)/p
 C

|α|=m
||D
α
A||
BMO
2
j
2
k(n+1)
[w
1
(B
j
)]
1/p−1



B
j
w
2
(y)
−1/(p−1)
dy

(p−1)/p
+ C

|α|=m
2
j
2
k(n+1)
|D
α
˜
A(x)|[w
1
(B
j
)]
1/p−1


B
j

w
2
(y)
−1/(p−1)
dy

(p−1)/p
.
Notice that w
2
∈ A
1
⊆ A
p
, w
2
satisfies
sup
Q

1
|Q|

Q
w
2
(x)dx

1
|Q|


Q
w
2
(x)
−1/(p−1)
dx

(p−1)/p
 C
60 Liu Lanzhe
and the reverse of H¨older’ inequality for some 1 <q<∞ (see [12]), thus
J
1
 C
∞

k=−∞
[w
1
(B
k
)]
1−1/p
k−1

j=−∞
|λ
j
|

2
j
2
k(n+1)
[w
1
(B
j
)]
−(1−1/p)
×


B
j
w
2
(y)
−1/(p−1)
dy

(p−1)/p
×

[w
2
(B
k
)]
1/p

+

|α|=m


B
k
|D
α
˜
A(x)|
p
w
2
(x)dx

1/p

 C
∞

k=−∞
[w
1
(B
k
)]
1−1/p
k−1


j=−∞
|λ
j
|
2
j
2
k(n+1)
[w
1
(B
j
)]
−(1−1/p)
×


B
j
w
2
(y)
−1/(p−1)
dy

(p−1)/p

[w
2
(B

k
)]
1/p
+

|α|=m

1
|B
k
|
×

B
k
|D
α
˜
A(x)|
pq

dx

1/pq


1
|B
k
|


B
k
w
2
(x)
q
dx

1/pq
|B
k
|
1/p

 C
∞

k=−∞
k−1

j=−∞
|λ
j
|
2
j
2
k(n+1)


w
1
(B
k
)
w
1
(B
j
)

1−1/p
×


B
j
w
2
(x)
−1/(p−1)
dx

(p−1)/p
[w
2
(B
k
)]
1/p

 C
∞

k=−∞
k−1

j=−∞
|λ
j
|
2
j
2
k(n+1)

w
1
(B
k
)
w
1
(B
j
)

1−1/p
×

w

2
(B
k
)
w
2
(B
j
)

1/p
×|B
j
|

1
|B
j
|

B
j
w
2
(x)dx

1/p

1
|B

j
|

B
j
w
2
(y)
−1/(p−1)
dy

(p−1)/p
 C
∞

j=−∞
|λ
j
|
∞

k=j+1
2
j
2
k(n+1)

w
1
(B

k
)
w
1
(B
j
)
|B
j
|
|B
k
|

1−1/p
×

w
2
(B
k
)
w
2
(B
j
)
|B
j
|

|B
k
|

1/p
|B
k
|
 C
∞

j=−∞
|λ
j
|
∞

k=j+1
2
j−k
 C
∞

j=−∞
|λ
j
|  C||f||
H
˙
K

p
(w
1
,w
2
)
.
This completes the proof of Theorem 2.

Weighted Estimates of Multilinear Singular Integral Operators 61
References
1. Bui Huy Qui, Weighted Hardy spaces, Math. Nachr. 103 (1981) 45–62.
2. A. P. Calder´on and A. Zygmund, On singular integrals with variable kernels,
Appl. Anal. 7 (1978) 221–238.
3. W. G. Chen and G. E. Hu, Weak type (
H
1
, L
1
) estimate for multilinear singular
integral operator, Adv. in Math. 30 (2001) 63–69 (Chinese).
4. F. Chiarenza, M. Frasca, and P. Longo, Interior
W
2,p
-estimates for nondivergence
elliptic equations with discontinuous coefficients, Ricer che Mat. 40 (1991) 149–
168.
5. J. Cohen, A sharp estimate for a multilinear singular integral on
R
n

, Indiana
Univ. Math. J. 30 (1981) 693–702.
6. J. Cohen and J. Gosselin, On multilinear singular integral operators on
R
n
,
Studia Math. 72 (1982) 199–223.
7. J. Cohen and J. Gosselin, A BMO estimate for multilinear singular integral op-
erators, Il linois J. Math. 30 (1986) 445–465.
8. R. Coifman and Y. Meyer, Wavelets, Calder´on-Zygmund and multilinear oper-
ators, Cambridge Studies in Advanced Math. 48, Cambridge University Press,
Cambridge, 1997.
9. R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces
in several variables, Ann. of Math. 103 (1976) 611–635.
10. G. Di Fazio and M. A. Ragusa, Interior estimates in Morrey spaces for strong so-
lutions to nondivergence form equations with discontinuous coefficients, J. Func.
Anal. 112 (1993) 241–256.
11. J. Garcia-Cuerva and M. L. Herrero, A theory of Hardy spaces associated to the
Herz spaces, Proc. London Math. Soc. 69 (1994) 605–628.
12. J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and
Related Topics, North-Holland Math.16, Amsterdam, 1985.
13. E. Harboure, C. Segovia, and J. L.Torra, Boundedness of commutators of frac-
tional and singular integrals for the extreme values of
p, Illinois J. Math. 41
(1997) 676–700.
14. S. Z. Lu and D. C.Yang, The decomposition of the weighted Herz spaces and its
applications, Sci. in China (ser. A) 38 (1995) 147–158.
15. S. Z.Lu and D. C.Yang, The weighted Herz type Hardy spaces and its applications,
Sci. in China (ser.A) 38 (1995) 662–673.
16. S. Z. Lu, D. C.Yang, and Z. S. Zhou, Oscillatory singular integral operators with

Calder´on-Zygmund kernels, Southeast Asian Bull . of Math. 23 (1999) 457–470.
17. E. M.Stein, Harmonic Analysis: Real Variable Methods, Orthogonality and Os-
cillatory Integrals, Princeton Univ. Press, Princeton NJ, 1993.