# Marcinkiewicz Integrals on Weighted Weak Hardy Spaces

Authors: Yue Hu; Yueshan Wang
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

We prove that, under the conditionΩ
∈
Lip
α, Marcinkiewicz integral μ
Ω is bounded from weighted weak Hardy space W
H
w
p
R
n to weighted weak Lebesgue space W
L
w
p
R
n for m
a
x
n
/
n
+
1
/
2
,
n
/
n
+
α
<
p
≤
1, where w belongs to the Muckenhoupt weight class. We also give weaker smoothness condition assumed on Ω to imply the boundedness of μ
Ω from W
H
w
1
ℝ
n to W
L
w
1
R
n.


---

## Body

## 1. Introduction and Results

Suppose thatS
n
-
1 is the unit sphere in R
n
(
n
≥
2
) equipped with Lebesgue measure d
σ
(
x
′
). Let Ω be homogeneous of degree zero on R
n satisfying Ω
∈
L
1
(
S
n
-
1
) and
(1)
∫
S
n
-
1
Ω
(
x
′
)
d
σ
(
x
′
)
=
0
,
where x
′
=
x
/
|
x
| for any x
≠
0. Then, the Marcinkiewicz integral operator μ
Ω is defined by
(2)
μ
Ω
(
f
)
(
x
)
=
(
∫
0
∞
|
F
Ω
,
t
(
f
)
(
x
)
|
2
d
t
t
3
)
1
/
2
,
where
(3)
F
Ω
,
t
(
f
)
(
x
)
=
∫
|
x
-
y
|
≤
t
Ω
(
x
-
y
)
|
x
-
y
|
n
-
1
f
(
y
)
d
y
.The operatorμ
Ω was originally introduced by Marcinkiewicz [1] in 1938 for n
=
1 and Ω
(
t
)
= sign t. In 1958, Stein [2] defined the Marcinkiewicz integral of higher dimensions and proved that if Ω
∈
Li
p
α
(
S
n
-
1
)
,
0
<
α
≤
1, then μ
Ω is of strong type (
p
,
p
) for 1
<
p
≤
2 and of weak type (
1,1
). We say that Ω
∈
Li
p
α
(
S
n
-
1
) if
(4)
|
Ω
(
x
′
)
-
Ω
(
y
′
)
|
≤
|
x
′
-
y
′
|
α
,
x
′
,
y
′
∈
S
n
-
1
.
In 1962, Benedek et al. [3] showed that if Ω is continuously differentiable on S
n
-
1, then μ
Ω is of strong type (
p
,
p
) for all 1
<
p
<
∞. In 1990, Torchinsky and Wang [4] considered the weighted case and proved that if Ω
∈
Li
p
α
(
S
n
-
1
)
,
0
<
α
≤
1, then, for all w
∈
A
p (Muckenhoupt weight class), μ
Ω is bounded on L
w
p
(
R
n
).In 1999, Ding et al. [5] improved Torchinsky and Wang's result, by ridding Ω of the smoothness condition, as follows.Theorem A.
LetΩ
∈
L
q
(
S
n
-
1
)
,
1
<
q
≤
∞, and satisfy (1). If w
q
′
∈
A
p
,
1
<
p
<
∞, then there is C
>
0 independent of f such that
(5)
∥
μ
Ω
(
f
)
∥
L
w
p
≤
C
∥
f
∥
L
w
p
,
where q
′ is the dual of q such that 1
/
q
+
1
/
q
′
=
1.In 2003, Ding et al. [6] discussed the boundedness of Marcinkiewicz integral on weighted Hardy space, getting the following results.Theorem B.
Let0
<
α
≤
1
,
β
=
min
⁡
{
α
,
1
/
2
}, and n
/
(
n
+
β
)
<
p
<
1. Let Ω
∈
Lip
⁡
α
(
R
n
) satisfy (1). If w
∈
A
p
(
1
+
β
/
n
), then there exists a constant C
>
0 independent of f such that ∥
μ
Ω
(
f
)
∥
L
w
p
≤
C
∥
f
∥
H
w
p.Theorem C.
LetΩ satisfy (1) and L
q condition (q
>
1
). If w
q
′
∈
A
1, then there is a C
>
0 independent of f such that ∥
μ
Ω
(
f
)
∥
L
w
1
≤
C
∥
f
∥
H
w
1.In 2007, C. C. Lin and Y. C. Lin [7] showed that under weaker condition assumed on Ω, which is called L
q-Dini type condition of order α, Marcinkiewicz integral operator μ
Ω is bounded from H
w
p to L
w
p.We say thatΩ satisfies the L
q-Dini condition if Ω
∈
L
q
(
S
n
-
1
)
,
q
≥
1, is homogenous of degree zero on R
n, and
(6)
∫
0
∞
w
q
(
δ
)
δ
d
δ
<
∞
,
where w
q
(
δ
) denotes the integral modulus of continuity q of Ω defined by
(7)
w
q
(
δ
)
=
sup
⁡
|
ρ
|
<
δ
(
∫
S
n
-
1
|
Ω
(
ρ
x
′
)
-
Ω
(
x
′
)
|
q
d
σ
(
x
′
)
)
1
/
q
,
and ρ is a rotation in R
n with |
ρ
|
=
∥
ρ
-
I
∥. If q
=
1, we let w
(
δ
)
=
w
q
(
δ
).In 2002, Ding et al. [8] considered the boundedness of Marcinkiewicz integral on weak Hardy space; they obtained the following result.Theorem D.
Suppose thatΩ
∈
L
1
(
S
n
-
1
) satisfies (1) and
(8)
∫
0
1
w
(
δ
)
δ
(
1
+
|
log
⁡
δ
|
)
σ
d
δ
<
∞
,
f
o
r
s
o
m
e
σ
>
1
.
Then, there exists a constant C
>
0 such that
(9)
∥
μ
Ω
(
f
)
∥
W
L
1
≤
C
∥
f
∥
W
H
1
.In this paper, we will investigate the boundedness of Marcinkiewicz integral on weighted weak Hardy space. We have the following conclusions.Theorem 1.
Suppose thatΩ
∈
Lip
⁡
α (
R
n
)
,
0
<
α
≤
1, satisfies (1) and β
=
min
⁡
{
α
,
1
/
2
}. If n
/
(
n
+
β
)
<
p
≤
1 and w
∈
A
p
(
1
+
β
/
n
), then there exists a constant C
>
0 independent of f such that ∥
μ
Ω
(
f
)
∥
W
L
w
p
≤
C
∥
f
∥
W
H
w
p.Theorem 2.
LetΩ
∈
L
q
(
S
n
-
1
)
,
1
<
q
≤
∞, satisfy (1) and
(10)
∫
0
1
w
q
(
δ
)
δ
(
1
+
|
log
⁡
δ
|
)
σ
d
δ
<
∞
,
f
o
r
s
o
m
e
σ
>
1
.
Ifw
q
′
∈
A
1, then there is a C
>
0 independent of f such that ∥
μ
Ω
(
f
)
∥
W
L
w
1
≤
C
∥
f
∥
W
H
w
1.Remark 3.
From [8, Remark 2], Condition (10) is weaker than Condition (4).


## 2. Some Preliminaries and Notations

We begin this section with some properties ofA
p weights which play an important role in the proofs of our main results.A weightw is a nonnegative, locally integrable function on R
n. Let B
=
B
(
x
0
,
r
B
) denote the ball with the center x
0 and radius r
B. For any ball B and λ
>
0
,
λ
B denotes the ball concentric with B whose radius is λ times as long. For a given weight function w and a measurable set E, we also denote the Lebesgue measure of E by |
E
| and set weighted measure w
(
E
)
=
∫
E
‍
w
(
x
)
d
x.Definition 4.
A weightw is said to belong to A
p for 1
<
p
<
∞, if there exists a constant C such that for every ball, B
⊂
R
n,
(11)
(
1
|
B
|
∫
B
w
(
x
)
d
x
)
(
1
|
B
|
∫
B
w
(
x
)
1
-
p
′
d
x
)
p
-
1
≤
C
.
The classA
1 is defined by replacing the above inequality with
(12)
1
|
B
|
∫
B
w
(
y
)
d
y
≤
C
·
ess
inf
⁡
x
∈
B
w
(
x
)
iii
for
every
ball
B
⊆
R
n
.
The classicalA
p weight theory was first introduced by Muckenhoupt in the study of weighted L
p boundedness of Hardy-Littlewood maximal functions in [9].
We know that ifw
∈
A
p with 1
<
p
<
∞, then w
∈
A
r for all r
>
p and w
∈
A
q for some 1
<
q
<
p. Therefore, we will use the notation q
w
≡
inf
⁡
{
q
>
1
:
w
∈
A
q
} to denote the critical index of w. Obviously, if w
∈
A
q, q
>
1, then we have 1
≤
q
w
<
q.
We state the following results aboutA
p class that will be used later on.Lemma 5 (see [10]).
Letw
∈
A
p, p
≥
1. Then, for any ball B and any λ
>
1, there exists an absolute constant C
>
0 such that
(13)
w
(
λ
B
)
≤
C
λ
n
p
w
(
B
)
,
where C depends neither on B nor on λ.Lemma 6 (see [10]).
Letw
∈
A
q with q
≥
1. Then, for any r
>
0, there exists an absolute C
>
1 independent of r such that
(14)
∫
|
x
-
x
0
|
≥
r
w
(
x
)
|
x
-
x
0
|
n
q
d
x
≤
C
r
-
n
q
w
(
B
(
x
0
,
2
r
)
)
.We also need the following result aboutL
q-Dini condition.Lemma 7 (see [11]).
Suppose thatq
>
1 and Ω satisfies L
q-Dini condition. Then, for |
y
|
≤
R
/
2,
(15)
(
∫
R
<
|
x
|
<
2
R
|
Ω
(
x
-
y
)
|
x
-
y
|
n
-
1
-
Ω
(
x
)
|
x
|
n
-
1
|
q
d
x
)
1
/
q
≤
C
R
n
/
q
-
n
+
1
(
|
y
|
R
+
∫
|
y
|
/
2
R
<
δ
<
|
y
|
/
R
w
q
(
δ
)
δ
d
δ
)
.Given a Muckenhoupt weight functionw on R
n, for 0
<
p
<
∞, we denote by L
w
p the space of all functions satisfying
(16)
∥
f
∥
L
w
p
=
(
∫
R
n
‍
|
f
(
x
)
|
p
w
(
x
)
d
x
)
1
/
p
<
∞
.
When q
=
∞, L
w
∞ will be taken to mean L
∞, and we set ∥
f
∥
L
w
∞
=
∥
f
∥
L
∞. And also the weighted weak Lebesgue spaces is defined by
(17)
∥
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. [12]. The atomic decomposition theory of weak H
1 space on R
n was given by Fefferman and Soria in [13]. In 2000, Quek and Yang [14] introduced the weighted weak Hardy spaces WH
w
p and established their atomic decompositions. Moreover, by using the atomic decomposition theory of WH
w
p, Quek and Yang 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 smooth 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
(18)
𝒜
N
,
w
=
{
φ
∈
𝒮
(
R
n
)
:
sup
⁡
x
∈
R
n
sup
⁡
|
α
|
≤
N
+
1
(
1
+
|
x
|
)
N
+
n
+
1
d
d
d
×
|
D
α
φ
(
x
)
|
≤
1
φ
∈
𝒮
(
R
n
)
:
sup
⁡
x
∈
R
n
sup
⁡
|
α
|
≤
N
+
1
(
1
+
|
x
|
)
N
+
n
+
1
}
,
where α
=
(
α
1
,
…
,
α
n
)
∈
(
ℕ
∪
{
0
}
)
n
,
|
α
|
=
α
1
+
⋯
+
α
n, and D
α
φ
=
∂
|
α
|
φ
/
(
∂
x
1
α
1
,
…
,
∂
x
n
α
n
).For𝒮
′
(
R
n
) the grand maximal function of f is defined by
(19)
G
w
(
f
)
(
x
)
=
sup
⁡
φ
∈
𝒜
N
,
w
sup
⁡
|
x
-
y
|
<
t
|
(
φ
t
*
f
)
(
y
)
|
.
Then, we can define the weighted weak Hardy space WH
w
p
(
R
n
) by
(20)
WH
w
p
(
R
n
)
=
{
𝒮
′
(
R
n
)
:
G
w
f
∈
WL
w
p
(
R
n
)
}
.Moreover, we set∥
f
∥
WH
w
p
=
∥
G
w
f
∥
WL
w
p.Lemma 8 (see [14]).
Let0
<
p
≤
1
a
n
d
w
∈
A
∞. For every f in 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 the following:
(a)
eachb
i
k is supported in a ball B
i
k with ∑
i
‍
w
(
B
i
k
)
≤
c
2
-
k
p and  ∑
i
‍
χ
B
i
k
(
x
)
≤
c. Here, χ
E denotes the characteristic function of the set E and c
~
∥
f
∥
W
H
w
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 (ii), then f
∈
W
H
w
p
(
R
n
). Moreover, we have ∥
f
∥
W
H
w
p
~
c.


## 3. Proof of Theorem1

It suffices to show that there exists a constantC
>
0, such that, for any f
∈
W
H
w
p
and
β
>
0,
(21)
β
p
w
(
{
x
∈
R
n
:
|
μ
Ω
(
f
)
(
x
)
|
>
β
}
)
≤
C
∥
f
∥
W
H
w
p
p
.For any givenβ
>
0, we choose k
0
∈
ℤ such that 2
k
0
≤
β
<
2
k
0
+
1. Then, by Lemma 8, for every f
∈
W
H
w
p, we can write
(22)
f
=
∑
k
=
-
∞
∞
‍
f
k
=
∑
k
=
-
∞
k
0
‍
f
k
+
∑
k
=
k
0
+
1
∞
‍
f
k
=
F
1
+
F
2
,
where
(23)
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
+
2
∞
∑
i
b
i
k
and {
b
i
k
} satisfies (a)–(c) in Lemma 8. Then,
(24)
β
p
(
{
x
∈
R
n
:
|
μ
Ω
(
f
)
(
x
)
|
>
β
}
)
≤
β
p
w
(
{
x
∈
R
n
:
|
μ
Ω
(
F
1
)
(
x
)
|
>
β
2
}
)
+
β
p
w
(
{
x
∈
R
n
:
|
μ
Ω
(
F
2
)
(
x
)
|
>
β
2
}
)
=
M
1
+
M
2
.We claim that the following inequality holds:(25)
∥
F
1
∥
L
w
2
≤
C
β
1
-
p
/
2
∥
f
∥
W
H
w
p
p
/
2
.
In fact, it follows from Minkowski's integral inequality and the bounded overlapping property of the cubes {
B
i
k
} 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
(
B
i
k
)
1
/
2
≤
C
∑
k
=
-
∞
k
0
2
k
(
∑
i
w
(
B
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
)
β
1
-
p
/
2
∥
f
∥
W
H
w
p
p
/
2
≤
C
β
1
-
p
/
2
∥
f
∥
W
H
w
p
p
/
2
.
Notice that w
∈
A
p
(
1
+
β
/
n
)
⊂
A
2; then from Chebyshev's inequality and the boundedness of μ
Ω on L
w
2 we have
(27)
M
1
≤
β
p
C
β
2
∥
μ
Ω
(
F
1
)
∥
L
w
2
2
≤
C
∥
f
∥
W
H
w
p
pWe set(28)
A
k
0
=
⋃
k
=
k
0
+
1
∞
‍
⋃
i
‍
B
i
k
~
,
where B
i
k
~
=
B
(
x
i
k
,
(
3
/
2
)
(
k
-
k
0
)
/
(
n
+
β
)
n
d
i
k
). Then, we get
(29)
M
2
≤
β
p
w
(
{
x
∈
A
k
0
:
|
μ
Ω
(
F
2
)
(
x
)
|
>
β
2
}
)
+
β
p
w
(
{
x
∈
(
A
k
0
)
c
:
|
μ
Ω
(
F
2
)
(
x
)
|
>
β
2
}
)
=
M
21
+
M
22
.
Since w
∈
A
p
(
1
+
β
/
n
), then, by Lemmas 5 and 8 and the fact 2
k
0
≤
β
<
2
k
0
+
1, we get
(30)
M
21
≤
β
p
w
(
A
k
0
)
≤
β
p
∑
k
=
k
0
+
2
∞
∑
i
w
(
B
i
k
~
)
≤
C
β
p
∑
k
=
k
0
+
2
∞
(
3
2
)
(
k
-
k
0
)
p
∑
i
w
(
B
i
k
)
≤
C
∥
f
∥
WH
w
p
p
∑
k
=
k
0
+
2
∞
(
3
4
)
(
k
-
k
0
)
p
≤
C
∥
f
∥
WH
w
p
p
.
An application of Chebyshev's inequality gives us that
(31)
M
22
≤
C
∫
(
A
k
0
)
c
|
μ
Ω
(
F
2
)
|
p
w
(
x
)
d
x
≤
C
∑
k
=
k
0
+
1
∞
∑
i
∫
(
B
i
k
~
)
c
|
μ
Ω
(
b
i
k
)
(
x
)
|
p
w
(
x
)
d
x
.
Below, we will give the estimate of integral I
=
∫
(
B
i
k
~
)
c
|
μ
Ω
(
b
i
k
)
(
x
)
|
p
w
(
x
)
d
x. We have
(32)
I
≤
∫
(
B
i
k
~
)
c
(
∫
0
|
x
-
x
i
k
|
+
n
d
i
k
|
∫
|
x
-
y
|
≤
t
Ω
(
x
-
y
)
|
x
-
y
|
n
-
1
b
i
k
(
y
)
d
y
|
2
d
t
t
3
)
p
/
2
v
v
i
×
w
(
x
)
d
x

+

∫
(
B
i
k
~
)
c
(
∫
|
x
-
x
i
k
|
+
n
d
i
k
∞
|
∫
|
x
-
y
|
≤
t
Ω
(
x

-

y
)
|
x

-

y
|
n
-
1
b
i
k
(
y
)
d
y
|
2

d
t
t
3
)
p
/
2
v
v
v
×
w
(
x
)
d
x
=
I
1
+
I
2
.Fory
∈
B
i
k and x
∈
(
B
i
k
~
)
c, we have |
x
-
y
|
≈
|
x
-
x
i
k
|
≈
|
x
-
x
i
k
|
+
n
d
i
k. Thus,
(33)
|
1
|
x
-
y
|
2
-
1
(
|
x
-
x
i
k
|
+
n
d
i
k
)
2
|
≤
C
d
i
k
|
x
-
y
|
3
.
By Minkowski's inequality for integral,
(34)
(
∫
0
|
x
-
x
i
k
|
+
n
d
i
k
|
∫
|
x
-
y
|
≤
t
Ω
(
x
-
y
)
|
x
-
y
|
n
-
1
b
i
k
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
≤
∫
B
i
k
|
Ω
(
x
-
y
)
|
|
x
-
y
|
n
-
1
|
b
i
k
(
y
)
|
(
∫
|
x
-
y
|
|
x
-
x
i
k
|
+
n
d
i
k
d
t
t
3
)
1
/
2
d
y
≤
C
2
k
(
d
i
k
)
n
+
1
/
2
|
x
-
x
i
k
|
n
+
1
/
2
.
From the fact that q
=
(
1
+
β
/
n
)
p
≤
(
1
+
1
/
2
n
)
p and Lemma 6, we have
(35)
I
1
≤
C
2
k
p
(
d
i
k
)
(
n
+
1
/
2
)
p
∫
(
B
i
k
~
)
c
w
(
x
)
|
x
-
x
i
k
|
(
n
+
1
/
2
)
p
d
x
≤
C
2
k
p
(
d
i
k
)
(
n
+
1
/
2
)
p
(
(
3
2
)
(
k
-
k
0
)
/
(
n
+
β
)
n
d
i
k
)
-
(
n
+
1
/
2
)
p
×
w
(
B
i
k
~
)
.
Since w
∈
A
(
1
+
1
/
2
n
)
p, then there exists a ϵ
>
0 such that w
∈
A
(
1
+
1
/
2
n
)
p
-
ϵ. Thus, we have
(36)
w
(
B
i
k
~
)
≤
(
3
2
)
(
(
k
-
k
0
)
/
(
n
+
β
)
)
(
(
n
+
1
/
2
)
p
-
ϵ
)
w
(
B
i
k
)
.
Hence,
(37)
I
1
≤
C
2
k
p
(
3
2
)
-
(
(
k
-
k
0
)
/
(
n
+
β
)
)
n
ϵ
w
(
B
i
k
)
.Forn
/
(
n
+
β
)
<
p
≤
1 and w
∈
A
q with q
=
(
1
+
β
/
n
)
p
>
1, we can see that [
n
(
q
w
/
p
-
1
)
]
=
0. Since t
≥
|
x
-
x
i
k
|
+
n
d
i
k, we have B
i
k
⊂
{
y
:
|
x
-
y
|
≤
t
}. By the vanishing moment condition of b
i
k,
(38)
I
2
=
∫
(
B
i
k
~
)
c
(
d
t
t
3
∫
|
x
-
x
i
k
|
+
n
d
i
k
∞
|
∫
B
i
k
‍
(
Ω
(
x
-
y
)
|
x
-
y
|
n
-
1
∫
|
x
-
x
i
k
|
+
n
d
i
k
∞
|
∫
B
i
k
‍
(
Ω
(
x
-
y
)
|
x
-
y
|
n
-
1
-
Ω
(
x
-
x
i
k
)
|
x
-
x
i
k
|
n
-
1
)
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
×
b
i
k
(
y
)
d
y
∫
B
i
k
‍
|
2
d
t
t
3
∫
|
x
-
x
i
k
|
+
n
d
i
k
∞
|
∫
B
i
k
‍
(
Ω
(
x
-
y
)
|
x
-
y
|
n
-
1
)
p
/
2
w
(
x
)
d
x
.
But
(39)
|
∫
B
i
k
(
Ω
(
x
-
y
)
|
x
-
y
|
n
-
1
-
Ω
(
x
-
x
i
k
)
|
x
-
x
i
k
|
n
-
1
)
b
i
k
(
y
)
d
y
|
≤
∫
B
i
k
|
1
|
x
-
y
|
n
-
1
-
1
|
x
-
x
i
k
|
n
-
1
|
|
Ω
(
x
-
y
)
b
i
k
(
y
)
|
d
y
+
∫
B
i
k
|
Ω
(
x
-
y
)
-
Ω
(
x
-
x
i
k
)
|
|
x
-
x
i
k
|
n
-
1
|
b
i
k
(
y
)
|
d
y
=
K
1
+
K
2
.
Note that for any y
∈
B
i
k and x
∈
(
B
i
k
~
)
c
,
|
x
-
y
|
≈
|
x
-
x
i
k
|. Thus,
(40)
K
1
≤
C
∥
Ω
∥
L
∞
∥
b
i
k
∥
L
∞
∫
B
i
k
|
1
|
x
-
y
|
n
-
1
-
1
|
x
-
x
i
k
|
n
-
1
|
d
y
≤
C
∥
b
i
k
∥
L
∞
(
d
i
k
)
n
+
1
|
x
-
x
i
k
|
n
.
From the fact that Ω
∈
Li
p
α
(
S
n
-
1
) and Ω is homogeneous of degree zero, we get
(41)
|
Ω
(
x
-
y
)
-
Ω
(
x
-
x
i
k
)
|
≤
|
Ω
(
x
-
y
|
x
-
y
|
)
-
Ω
(
x
-
x
i
k
|
x
-
x
i
k
|
)
|
≤
C
|
x
-
y
|
x
-
y
|
-
x
-
x
i
k
|
x
-
x
i
k
|
|
α
≤
C
(
|
y
-
x
i
k
|
|
x
-
x
i
k
|
)
α
.
Then,
(42)
K
2
≤
C
∥
b
i
k
∥
L
∞
(
d
i
k
)
n
+
α
|
x
-
x
i
k
|
n
-
1
+
α
.
For x
∈
(
B
i
k
~
)
c, we have |
x
-
x
i
k
|
≈
|
x
-
x
i
k
|
+
n
d
i
k. Then,
(43)
∫
|
x
-
x
i
k
|
+
n
d
i
k
∞
d
t
t
3
≤
C
|
x
-
x
i
k
|
2
.
Hence,
(44)
I
2
≤
C
2
k
p
(
d
i
k
)
(
n
+
1
)
p
∫
(
B
i
k
~
)
c
w
(
x
)
|
x
-
x
i
k
|
(
n
+
1
)
p
d
x
+
C
2
k
p
(
d
i
k
)
(
n
+
α
)
p
∫
(
B
i
k
~
)
c
w
(
x
)
|
x
-
x
i
k
|
(
n
+
α
)
p
d
x
=
L
1
+
L
2
.
Since w
∈
A
p
(
1
+
β
/
n
)
and
β
=
min
⁡
{
α
,
1
/
2
}, then w
∈
A
p
(
1
+
1
/
n
) and w
∈
A
p
(
1
+
1
/
n
)
-
ϵ for some ϵ
>
0. From Lemma 6,
(45)
L
1
≤
C
2
k
p
(
3
2
)
-
(
(
k
-
k
0
)
/
(
n
+
β
)
)
p
(
n
+
1
)
w
(
B
i
k
~
)
≤
C
2
k
p
(
3
2
)
-
(
(
k
-
k
0
)
/
(
n
+
β
)
)
n
ϵ
w
(
B
i
k
)
.
Analogously, we have
(46)
L
2
≤
C
2
k
p
(
3
2
)
-
(
(
k
-
k
0
)
/
(
n
+
β
)
)
n
ϵ
w
(
B
i
k
)
.
Then,
(47)
I
2
≤
C
2
k
p
(
3
2
)
-
(
(
k
-
k
0
)
/
(
n
+
β
)
)
n
ϵ
w
(
B
i
k
)
.
Hence,
(48)
M
22
≤
C
∑
k
=
k
0
+
1
∞
∑
i
2
k
p
(
3
2
)
-
(
(
k
-
k
0
)
/
(
n
+
β
)
)
n
ϵ
w
(
B
i
k
)
≤
C
∥
f
∥
W
H
w
p
p
∑
k
=
k
0
+
1
∞
(
3
2
)
-
(
(
k
-
k
0
)
/
(
n
+
β
)
)
n
ϵ
≤
C
∥
f
∥
W
H
w
p
p
.Combining the estimate ofM
21 and M
22, we get M
2
≤
C
∥
f
∥
WH
w
p
p. This completes the proof of the Theorem 1.


## 4. Proof of Theorem2

We follow the strategy of the proof of Theorem1. For any given λ
>
0, we may choose k
0
∈
ℤ such that 2
k
0
≤
λ
<
2
k
0
+
1. For every f
∈
W
H
w
1, and then by Lemma 8, we can write
(49)
f
=
∑
k
=
-
∞
∞
f
k
=
∑
k
=
-
∞
k
0
f
k
+
∑
k
=
k
0
+
2
∞
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
+
2
∞
f
k
=
∑
k
=
k
0
+
2
∞
∑
i
b
i
k, and {
b
i
k
} satisfies (a)–(c) in Lemma 8. Then,
(50)
λ
w
(
{
x
∈
R
n
:
|
μ
Ω
(
f
)
(
x
)
|
>
λ
}
)
≤
λ
w
(
{
x
∈
R
n
:
|
μ
Ω
(
F
1
)
(
x
)
|
>
λ
2
}
)
+
λ
w
(
{
x
∈
R
n
:
|
μ
Ω
(
F
2
)
(
x
)
|
>
λ
2
}
)
=
P
1
+
P
2
.Similar to the estimate ofM
1 in the proof of Theorem 1, we obtain
(51)
P
1
≤
C
∥
f
∥
W
H
w
1
.We set(52)
A
k
0
=
⋃
k
=
k
0
+
1
∞
‍
⋃
i
‍
B
i
k
~
,
where B
i
k
~
=
B
(
x
i
k
,
(
3
/
2
)
(
k
-
k
0
)
/
n
n
d
i
k
). Then, we get
(53)
P
2
≤
λ
w
(
{
x
∈
A
k
0
:
|
μ
Ω
(
F
2
)
(
x
)
|
>
λ
2
}
)
+
λ
w
(
{
x
∈
(
A
k
0
)
c
:
|
μ
Ω
(
F
2
)
(
x
)
|
>
λ
2
}
)
=
P
21
+
P
22
.
Since w
∈
A
1, and then by Lemmas 5 and 8 and the fact 2
k
0
≤
λ
<
2
k
0
+
1, similar to the estimate in the proof of Theorem 1, we get
(54)
P
21
≤
C
∥
f
∥
W
H
w
1
.
An application of Chebyshev's inequality gives us that
(55)
P
22
≤
C
∫
(
A
k
0
)
c
|
μ
Ω
(
F
2
)
|
w
(
x
)
d
x
≤
C
∑
k
=
k
0
+
2
∞
∑
i
∫
(
B
i
k
~
)
c
|
μ
Ω
(
b
i
k
)
(
x
)
|
w
(
x
)
d
x
.
Below, we give the estimate of integral J
=
∫
(
B
i
k
~
)
c
|
μ
Ω
(
b
i
k
)
(
x
)
|
w
(
x
)
d
x. We have
(56)
J
≤
∫
(
B
i
k
~
)
c
(
∫
0
|
x
-
x
i
k
|
+
n
d
i
k
|
∫
|
x
-
y
|
≤
t
Ω
(
x
-
y
)
|
x
-
y
|
n
-
1
b
i
k
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
v
v
i
×
w
(
x
)
d
x

+

∫
(
B
i
k
~
)
c
(
∫
|
x
-
x
i
k
|
+
n
d
i
k
∞
|
∫
|
x
-
y
|
≤
t
Ω
(
x

-

y
)
|
x

-

y
|
n
-
1
b
i
k
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
v
v
v
i
×
w
(
x
)
d
x
=
J
1
+
J
2
.
For y
∈
B
i
k and x
∈
(
B
i
k
~
)
c, we have |
x
-
y
|
≈
|
x
-
x
i
k
|
≈
|
x
-
x
i
k
|
+
n
d
i
k. Thus,
(57)
|
1
|
x
-
y
|
2
-
1
(
|
x
-
x
i
k
|
+
n
d
i
k
)
2
|
≤
C
d
i
k
|
x
-
y
|
3
.
Then, Minkowski's inequality for integral gives us that
(58)
J
1
≤
∫
(
B
i
k
~
)
c
∫
B
i
k
|
Ω
(
x
-
y
)
|
|
x
-
y
|
n
-
1
|
b
i
k
(
y
)
|
vvvvvvv
v
×
(
∫
|
x
-
y
|
|
x
-
x
i
k
|
+
n
d
i
k
d
t
t
3
)
1
/
2
d
y
w
(
x
)
d
x
≤
C
(
d
i
k
)
1
/
2
∫
B
i
k
(
∫
(
B
i
k
~
)
c
|
Ω
(
x
-
y
)
|
|
x
-
y
|
n
+
1
/
2
w
(
x
)
d
x
)
v
v
v
v
v
v
vv
×
|
b
i
k
(
y
)
|
d
y
.
Since w
q
′
∈
A
1, then w
∈
A
1
⊂
A
1
+
1
/
2
n. From Lemmas 6 and 5, we have
(59)
(
∫
(
B
i
k
~
)
c
w
(
x
)
q
′
|
x
-
y
|
n
+
1
/
2
d
x
)
1
/
q
′
ii
≤
C
(
(
3
2
)
(
k
-
k
0
)
/
n
d
i
k
)
-
n
/
q
′
-
1
/
2
q
′
(
w
q
′
(
B
i
k
~
)
)
1
/
q
′
ii
≤
C
(
(
3
2
)
(
k
-
k
0
)
/
n
d
i
k
)
-
n
/
q
′
-
1
/
2
q
′
×
(
(
3
2
)
(
k
-
k
0
)
/
n
)
n
/
q
′
(
w
q
′
(
B
i
k
)
)
1
/
q
′
ii
≤
C
(
(
3
2
)
(
k
-
k
0
)
/
n
d
i
k
)
-
1
/
2
q
′
inf
⁡
x
∈
B
i
k
w
(
x
)
.
Hence, for y
∈
B
i
k, Hölder's inequality and Ω
∈
L
q
(
S
n
-
1
) imply that
(60)
∫
(
B
i
k
~
)
c
|
Ω
(
x
-
y
)
|
|
x
-
y
|
n
+
1
/
2
w
(
x
)
d
x
≤
(
∫
(
B
i
k
~
)
c
|
Ω
(
x
-
y
)
|
q
|
x
-
y
|
n
+
1
/
2
d
x
)
1
/
q
v
×
(
∫
(
B
i
k
~
)
c
w
(
x
)
q
′
|
x
-
y
|
n
+
1
/
2
d
x
)
1
/
q
′
iii
i
≤
(
∫
(
3
/
2
)
(
k
-
k
0
)
/
n
n
d
i
k
∞
∫
S
n
-
1
|
Ω
(
x
′
)
|
q
r
n
+
1
/
2
r
n
-
1
d
σ
(
x
′
)
d
r
)
1
/
q
v
×
(
(
3
2
)
(
k
-
k
0
)
/
n
d
i
k
)
-
1
/
2
q
′
inf
⁡
x
∈
B
i
k
w
(
x
)
iii
i
≤
C
(
(
3
2
)
(
k
-
k
0
)
/
n
d
i
k
)
-
1
/
2
q
v
×
(
(
3
2
)
-
(
k
-
k
0
)
/
n
d
i
k
)
-
1
/
2
q
′
inf
⁡
x
∈
B
i
k
w
(
x
)
iii
i
≤
C
(
(
3
2
)
(
k
-
k
0
)
/
n
d
i
k
)
-
1
/
2
inf
⁡
x
∈
B
i
k
w
(
x
)
.
Then,
(61)
J
1
≤
C
2
k
(
d
i
k
)
1
/
2
(
(
3
2
)
(
k
-
k
0
)
/
n
d
i
k
)
-
1
/
2
inf
⁡
x
∈
B
i
k
w
(
x
)
|
B
i
k
|
≤
C
2
k
w
(
B
i
k
)
(
2
3
)
(
k
-
k
0
)
/
2
n
.We now estimateJ
2. Since t
≥
|
x
-
x
i
k
|
+
n
d
i
k, we have B
i
k
⊂
{
y
:
|
x
-
y
|
≤
t
} and hence
(62)
J
2
=
∫
(
B
i
k
~
)
c
(
∫
|
x
-
x
i
k
|
+
n
d
i
k
∞
|
∫
B
i
k
(
Ω
(
x
-
y
)
|
x
-
y
|
n
-
1
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
v
-
Ω
(
x
-
x
i
k
)
|
x
-
x
i
k
|
n
-
1
)
b
i
k
(
y
)
d
y
∫
B
i
k
(
Ω
(
x
-
y
)
|
x
-
y
|
n
-
1
|
2
d
t
t
3
∫
|
x
-
x
i
k
|
+
n
d
i
k
∞
)
1
/
2
ddddddd
×
w
(
x
)
d
x
.DenoteR
j
k
=
2
j
(
3
/
2
)
(
k
-
k
0
)
/
n
n. Minkowski's inequality for integrals and the fact |
x
-
x
i
k
|
+
n
d
i
k
≈
|
x
-
x
i
k
| for x
∈
(
B
i
k
~
)
c give us
(63)
J
2
≤
∫
(
B
i
k
~
)
c
∫
B
i
k
|
Ω
(
x
-
y
)
|
x
-
y
|
n
-
1
-
Ω
(
x
-
x
i
k
)
|
x
-
x
i
k
|
n
-
1
|
|
b
i
k
(
y
)
|
v
v
v
v
v
v
i
×
(
∫
|
x
-
x
i
k
|
+
n
d
i
k
∞
d
t
t
3
)
1
/
2
d
y
w
(
x
)
d
x
≤
∫
(
B
i
k
~
)
c
∫
B
i
k
|
Ω
(
x
-
y
)
|
x
-
y
|
n
-
1
-
Ω
(
x
-
x
i
k
)
|
x
-
x
i
k
|
n
-
1
|
v
v
v
v
v
v
i
i
i
i
×
|
b
i
k
(
y
)
|
|
x
-
x
i
k
|
d
y
w
(
x
)
d
x
≤
C
∫
B
i
k
∑
j
=
0
∞
1
R
j
k
d
i
k
×
∫
R
j
k
d
i
k
≤
|
x
-
x
i
k
|
≤
R
j
+
1
k
d
i
k
|
Ω
(
x
-
x
i
k
)
|
x
-
x
i
k
|
n
-
1
Ω
(
x
-
y
)
|
x
-
y
|
n
-
1
-
Ω
(
x
-
x
i
k
)
|
x
-
x
i
k
|
n
-
1
|
v
v
v
v
v
v
v
v
v
v
v
v
v
v
×
w
(
x
)
d
x
|
b
i
k
(
y
)
|
d
y
.Notice that(64)
(
w
q
′
(
R
j
+
1
k
B
i
k
)
)
1
/
q
′
≤
C
(
R
j
k
)
n
/
q
′
(
d
i
k
)
n
/
q
′
inf
⁡
x
∈
B
i
k
w
(
x
)
.By Hölder's inequality and Lemma7, we have
(65)
∫
R
j
k
d
i
k
≤
|
x
-
x
i
k
|
≤
R
j
+
1
k
d
i
k
|
Ω
(
x
-
y
)
|
x
-
y
|
n
-
1
-
Ω
(
x
-
x
i
k
)
|
x
-
x
i
k
|
n
-
1
|
w
(
x
)
d
x
≤
(
d
x
∫
‍
R
j
k
d
i
k
≤
|
x
-
x
i
k
|
≤
R
j
+
1
k
d
i
k
|
Ω
(
x
-
y
)
|
x
-
y
|
n
-
1
∫
R
j
k
d
i
k
≤
|
x
-
x
i
k
|
≤
R
j
+
1
k
d
i
k
|
Ω
(
x
-
y
)
|
x
-
y
|
n
-
1
-
Ω
(
x
-
x
i
k
)
|
x
-
x
i
k
|
n
-
1
|
q
d
x
)
1
/
q
×
(
w
q
′
(
R
j
+
1
k
B
i
k
)
)
1
/
q
′
≤
C
(
R
j
k
d
i
k
)
1
-
n
+
n
/
q
×
(
|
y
-
x
i
k
|
R
j
k
d
i
k
+
∫
|
y
-
x
i
k
|
/
R
j
+
1
k
d
i
k
≤
δ
≤
|
y
-
x
i
k
|
/
R
j
k
d
i
k
ω
q
(
δ
)
δ
d
δ
|
y
-
x
i
k
|
R
j
k
d
i
k
)
iii
i
×
(
w
q
′
(
R
j
+
1
k
B
i
k
)
)
1
/
q
′
≤
C
R
j
k
d
i
k
inf
⁡
x
∈
B
i
k
w
(
x
)
×
(
|
y
-
x
i
k
|
R
j
k
d
i
k
+
∫
|
y
-
x
i
k
|
/
R
j
+
1
k
d
i
k
≤
δ
≤
|
y
-
x
i
k
|
/
R
j
k
d
i
k
ω
q
(
δ
)
δ
d
δ
|
y
-
x
i
k
|
R
j
k
d
i
k
)
.
Then,
(66)
J
2
≤
C
2
k
inf
⁡
x
∈
B
i
k
w
(
x
)
|
B
i
k
|
×
∑
j
=
0
∞
(
1
2
j
(
2
3
)
(
k
-
k
0
)
/
n
iiiiiii
iii
+
∫
|
y
-
x
i
k
|
/
R
j
+
1
k
d
i
k
≤
δ
≤
|
y
-
x
i
k
|
/
R
j
k
d
i
k
ω
q
(
δ
)
δ
d
δ
)
≤
C
2
k
w
(
B
i
k
)
(
(
2
3
)
(
k
-
k
0
)
/
n
+
∫
0
(
2
/
3
)
(
k
-
k
0
)
/
n
ω
q
(
δ
)
δ
d
δ
)
.
But(67)
∫
0
(
2
/
3
)
(
k
-
k
0
)
/
n
ω
q
(
δ
)
δ
d
δ
≤
C
[
1
+
(
k
-
k
0
)
log
⁡
(
3
/
2
)
]
σ
∫
0
1
w
q
(
δ
)
δ
(
1
+
|
log
⁡
δ
|
)
σ
d
δ
<
C
(
k
-
k
0
)
σ
,
from which we get
(68)
J
2
≤
C
2
k
w
(
B
i
k
)
(
(
2
3
)
(
k
-
k
0
)
/
n
+
C
(
k
-
k
0
)
σ
)
.
Thus,
(69)
P
22
≤
C
∑
k
=
k
0
+
2
∞
∑
i
∫
(
B
i
k
~
)
c
|
μ
Ω
(
b
i
k
)
(
x
)
|
w
(
x
)
d
x
≤
C
∑
k
=
k
0
+
2
∞
2
k
∑
i
w
(
B
i
k
)
(
(
2
3
)
(
k
-
k
0
)
/
n
+
C
(
k
-
k
0
)
σ
)
≤
C
∥
f
∥
W
H
w
1
∑
k
=
k
0
+
2
∞
(
(
2
3
)
(
k
-
k
0
)
/
n
+
C
(
k
-
k
0
)
σ
)
≤
C
∥
f
∥
W
H
w
1
.This completes the proof of Theorem2.



---

*Source: 765984-2014-04-03.xml*