# Boundedness of Singular Integral Operators with Variable Kernels on Weighted Weak Hardy Spaces

Authors: Hua Wang
Journal: Chinese Journal of Mathematics
 (2014)

---

## Abstract

LetT
Ω be the singular integral operator with variable kernel Ω
(
x
,
z
). In this paper, by using the atomic decomposition theory of weighted weak Hardy spaces, we will obtain the boundedness properties of T
Ω on these spaces, under some Dini type conditions imposed on the variable kernel Ω
(
x
,
z
).


---

## Body

## 1. Introduction

LetS
n
-
1 be the unit sphere in ℝ
n (n
≥
2) equipped with the normalized Lebesgue measure d
σ. A function Ω
(
x
,
z
) defined on ℝ
n
×
ℝ
n is said to belong to L
∞
(
ℝ
n
)
×
L
r
(
S
n
-
1
), r
≥
1, if it satisfies the following conditions:(A)
for allλ
>
0 and x
,
z
∈
ℝ
n, Ω
(
x
,
λ
z
)
=
Ω
(
x
,
z
);
(B)
for anyx
∈
ℝ
n, ∫
S
n
-
1
‍
Ω
(
x
,
z
′
)
d
σ
(
z
′
)
=
0;
(C)
∥
Ω
∥
L
∞
(
ℝ
n
)
×
L
r
(
S
n
-
1
)

:
=

sup
⁡
x
∈
ℝ
n
⁡
(
∫
S
n
-
1
‍
|
Ω
(
x
,
z
′
)
|
r
d
σ
(
z
′
)
)
1
/
r
<
∞,wherez
′
=
z
/
|
z
| for any z
∈
ℝ
n
∖
{
0
}. Set K
(
x
,
z
)
=
Ω
(
x
,
z
′
)
/
|
z
|
n. In this paper, we consider the singular integral operator with variable kernel which is defined by
(1)
T
Ω
f
(
x
)
=
P
.
V
.
∫
ℝ
n
‍
K
(
x
,
x
-
y
)
f
(
y
)
d
y
.In [1, 2], Calderón and Zygmund investigated the L
p boundedness of singular integral operators with variable kernels. They found that these operators T
Ω are closely related to the problem about second order elliptic partial differential equations with variable coefficients. We will denote the conjugate exponent of p
>
1 by p
′
=
p
/
(
p
-
1
). In [3], Calderón and Zygmund proved the following theorem.Theorem A (see [3]).
Let1
<
p, r
<
∞ satisfy(i)
1
/
r
<
(
1
/
p
′
)
+
(
1
/
p
′
(
n
-
1
)
) if 1
<
p
≤
2, or
(ii)
1
/
r
<
(
1
/
p
′
)
+
(
1
/
p
(
n
-
1
)
) if 2
≤
p
<
∞.
Suppose thatΩ
(
x
,
z
)
∈
L
∞
(
ℝ
n
)
×
L
r
(
S
n
-
1
). Then there exists a constant C
>
0 independent of f such that
(2)
∥
T
Ω
(
f
)
∥
L
p
≤
C
∥
f
∥
L
p
.
In particular, T
Ω is bounded on L
p
(
ℝ
n
) for all p
≥
r
′.In 1971, Muckenhoupt and Wheeden [4] studied the weighted norm inequalities for T
Ω with power weights. In 2008, Lee et al. [5] also considered the weighted boundedness of T
Ω with more general weights and showed that if the kernel K
(
x
,
y
) satisfies the L
r-Hörmander condition with respect to x and y variables, respectively, then T
Ω is bounded on L
w
p
(
ℝ
n
). More precisely, they proved the following.Theorem B (see [5]).
Let1
<
r
<
∞. Suppose that Ω
(
x
,
z
)
∈
L
∞
(
ℝ
n
)
×
L
r
(
S
n
-
1
) such that the following two inequalities
(3)
sup
⁡
x
∈
ℝ
n
0
<
|
y
|
<
R
⁡
∑
k
=
1
∞
(

2
k
R

)
n
/
r
′

(

∫
2
k
R
≤
|
z
|
<
2
k
+
1
R

|
K
(
x
,
z

-

y
)

-

K
(
x
,
z
)
|
r
d
z

)
1
/
r
<
∞
,
sup
⁡
x
,
y
∈
ℝ
n
0
<
|
x
-
y
|
<
R
∑
k
=
1
∞
(
2
k
R
)
n
/
r
′

(

∫
2
k
R
≤
|
z
|
<
2
k
+
1
R

|
K
(
x
,
z
)

-

K
(
y
,
z
)
|
r
d
z

)
1
/
r
<
∞
hold for all R
>
0. If r
′
≤
p
<
∞ and w
∈
A
p
/
r
′, then T
Ω is bounded on L
w
p
(
ℝ
n
).It should be pointed out that the aboveL
r-Hörmander conditions on the variable kernels were also considered by Rubio de Francia et al. in [6].In [7, 8], Ding et al. introduced some definitions about the variable kernel Ω
(
x
,
z
) when they studied the H
1
-
L
1 boundedness of Marcinkiewicz integral. Replacing the condition (C) mentioned above, they strengthened it to the condition(
C
′
)
sup
⁡
x
∈
ℝ
n
ρ
≥
0
(
∫
S
n
-
1
‍
|
Ω
(
x
+
ρ
z
′
,
z
′
)
|
r
d
σ
(
z
′
)
)
1
/
r
<
∞.Forr
≥
1, a function Ω
(
x
,
z
) is said to satisfy the L
r-Dini condition if the conditions (A), (B), and (
C
′
) hold and
(4)
∫
0
1
‍
ω
r
(
δ
)
δ
d
δ
<
∞
,
where ω
r
(
δ
) is the integral modulus of continuity of order r of Ω defined by
(5)
ω
r
(
δ
)
≔
sup
⁡
x
∈
ℝ
n
ρ
≥
0
(
∫
S
n
-
1
sup
⁡
y
′
∈
S
n
-
1
|
y
′
-
z
′
|
≤
δ
|
Ω
(
x
+
ρ
z
′
,
y
′
)
222222222222222222222
-
Ω
(
x
+
ρ
z
′
,
z
′
)
|
r
d
σ
(
z
′
)
∫
S
n
-
1
sup
⁡
|
y
′
-
z
′
|
≤
δ
y
′
∈
S
n
-
1
|
Ω
(
x
+
ρ
z
′
,
y
′
)

)
1
/
r
.In order to obtain theH
w
p
-
L
w
p boundedness of T
Ω, Lee et al. [5] generalized the L
r-Dini condition by replacing (4) to the following stronger condition (see also [9]):
(6)
∫
0
1
‍
ω
r
(
δ
)
δ
1
+
α
d
δ
<
∞
,
0
≤
α
≤
1
.IfΩ satisfies (6) for some r
≥
1 and 0
≤
α
≤
1, we say that it satisfies the L
r
,
α-Dini condition. For the special case α
=
0, it reduces to the L
r-Dini condition. For 0
≤
β
<
α
≤
1, if Ω satisfies the L
r
,
α-Dini condition, then it also satisfies the L
r
,
β-Dini condition. We thus denote by Di
n
α
r
(
S
n
-
1
) the class of all functions which satisfy the L
r
,
β-Dini condition for all 0
<
β
<
α.Theorem C (see [5]).
Let0
<
α
≤
1 and n
/
(
n
+
α
)
<
p
<
1. Suppose Ω
∈
Din
α
r
(
S
n
-
1
) such that (3) hold for a certain large number r
>
1. If w
r
′
∈
A
(
p
+
(
p
α
/
n
)
-
(
1
/
r
)
)
r
′, then there exists a constant C
>
0 independent of f such that
(7)
∥
T
Ω
(
f
)
∥
L
w
p
≤
C
∥
f
∥
H
w
p
.It is easy to check that(8)
Di
n
α
r
(
S
n
-
1
)
⊂
Di
n
α
s
(
S
n
-
1
)
,
if
1
≤
s
<
r
<
∞
.
Then for 0
<
α
≤
1, we define
(9)
Di
n
α
∞
(
S
n
-
1
)
=
⋂
r
≥
1
‍
Di
n
α
r
(
S
n
-
1
)
.The main purpose of this paper is to study the corresponding estimates ofT
Ω on the weighted weak Hardy spaces W
H
w
p
(
ℝ
n
) (see Section 2 for its definition). We now present our main result as follows.Theorem 1.
Let0
<
α
≤
1, n
/
(
n
+
α
)
<
p
≤
1, and w
∈
A
p
(
1
+
(
α
/
n
)
). Suppose Ω
∈
Din
α
∞
(
S
n
-
1
) such that (3) hold. Then there exists a constant C
>
0 independent of f such that
(10)
∥
T
Ω
(
f
)
∥
W
L
w
p
≤
C
∥
f
∥
W
H
w
p
.


## 2. Notations and Preliminaries

The definition ofA
p class was first used by Muckenhoupt [10], Hunt et al. [11], and Coifman and Fefferman [12] in the study of weighted L
p boundedness of Hardy-Littlewood maximal functions and singular integrals. Let w be a nonnegative, locally integrable function defined on ℝ
n; all cubes are assumed to have their sides parallel to the coordinate axes. We say that w
∈
A
p, 1
<
p
<
∞, if
(11)
(
1
|
Q
|
∫
Q
‍
w
(
x
)
d
x
)
(
1
|
Q
|
∫
Q
‍
w
(
x
)
-
1
/
(
p
-
1
)
d
x
)
p
-
1
≤
C
,
2222222222222
l
22222222222
for
every
cube
Q
⊆
ℝ
n
,
where C is a positive constant which is independent of the choice of Q. For the case p
=
1, w
∈
A
1, if
(12)
1
|
Q
|
∫
Q
‍
w
(
x
)
d
x
≤
C
·
ess
inf
⁡
x
∈
Q
w
(
x
)
,
2222222
l
222
for
every
cube
Q
⊆
ℝ
n
.
The smallest value of C
>
0 such that the above inequalities hold is called the A
p characteristic constant of w and denoted by [
w
]
A
p. For the case p
=
∞, w
∈
A
∞, if it satisfies the A
p condition for some 1
<
p
<
∞.A weight functionw is said to belong to the reverse Hölder class R
H
s if there exist two constants s
>
1 and C
>
0 such that the following reverse Hölder inequality holds:
(13)
(
1
|
Q
|
∫
Q
‍
w
(
x
)
s
d
x
)
1
/
s
≤
C
(
1
|
Q
|
∫
Q
‍
w
(
x
)
d
x
)
,
2222222222222222222
for
every
cube
Q
⊆
ℝ
n
.It is well known that ifw
∈
A
p with 1
<
p
<
∞, then w
∈
A
r for all r
>
p and w
∈
A
q for some 1
<
q
<
p. We thus write q
w
≡
inf
⁡
{
q
>
1
:
w
∈
A
q
} to denote the critical index of w. Moreover, if w
∈
A
p with 1
≤
p
<
∞, then there exists s
>
1 such that w
∈
R
H
s. It follows directly from Hölder’s inequality that w
∈
R
H
r implies w
∈
R
H
s for all 1
<
s
<
r.Given a cubeQ and λ
>
0, λ
Q stands for the cube with the same center as Q whose side length is λ times that of Q. Q
=
Q
(
x
0
,
r
) denotes the cube centered at x
0 with side length r. For a weight function w and a measurable set E, we denote the Lebesgue measure of E by |
E
| and set the weighted measure w
(
E
)
=
∫
E
‍
w
(
x
)
d
x.We give the following results that will be used in the sequel.Lemma 2 (see [13]).
Letw
∈
A
q with q
≥
1. Then, for any cube Q, there exists an absolute constant C
>
0 such that
(14)
w
(
2
Q
)
≤
C
w
(
Q
)
.
In general, for any λ
>
1, one has
(15)
w
(
λ
Q
)
≤
C
·
λ
n
q
w
(
Q
)
,
where C does not depend on Q or λ.Given a weight functionw on ℝ
n, for 0
<
p
<
∞, we denote by L
w
p
(
ℝ
n
) the weighted space of all functions f satisfying
(16)
∥
f
∥
L
w
p
=
(
∫
ℝ
n
‍
|
f
(
x
)
|
p
w
(
x
)
d
x
)
1
/
p
<
∞
.
When p
=
∞, L
w
∞
(
ℝ
n
) will be taken to mean L
∞
(
ℝ
n
) and
(17)
∥
f
∥
L
w
∞
=
∥
f
∥
L
∞
=
ess
sup
⁡
x
∈
ℝ
n
|
f
(
x
)
|
.
We also denote by W
L
w
p
(
ℝ
n
) the weighted weak L
p space which is formed by all measurable functions f satisfying
(18)
∥
f
∥
W
L
w
p
=
sup
⁡
λ
>
0
λ
·
w
(
{
x
∈
ℝ
n
:
|
f
(
x
)
|
>
λ
}
)
1
/
p
<
∞
.Let us now turn to the weighted weak Hardy spaces. The (unweighted) weakH
p spaces have first appeared in the work of Fefferman et al. [14], which are the intermediate spaces between two Hardy spaces through the real method of interpolation. The atomic decomposition characterization of weak H
1 space on ℝ
n was given by Fefferman and Soria in [15]. Later, Liu [16] established the weak H
p spaces on homogeneous groups for the whole range 0
<
p
≤
1. The corresponding results related to ℝ
n can be found in [17]. For the boundedness properties of some operators on weak Hardy spaces, we refer the readers to [18–24]. In 2000, Quek and Yang [25] introduced the weighted weak Hardy spaces W
H
w
p
(
ℝ
n
) and established their atomic decompositions. Moreover, by using the atomic decomposition theory of W
H
w
p
(
ℝ
n
), Quek and Yang [25] also obtained the boundedness of Calderón-Zygmund type operators on these weighted spaces.We write𝒮
(
ℝ
n
) to denote the Schwartz space of all rapidly decreasing infinitely differentiable functions and 𝒮
′
(
ℝ
n
) to denote the space of all tempered distributions, that is, the topological dual of 𝒮
(
ℝ
n
). Let w
∈
A
∞, 0
<
p
≤
1 and N
=
[
n
(
q
w
/
p
-
1
)
]. Define
(19)
𝒜
N
,
w
=
{
sup
⁡
|
α
|
≤
N
+
1
(
1
+
|
x
|
)
N
+
n
+
1
φ
∈
𝒮
(
ℝ
n
)
:
222222222
sup
⁡
x
∈
ℝ
n
sup
⁡
|
α
|
≤
N
+
1
(
1
+
|
x
|
)
N
+
n
+
1
|
D
α
φ
(
x
)
|
≤
1
}
,
where α
=
(
α
1
,
…
,
α
n
)
∈
(
ℕ
∪
{
0
}
)
n, |
α
|
=
α
1
+
⋯
+
α
n and
(20)
D
α
φ
=
∂
|
α
|
φ
∂
x
1
α
1
⋯
∂
x
n
α
n
.
For any given f
∈
𝒮
′
(
ℝ
n
), the grand maximal function of f is defined by
(21)
G
w
f
(
x
)
=
sup
⁡
φ
∈
𝒜
N
,
w
sup
⁡
|
y
-
x
|
<
t
|
(
φ
t
*
f
)
(
y
)
|
.
Then we can define the weighted weak Hardy space W
H
w
p
(
ℝ
n
) by W
H
w
p
(
ℝ
n
)
=
{
f
∈
𝒮
′
(
ℝ
n
)
:
G
w
f
∈
W
L
w
p
(
ℝ
n
)
}. Moreover, we set ∥
f
∥
W
H
w
p
=
∥
G
w
f
∥
W
L
w
p.Theorem 3 (see [25]).
Let0
<
p
≤
1 and w
∈
A
∞. For every f
∈
W
H
w
p
(
ℝ
n
), there exists a sequence of bounded measurable functions {
f
k
}
k
=
-
∞
∞ such that(i)
f
=
∑
k
=
-
∞
∞
‍
f
k in the sense of distributions;
(ii)
eachf
k can be further decomposed into f
k
=
∑
i
‍
b
i
k, where {
b
i
k
} satisfies that
(a)
eachb
i
k is supported in a cube Q
i
k with ∑
i
‍
w
(
Q
i
k
)
≤
c
2
-
k
p and ∑
i
‍
χ
Q
i
k
(
x
)
≤
c. Here χ
E denotes the characteristic function of the set E and c
~
∥
f
∥
W
H
w
p
p;
(b)
∥
b
i
k
∥
L
∞
≤
C
2
k, where C
>
0 is independent of i and k;
(c)
∫
ℝ
n
‍
b
i
k
(
x
)
x
α
d
x
=
0 for every multi-index α with |
α
|
≤
[
n
(
q
w
/
p
-
1
)
].
Conversely, iff
∈
𝒮
′
(
ℝ
n
) has a decomposition satisfying (
i
) and (
i
i
), then f
∈
W
H
w
p
(
ℝ
n
). Moreover, one has ∥
f
∥
W
H
w
p
p
~
c.Throughout this paperC always denotes a positive constant, which is independent of the main parameters and not necessarily the same at each occurrence.


## 3. Proof of Theorem1

Following the same arguments as in the proof of Lemma  5 in [26], we can also establish the following lemma on the variable kernel Ω
(
x
,
z
) (see [5, 8]).Lemma 4.
Letr
≥
1. Suppose that Ω
(
x
,
z
)
∈
L
∞
(
ℝ
n
)
×
L
r
(
S
n
-
1
) satisfies the L
r-Dini condition in Section 1. If there exists a constant 0
<
γ
≤
1
/
2 such that |
y
|
<
γ
R, then for any x
0
∈
ℝ
n, one has
(22)
(
∫
R
≤
|
x
|
<
2
R
‍
|
K
(
x
+
x
0
,
x
-
y
)
-
K
(
x
+
x
0
,
x
)
|
r
d
x
)
1
/
r
≤
C
·
R
-
n
/
r
′
(
|
y
|
R
+
∫
|
y
|
/
2
R
|
y
|
/
R
‍
ω
r
(
δ
)
δ
d
δ
)
,
where the constant C
>
0 is independent of R and y.We are now in a position to give the proof of Theorem1.Proof of Theorem1.
For any givenλ
>
0, we may choose k
0
∈
ℤ such that 2
k
0
≤
λ
<
2
k
0
+
1. For every f
∈
W
H
w
p
(
ℝ
n
), then by Theorem 3, we can write
(23)
f
=
∑
k
=
-
∞
∞
‍
f
k
=
∑
k
=
-
∞
k
0
‍
f
k
+
∑
k
=
k
0
+
1
∞
‍
f
k
:
=
F
1
+
F
2
,
where F
1
=
∑
k
=
-
∞
k
0
‍
f
k
=
∑
k
=
-
∞
k
0
‍
∑
i
‍
b
i
k, F
2
=
∑
k
=
k
0
+
1
∞
‍
f
k
=
∑
k
=
k
0
+
1
∞
‍
∑
i
‍
b
i
k, and {
b
i
k
} satisfies (a)–(c) in Theorem 3. Then we have
(24)
λ
p
·
w
(
{
x
∈
ℝ
n
:
|
T
Ω
(
f
)
(
x
)
|
>
λ
}
)
≤
λ
p
·
w
(
{
x
∈
ℝ
n
:
|
T
Ω
(
F
1
)
(
x
)
|
>
λ
2
}
)
+
λ
p
·
w
(
{
x
∈
ℝ
n
:
|
T
Ω
(
F
2
)
(
x
)
|
>
λ
2
}
)
=
I
1
+
I
2
.
First we claim that the following inequality holds:
(25)
∥
F
1
∥
L
w
2
≤
C
·
λ
1
-
p
/
2
∥
f
∥
W
H
w
p
p
/
2
.
In fact, since supp
⁡
b
i
k
⊆
Q
i
k
=
Q
(
x
i
k
,
r
i
k
) and ∥
b
i
k
∥
L
∞
≤
C
2
k according to Theorem 3, then it follows directly from Minkowski’s inequality that
(26)
∥
F
1
∥
L
w
2
≤
∑
k
=
-
∞
k
0
‍
∑
i
‍
∥
b
i
k
∥
L
w
2
≤
∑
k
=
-
∞
k
0
‍
∑
i
‍
∥
b
i
k
∥
L
∞
w
(
Q
i
k
)
1
/
2
.
For each k
∈
ℤ, by using the bounded overlapping property of the cubes {
Q
i
k
} and the fact that 1
-
p
/
2
>
0, we thus obtain
(27)
∥
F
1
∥
L
w
2
≤
C
∑
k
=
-
∞
k
0
‍
2
k
(
∑
i
‍
w
(
Q
i
k
)
)
1
/
2
≤
C
∑
k
=
-
∞
k
0
‍
2
k
(
1
-
p
/
2
)
∥
f
∥
W
H
w
p
p
/
2
≤
C
∑
k
=
-
∞
k
0
‍
2
(
k
-
k
0
)
(
1
-
p
/
2
)
·
λ
1
-
(
p
/
2
)
∥
f
∥
W
H
w
p
p
/
2
≤
C
·
λ
1
-
p
/
2
∥
f
∥
W
H
w
p
p
/
2
.
Since w
∈
A
p
(
1
+
(
α
/
n
)
) and p
(
1
+
(
α
/
n
)
)
≤
1
+
(
α
/
n
)
<
2, then we have w
∈
A
2. In this case, we know that there exists a number s
>
1 such that w
∈
R
H
s. More specifically, by using the sharp reverse Hölder’s inequality for A
2 weights obtained recently in [27], we find that for w
∈
A
2,
(28)
w
∈
R
H
s
with
s
=
1
+
1
2
n
+
5
[
w
]
A
2
.
Observe that Ω
∈
Di
n
α
∞
(
S
n
-
1
), then we are able to find a positive number r
>
1 large enough such that r
>
max
⁡
{
s
′
,
(
2
n
)
/
(
n
-
α
)
} and Ω
∈
Di
n
α
r
(
S
n
-
1
). By the choice of r, we can easily check that 2
/
r
′
>
1
+
α
/
n
≥
p
(
1
+
α
/
n
), which implies w
∈
A
2
/
r
′. Hence, by using Theorem B, we know that T
Ω is bounded on L
w
2
(
ℝ
n
). This fact together with Chebyshev’s inequality and (25) yields
(29)
I
1
≤
λ
p
·
4
λ
2
∥
T
Ω
(
F
1
)
∥
L
w
2
2
≤
C
·
λ
p
-
2
∥
F
1
∥
L
w
2
2
≤
C
∥
f
∥
W
H
w
p
p
.
We now turn our attention to the estimate of I
2. Setting
(30)
A
k
0
=
⋃
k
=
k
0
+
1
∞
‍
⋃
i
‍
Q
i
k
~
,
where Q
i
k
~
=
Q
(
x
i
k
,
τ
(
k
-
k
0
)
/
(
n
+
α
)
(
2
n
)
r
i
k
) and τ is a fixed positive number such that 1
<
τ
<
2. Thus, we can further decompose I
2 as
(31)
I
2
≤
λ
p
·
w
(
{
x
∈
A
k
0
:
|
T
Ω
(
F
2
)
(
x
)
|
>
λ
2
}
)
≤
+
λ
p
·
w
(
{
x
∈
(
A
k
0
)
c
:
|
T
Ω
(
F
2
)
(
x
)
|
>
λ
2
}
)
=
I
2
′
+
I
2
′′
.
Let us first deal with the term I
2
′. Since w
∈
A
p
(
1
+
(
α
/
n
)
), then by Lemma 2, we can deduce that
(32)
I
2
′
≤
λ
p
∑
k
=
k
0
+
1
∞
‍
∑
i
‍
w
(
Q
i
k
~
)
≤
C
·
λ
p
∑
k
=
k
0
+
1
∞
‍
τ
(
k
-
k
0
)
p
∑
i
‍
w
(
Q
i
k
)
≤
C
∥
f
∥
W
H
w
p
p
∑
k
=
k
0
+
1
∞
‍
(
τ
2
)
(
k
-
k
0
)
p
≤
C
∥
f
∥
W
H
w
p
p
.
On the other hand, it follows immediately from Chebyshev’s inequality that
(33)
I
2
′′
≤
2
p
∫
(
A
k
0
)
c
‍
|
T
Ω
(
F
2
)
(
x
)
|
p
w
(
x
)
d
x
≤
2
p
∑
k
=
k
0
+
1
∞
∑
i
∫
(
Q
i
k
~
)
c
‍
|
T
Ω
(
b
i
k
)
(
x
)
|
p
w
(
x
)
d
x
=
2
p
∑
k
=
k
0
+
1
∞
‍
∑
i
‍
J
i
k
.
Now denote τ
i
,
ℓ
k
=
2
ℓ
-
1
τ
(
k
-
k
0
)
/
(
n
+
α
)
n
r
i
k, Q
i
,
ℓ
k
~
=
Q
(
x
i
k
,
τ
i
,
ℓ
k
), and
(34)
E
i
,
ℓ
k
=
{
x
∈
ℝ
n
:
τ
i
,
ℓ
k
≤
|
x
-
x
i
k
|
<
2
τ
i
,
ℓ
k
}
,
ℓ
=
1,2
,
…
.
An application of Hölder’s inequality gives us that
(35)
J
i
k
≤
∑
ℓ
=
1
∞
‍
∫
E
i
,
ℓ
k
‍
|
T
Ω
(
b
i
k
)
(
x
)
|
p
w
(
x
)
d
x
≤
∑
ℓ
=
1
∞
‍
(
∫
E
i
,
ℓ
k
‍
w
(
x
)
d
x
)
1
-
p
(
∫
E
i
,
ℓ
k
‍
|
T
Ω
(
b
i
k
)
(
x
)
|
w
(
x
)
d
x
)
p
.
Let q
=
p
(
1
+
(
α
/
n
)
) for simplicity. Then for any n
/
(
n
+
α
)
<
p
≤
1 and w
∈
A
q with q
>
1, we can easily see that [
n
(
q
w
/
p
-
1
)
]
=
0. Hence, by the cancellation condition of b
i
k
∈
L
∞
(
ℝ
n
), we get
(36)
∫
E
i
,
ℓ
k
‍
|
T
Ω
(
b
i
k
)
(
x
)
|
w
(
x
)
d
x
=
∫
E
i
,
ℓ
k

|
∫
Q
i
k

[
K
(
x
,
x
-
y
)

-

K
(
x
,
x
-
x
i
k
)
]
b
i
k

(
y
)
d
y
|
w

(
x
)
d
x
≤
∫
Q
i
k

{

∫
E
i
,
ℓ
k

|
K
(
x
,
x
-
y
)

-

K
(
x
,
x
-
x
i
k
)
|
w

(
x
)
d
x

}
|
b
i
k

(
y
)
|
d
y
≤
∥
b
i
k
∥
L
∞

|
Q
i
k
|

(
∫
E
i
,
ℓ
k
|
K
(
x
,
x
-
y
)

-

K
(
x
,
x
-
x
i
k
)
|
w

(
x
)
d
x
)
.
When y
∈
Q
i
k and x
∈
(
Q
i
k
~
)
c, then a trivial computation shows that
(37)
|
x
-
x
i
k
|
≥
τ
(
k
-
k
0
)
/
(
n
+
α
)
n
r
i
k
>
n
r
i
k
≥
2
|
y
-
x
i
k
|
.
We also observe that w
∈
R
H
s and r
>
s
′, then w
∈
R
H
r
′. Using Hölder’s inequality, the estimate (37), and Lemma 4, we can see that for any y
∈
Q
i
k, the integral of the above expression is dominated by
(38)
(
∫
E
i
,
ℓ
k
‍
|
K
(
x
,
x
-
y
)
-
K
(
x
,
x
-
x
i
k
)
|
r
d
x
)
1
/
r
×
(
∫
E
i
,
ℓ
k
‍
w
(
x
)
r
′
d
x
)
1
/
r
′
≤
C
·
w
(
Q
i
,
ℓ
+
1
k
~
)
|
Q
i
,
ℓ
+
1
k
~
|
1
/
r
(
∫
τ
i
,
ℓ
k
≤
|
x
|
<
2
τ
i
,
ℓ
k
‍
|
K
(
x
+
x
i
k
,
x
-
(
y
-
x
i
k
)
)
2222222222222222
l
2222222222
-
K
(
x
+
x
i
k
,
x
)
|
r
d
x
∫
τ
i
,
ℓ
k
≤
|
x
|
<
2
τ
i
,
ℓ
k
‍
|
K
(
x
+
x
i
k
,
x
-
(
y
-
x
i
k
)
)
)
1
/
r
≤
C
·
w
(
Q
i
,
ℓ
+
1
k
~
)
|
Q
i
,
ℓ
+
1
k
~
|
1
/
r
·
(
τ
i
,
ℓ
k
)
-
n
/
r
′
×
(
|
y
-
x
i
k
|
τ
i
,
ℓ
k
+
∫
|
y
-
x
i
k
|
/
2
τ
i
,
ℓ
k
|
y
-
x
i
k
|
/
τ
i
,
ℓ
k
‍
ω
r
(
δ
)
δ
d
δ
)
≤
C
·
w
(
Q
i
,
ℓ
+
1
k
~
)
|
Q
i
,
ℓ
+
1
k
~
|
1
/
r
·
(
τ
i
,
ℓ
k
)
-
n
/
r
′
×
(
|
y
-
x
i
k
|
τ
i
,
ℓ
k
+
|
y
-
x
i
k
|
α
(
τ
i
,
ℓ
k
)
α
×
∫
|
y
-
x
i
k
|
/
2
τ
i
,
ℓ
k
|
y
-
x
i
k
|
/
τ
i
,
ℓ
k
‍
ω
r
(
δ
)
δ
1
+
α
d
δ
)
≤
C
·
w
(
Q
i
,
ℓ
+
1
k
~
)
|
Q
i
,
ℓ
+
1
k
~
|
·
(
1
2
ℓ
τ
(
k
-
k
0
)
/
(
n
+
α
)
+
[
1
2
ℓ
τ
(
k
-
k
0
)
/
(
n
+
α
)
]
α
∫
0
1
‍
ω
r
(
δ
)
δ
1
+
α
d
δ
)
≤
C
·
(
1
+
∫
0
1
‍
ω
r
(
δ
)
δ
1
+
α
d
δ
)
·
w
(
Q
i
,
ℓ
+
1
k
~
)
|
Q
i
,
ℓ
+
1
k
~
|
(
1
2
ℓ
τ
(
k
-
k
0
)
/
(
n
+
α
)
)
α
.
Recall that ∥
b
i
k
∥
L
∞
≤
C
2
k. From the above estimate (38), it follows that
(39)
J
i
k
≤
C
·
2
k
p
∑
ℓ
=
1
∞
‍
w
(
Q
i
,
ℓ
+
1
k
~
)
(
|
Q
i
k
|
|
Q
i
,
ℓ
+
1
k
~
|
)
p
22222222222
×
(
1
2
ℓ
τ
(
k
-
k
0
)
/
(
n
+
α
)
)
α
p
.
In addition, for w
∈
A
q with q
>
1, then we can take a sufficiently small number ɛ
>
0 such that w
∈
A
q
-
ɛ. Thus, by using Lemma 2 again, we finally obtain
(40)
J
i
k
≤
C
·
2
k
p
w
(
Q
i
k
)
≤
×
∑
ℓ
=
1
∞
‍
(
2
ℓ
τ
(
k
-
k
0
)
/
(
n
+
α
)
n
)
n
(
q
-
ɛ
)
-
n
p
2222222
×
(
1
2
ℓ
τ
(
k
-
k
0
)
/
(
n
+
α
)
)
α
p
≤
C
·
2
k
p
w
(
Q
i
k
)
∑
ℓ
=
1
∞
‍
(
2
ℓ
τ
(
k
-
k
0
)
/
(
n
+
α
)
)
-
n
ɛ
≤
C
·
2
k
p
w
(
Q
i
k
)
(
τ
(
k
-
k
0
)
/
(
n
+
α
)
)
-
n
ɛ
.
Therefore
(41)
I
2
′′
≤
C
∑
k
=
k
0
+
1
∞
∑
i
2
k
p
(
τ
(
k
-
k
0
)
/
(
n
+
α
)
)
-
n
ɛ
w
(
Q
i
k
)
≤
C
∥
f
∥
W
H
w
p
p
∑
k
=
k
0
+
1
∞
‍
(
τ
(
k
-
k
0
)
/
(
n
+
α
)
)
-
n
ɛ
≤
C
∥
f
∥
W
H
w
p
p
.
Combining the above inequality (41) with (29) and (32), and then taking the supremum over all λ
>
0, we conclude the proof of Theorem 1.



---

*Source: 357135-2014-01-23.xml*