# The Boundedness of Marcinkiewicz Integrals Associated with Schrödinger Operator on Morrey Spaces

Authors: Dongxiang Chen; Fangting Jin
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

LetL
=
-
Δ
+
V be a Schrödinger operator, where V belongs to some reverse Hölder class. The authors establish the boundedness of Marcinkiewicz integrals associated with Schrödinger operators and their commutators on Morrey spaces.


---

## Body

## 1. Introduction

In this paper, we consider the Schrödinger operator(1)
L
=
-
Δ
+
V
(
x
)
,
in R
d, d
≥
3, where V
(
x
) is a nonnegative potential belonging to the reverse Hölder class RH
q for some exponent q
>
d
/
2.A nonnegative locallyL
q integrable function V
(
x
) on R
d is said to belong to RH
q
(
q
>
1
) if there exists C
>
0 such that the reverse Hölder inequality,
(2)
(
1
|
B
|
∫
B
V
(
y
)
q
d
y
)
1
/
q
≤
C
|
B
|
∫
B
V
(
y
)
d
y
,
holds true for every ball B
⊂
R
d. We introduce the definition of the reverse Hölder index of V as q
0
=
sup
⁡
{
q
:
V
∈
RH
q
}. It is known that V
∈
RH
q implies V
∈
RH
q
+
ϵ for some ϵ
>
0.The Marcinkiewicz integral operatorμ is defined by
(3)
μ
f
(
x
)
=
(
∫
0
∞
|
∫
|
x
-
y
|
≤
t
Ω
(
x
-
y
)
|
x
-
y
|
d
-
1
f
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
.The above operator was introduced by Stein in [1] as an extension of the notion of Marcinkiewicz function from one dimension to higher dimensions.Similar to the classical Marcinkiewicz functionμ, we define the Marcinkiewicz function μ
j
L associated with the Schrödinger operator L by
(4)
μ
j
L
f
(
x
)
=
(
∫
0
∞
|
∫
|
x
-
y
|
≤
t
K
j
L
(
x
,
y
)
f
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
,
where K
j
L
(
x
,
y
)
=
K
~
j
L
(
x
,
y
)
|
x
-
y
| and K
~
j
L
(
x
,
y
) is the kernel of R
j
L
=
(
∂
/
∂
x
j
)
L
-
1
/
2, j
=
1
,
…
,
d. In particular, when V
=
0, K
j
Δ
(
x
,
y
)
=
K
j
Δ
~
(
x
,
y
)
|
x
-
y
|
=
(
(
x
j
-
y
j
)
/
|
x
-
y
|
)
/
|
x
-
y
|
d
-
1 and K
j
Δ
~
(
x
,
y
) is the kernel of R
j
=
(
∂
/
∂
x
j
)
Δ
-
1
/
2, j
=
1
,
…
,
d. So, μ
j
Δ is defined by
(5)
μ
j
Δ
f
(
x
)
=
(
∫
0
∞
|
∫
|
x
-
y
|
≤
t
K
j
Δ
(
x
,
y
)
f
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
.Lemma 1 (see [2]).
For anyl
>
0, there exists C
l
>
0 such that
(6)
|
K
j
L
(
x
,
y
)
|
≤
C
l
(
1
+
|
x
-
y
|
/
ρ
(
y
)
)
l
1
|
x
-
y
|
d
-
1
,
|
K
j
L
(
x
,
y
)
-
K
j
(
x
,
y
)
|
≤
C
ρ
(
x
)
|
x
-
y
|
d
-
2
,
where ρ is the auxiliary function as follows.Gao and Tang [2] have shown that Marcinkiewicz integral μ
j
L is bounded on L
p
(
R
d
) for 1
<
p
<
∞ and bounded from L
1
(
R
d
) to weak L
1
(
R
d
). Meanwhile they also proved μ
j
L are bounded on BMO
L
(
R
d
) and also mapped from H
L
1
(
R
d
) to L
1
(
R
d
) under the assumption that K
j
L satisfy the condition in Lemma 1.WhenK
j
L satisfies the estimates in Lemma 8 in Section 2 below, Chen and Zou [3] also proved that the Marcinkiewicz integral μ
j
L has the same boundedness.Now we give the definition of the commutator generalized byμ
j
L and b by
(7)
[
b
,
μ
j
L
]
(
f
)
(
x
)
=
(
∫
0
∞
|
∫
|
x
-
y
|
≤
t
K
j
L
(
x
,
y
)
(
b
(
x
)
-
b
(
y
)
)
f
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
,
and the definition of the commutator generalized by μ
j
Δ and b by
(8)
[
b
,
μ
j
Δ
]
(
f
)
(
x
)
=
(
∫
0
∞
|
∫
|
x
-
y
|
≤
t
K
j
Δ
(
x
,
y
)
(
b
(
x
)
-
b
(
y
)
)
f
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
.Letρ
(
x
) be the auxiliary function defined by
(9)
ρ
(
x
)
=
1
m
V
(
x
)
=
sup
⁡
{
r
>
0
:
1
r
d
-
2
∫
B
(
x
,
r
)
V
≤
1
}
,
x
∈
R
d
.Obviously,0
<
m
V
(
x
)
<
∞ if V
≠
0. In particular, m
V
(
x
)
=
1 with V
=
1 and m
V
(
x
)
~
(
1
+
|
x
|
) with V
=
|
x
|
2.In this paper, we writeΨ
(
B
)
=
(
1
+
r
m
V
(
B
)
)
θ where m
V
(
B
)
=
(
1
/
|
B
|
)
∫
B
m
V
(
x
)
d
x and θ
>
0, and r denotes the radius of B.The maximal operatorM
V
f
(
x
) is defined by
(10)
M
V
f
(
x
)
=
sup
⁡
x
∈
B
1
Ψ
(
B
)
|
B
|
∫
B
|
f
(
x
)
|
d
x
.
When V
=
0, we denote M
0
f
(
x
) by M
f
(
x
) (the standard Hardy-Littlewood maximal function). It is easy to see that |
f
(
x
)
|
≤
M
V
f
(
x
)
≤
M
f
(
x
) for a.e x
∈
R
d. This information can be found in [4].Proposition 2 (see [5]).
There existsk
0
>
0 such that
(11)
1
C
(
1
+
|
x
-
y
|
ρ
(
x
)
)
-
k
0
≤
ρ
(
y
)
ρ
(
x
)
≤
C
(
1
+
|
x
-
y
|
ρ
(
x
)
)
k
0
/
(
k
0
+
1
)
.
In particular,ρ
(
x
)
~
ρ
(
y
) if |
x
-
y
|
<
C
ρ
(
x
).Proposition 3 (see [6]).
There exists a sequence of pointsx
j, j
≥
1, in R
d, so that the family Q
j
=
B
(
x
j
,
ρ
(
x
j
)
), j
≥
1, satisfies the following. (i)
⋃
j
Q
j
=
R
d.
(ii)
For everyσ
≥
1 there exist constants C and N
1 such that ∑
j
χ
σ
Q
j
≤
C
σ
N
1.
Tang and Dong [7] first introduced some type Morrey space associated with Schrödinger operators. Meanwhile they obtained the strong and weak boundedness of singular integral, fractional integral, and their commutators in Morrey spaces. Inspired by their work, we give another type of Morrey space, but in fact these two kinds of Morrey spaces are the same when their exponents are restricted in some exceeding. Recently, plenty of famous results on Schrödinger operators have appeared;we refer to [8–13].We now present the definition of Morrey spaces associated with Schrödinger operators which we needed in this paper.Definition 4.
Letf
∈
L
loc
⁡
q
(
R
d
), 1
≤
p
≤
q
<
+
∞; we say f
∈
M
p
q
(
R
d
) (Morrey spaces) provided that
(12)
∥
f
∥
M
p
q
(
R
d
)
=
sup
⁡
B
⊂
R
d
|
B
|
1
/
q
-
1
/
p
(
∫
B
|
f
(
x
)
|
p
d
x
)
1
/
p
<
+
∞
,
where B
=
B
(
x
0
,
r
).Obviously, whenp
=
q, the space M
p
q
(
R
d
) is the class Lebesgue space L
p
(
R
d
).Definition 5.
Letf
∈
L
loc
⁡
q
(
R
d
), 1
≤
p
≤
q
<
+
∞, α
∈
(
-
∞
,
+
∞
), and V
∈
RH
q
(
q
>
1
); we say f
∈
M
p
,
V
q
,
α
(
R
d
) (Morrey space associated with Schrödinger operators) provided that
(13)
∥
f
∥
M
p
,
V
q
,
α
(
R
d
)
=
sup
⁡
B
⊂
R
d
(
1
+
r
ρ
(
x
0
)
)
α
|
B
|
1
/
q
-
1
/
p
×
(
∫
B
‍
|
f
(
x
)
|
p
d
x
)
1
/
p
<
+
∞
,
where B
=
B
(
x
0
,
r
).In this note we will investigate the boundedness for the commutators of Marcinkiewicz inegral associated with Schrödinger operators on Morrey spaces given in Definition5.Our results can be formulated as follows.Theorem 6.
Supposeα
∈
(
-
∞
,
+
∞
) and V
∈
R
H
q
(
q
>
1
).(i)
If1
<
p
≤
q
<
+
∞, then
(14)
∥
μ
j
L
f
∥
M
p
,
V
q
,
α
(
R
d
)
≤
C
∥
f
∥
M
p
,
V
q
,
α
(
R
d
)
,
whereC is independent of f.
(ii)
If1
=
p
≤
q
<
+
∞, then, for any λ
>
0,
(15)
λ
(
1
+
r
ρ
(
x
)
)
α
|
{
y
∈
B
(
x
,
r
)
:
|
μ
j
L
f
(
y
)
|
>
λ
}
|
≤
C
|
B
(
x
,
r
)
|
1
-
1
/
q
∥
f
∥
M
p
,
V
q
,
α
(
R
d
)
holds for all ballsB
(
x
,
r
), where C is independent of x, r, λ, and f.Forθ
>
0, we define the class BMO
θ
(
ρ
) of locally integrable functions b such that
(16)
1
|
B
(
x
,
r
)
|
∫
B
(
x
,
r
)
|
b
(
y
)
-
b
B
|
d
y
≤
C
(
1
+
r
ρ
(
x
)
)
θ
,
for all x
∈
R
d and r
>
0, where b
B
=
(
1
/
|
B
|
)
∫
B
b. A norm for b
∈
BMO
θ
(
ρ
), denoted by [
b
]
θ, is given by the infimum of the constants in the inequalities above. Notice that if we let θ
=
0, we obtain the John-Nirenberg space BMO.Theorem 7.
Supposeb
∈
B
M
O
θ
(
ρ
), θ
>
0, α
∈
(
-
∞
,
+
∞
).(i)
If1
<
p
≤
q
<
+
∞, then
(17)
∥
[
b
,
μ
j
L
]
f
∥
M
p
,
V
q
,
α
(
R
d
)
≤
C
[
b
]
θ
∥
f
∥
M
p
,
V
q
,
α
(
R
d
)
,
whereC is independent of f.
(ii)
There exists a constantC
>
0, such that, for all λ
>
0,
(18)
λ
(
1
+
r
ρ
(
x
)
)
α
|
{
y
∈
R
d
:
|
[
b
,
μ
j
L
]
f
(
y
)
|
>
λ
}
|
≤
C
|
B
(
x
,
r
)
|
1
-
1
/
q
∫
R
d
‍
|
f
(
y
)
|
λ
(
1
+
log
⁡
+
|
f
(
y
)
|
λ
)
d
y
.Throughout this paper,C denotes the constants that are independent of the main parameters involved but whose value may differ from line to line. By A
~
B, we mean that there exists a constant C
>
1 such that 1
/
C
≤
A
/
B
≤
C.


## 2. Notation and Preliminaries

Shen [5] gave the following kernel estimate that we need.Lemma 8.
IfV
∈
R
H
q
(
q
>
1
), then one has(i)
for everyN there exists a constant C such that
(19)
|
K
j
L
(
x
,
z
)
|
≤
C
(
1
+
|
x
-
z
|
/
ρ
(
x
)
)
-
N
|
x
-
z
|
d
-
1
,
(ii)
for everyN and 0
<
δ
<
min
⁡
{
1,1
-
d
/
q
0
} there exists a constant C such that
(20)
|
K
j
L
(
x
,
z
)
-
K
j
L
(
y
,
z
)
|
≤
C
|
x
-
y
|
δ
(
1
+
|
x
-
z
|
/
ρ
(
x
)
)
-
N
|
x
-
z
|
d
-
1
+
δ
,
where|
x
-
y
|
<
(
2
/
3
)
|
x
-
z
|,
(iii)
for every0
<
σ
<
2
-
d
/
q
0, one has
(21)
|
K
j
L
(
x
,
z
)
-
K
j
Δ
(
x
,
z
)
|
≤
C
|
x
-
z
|
d
-
1
(
|
x
-
z
|
ρ
(
z
)
)
σ
,
whereK
j
Δ
(
x
,
z
)
=
(
(
x
j
-
z
j
)
/
|
x
-
z
|
)
/
|
x
-
z
|
d
-
1.Lemma 9 (see [14]).
Letθ
>
0 and 1
≤
s
<
∞. If b
∈
B
M
O
θ
(
ρ
), then
(22)
(
1
|
B
|
∫
B
‍
|
b
-
b
B
|
s
)
1
/
s
≤
C
[
b
]
θ
(
1
+
r
ρ
(
x
)
)
θ
′
,
for all B
=
B
(
x
,
r
), with x
∈
R
d and r
>
0, where θ
′
=
(
k
0
+
1
)
θ and k
0 is the constant appearing in Proposition 2.Corollary 10 (see [14]).
Letb
∈
B
M
O
θ
(
ρ
), B
=
B
(
x
0
,
r
), and s
≥
1; then
(23)
(
1
|
2
k
B
|
∫
2
k
B
‍
|
b
-
b
B
|
s
)
1
/
s
≤
C
[
b
]
θ
k
(
1
+
2
k
r
ρ
(
x
0
)
)
θ
′
,
for all k
∈
N, with θ
′ as in Lemma 9.From Lemma9, the author [15] proved the John-Nirenberg inequality for BMO
θ
(
ρ
).Proposition 11 (see [15]).
Suppose thatf is in B
M
O
θ
(
ρ
). There exist positive constants γ and C such that
(24)
sup
⁡
B
1
|
B
|
∫
B
exp
⁡
{
γ
[
f
]
θ
Ψ
θ
′
(
B
)
|
f
(
x
)
-
f
B
|
}
d
x
≤
C
,
where f
B
=
(
1
/
|
B
|
)
∫
B
f
(
y
)
d
y and Ψ
θ
′
(
B
)
=
(
1
+
r
/
ρ
(
x
0
)
)
θ
′, B
=
B
(
x
0
,
r
), and θ
′
=
(
k
0
+
1
)
θ.The dyadic maximal operatorM
V
Δ
f
(
x
) is defined by
(25)
M
V
Δ
f
(
x
)
=
sup
⁡
x
∈
Q
1
Ψ
(
Q
)
|
Q
|
∫
Q
|
f
(
x
)
|
d
x
,
where Q is a dyadic cube.The dyadic sharp maximal operatorM
V
♯
f
(
x
) is defined by
(26)
M
V
♯
f
(
x
)
=
sup
⁡
x
∈
Q
x
0
,
r
<
ρ
(
x
0
)
1
|
Q
x
0
|
∫
Q
x
0
|
f
(
y
)
-
f
Q
x
0
|
d
y
+
sup
⁡
x
∈
Q
x
0
,
r
≥
ρ
(
x
0
)
1
Ψ
(
Q
x
0
)
|
Q
x
0
|
∫
Q
x
0
|
f
(
y
)
|
d
y
≃
sup
⁡
x
∈
Q
x
0
,
r
<
ρ
(
x
0
)
inf
⁡
C
1
|
Q
x
0
|
∫
Q
x
0
|
f
(
y
)
-
C
|
d
y
+
sup
⁡
x
∈
Q
x
0
,
r
≥
ρ
(
x
0
)
1
Ψ
(
Q
x
0
)
|
Q
x
0
|
∫
Q
x
0
|
f
(
y
)
|
d
y
,
where Q
x
0 denotes dyadic cubes Q
(
x
0
,
r
) and f
Q
=
(
1
/
|
Q
|
)
∫
Q
f
(
x
)
d
x.A variant of dyadic maximal operator and dyadic sharp maximal operator is defined as following:(27)
M
δ
,
V
Δ
f
(
x
)
=
M
V
Δ
(
|
f
|
δ
)
1
/
δ
(
x
)
,
M
δ
,
V
♯
f
(
x
)
=
M
V
♯
(
|
f
|
δ
)
1
/
δ
(
x
)
.In our paper, we need the following proposition whenω
=
1.Proposition 12 (see [15]).
Let1
<
p
<
∞ and suppose that ω
∈
A
p
ρ. If p
<
p
1
<
∞, then the equality
(28)
∫
R
d
|
M
V
f
(
x
)
|
p
1
ω
(
x
)
d
x
≤
C
p
∫
R
d
|
f
(
x
)
|
p
1
ω
(
x
)
d
x
.
Further, let 1
≤
p
<
∞, ω
∈
A
p
ρ, if and only if
(29)
ω
(
{
x
∈
R
d
:
M
V
f
(
x
)
>
λ
}
)
≤
C
p
λ
p
∫
R
d
|
f
(
x
)
|
p
ω
(
x
)
d
x
.A functionA
:
[
0
,
∞
)
→
[
0
,
∞
) is said to be a Young function if it is continuous, convex, and increasing satisfying A
(
0
)
=
0 and A
(
t
)
→
∞ as t
→
∞. We defined the A-average of a function f over a cube Q by means of the following Luxemburg norm:
(30)
∥
f
∥
A
,
Q
=
inf
⁡
⁡
{
λ
>
0
:
1
|
Q
|
∫
Q
A
(
|
f
(
y
)
|
λ
)
d
y
≤
1
}
,
and the maximal operator M
A associated to ∥
·
∥
A
,
Q by M
A
f
(
x
)
=
sup
⁡
x
∈
Q
∥
f
∥
A
,
Q. Then the following generalized Hölder inequality holds:
(31)
1
|
Q
|
∫
Q
|
f
(
y
)
g
(
y
)
|
d
y
≤
∥
f
∥
A
,
Q
∥
g
∥
A
¯
,
Q
,
where A
¯ is the complementary Young function of A.We define the corresponding maximal function(32)
M
A
f
(
x
)
=
sup
⁡
x
∈
Q
∥
f
∥
A
,
Q
,
M
V
,
A
f
(
x
)
=
sup
⁡
x
∈
Q
Ψ
(
Q
)
-
1
∥
f
∥
A
,
Q
.In [16], one has given a general result that can be applied to prove the boundedness of the localized classical operators. One considers a covering of balls {
Q
j
} such that the family of a fixed dilation of them, {
Q
~
j
}, has bounded overlapping (e.g. a covering associated to ρ like in Proposition 3).An operatorS is defined by
(33)
S
(
f
)
=
∑
j
‍
χ
Q
j
|
S
(
f
χ
Q
~
j
)
|
.Proposition 13 (see [16]).
Let1
≤
p
≤
ν
<
∞, and a weight ω on R
d with the following property: for each j, ω
|
Q
~
j, admits an extension ω
j to R
d such that
(34)
S
:
L
p
(
ω
j
)
⟼
L
ν
(
ω
j
ν
/
p
)
bound with a constant independent of j. Then,
(35)
S
0
:
L
p
(
ω
j
)
⟼
L
ν
(
ω
j
ν
/
p
)
continuously. If p
=
1, the assumption of S is changed by weak type (
1
,
ν
), and the corresponding weak type can be concluded for S
0.In our paper, setS
=
μ
j
Δ and ω
=
1. As we know, μ
j
Δ is bounded from L
p to L
p. So, Proposition 13 also holds for μ
j
Δ and ω
=
1.From [16], we have the following result.Lemma 14.
Let(
μ
j
Δ
)
loc
⁡
(
f
)
(
x
)
=
μ
j
Δ
(
f
χ
B
(
x
,
ρ
(
x
)
)
)
(
x
). Assume that 1
<
p
<
∞. Then there exists a constant C
>
0 such that
(36)
∫
R
d
|
(
μ
j
Δ
)
loc
⁡
(
f
)
(
x
)
|
p
d
x
≤
C
∫
R
d
|
f
(
x
)
|
p
d
x
.
Furthermore, when p
=
1, there exists a constant C such that, for all λ
>
0,
(37)
|
{
x
∈
R
d
:
(
μ
j
Δ
)
loc
⁡
(
f
)
(
x
)
>
λ
}
|
≤
C
λ
∫
R
d
|
f
(
x
)
|
d
x
.Proof.
Letσ
=
C
2
k
0
/
(
k
0
+
1
), with C and k
0 as in Proposition 2. Let {
Q
j
} be the family given by Proposition 3 and set Q
~
j
=
σ
Q
j.
Clearly, we have(38)
⋃
x
∈
Q
j
B
(
x
,
ρ
(
x
)
)
⊂
Q
~
j
.
First, forx
∈
Q
j, Minkowski’s inequality says that
(39)
|
(
μ
j
Δ
)
loc
⁡
(
f
)
(
x
)
-
μ
j
Δ
(
χ
Q
~
j
f
)
(
x
)
|
≤
C
(
×
(
χ
Q
~
j
-
χ
B
(
x
,
ρ
(
x
)
)
)
(
y
)
d
y
∫
|
x
-
y
|
≤
t
‍
|
2
∫
0
∞
‍
|
∫
|
x
-
y
|
≤
t
‍
K
j
Δ
(
x
,
y
)
f
(
y
)
×
(
χ
Q
~
j
-
χ
B
(
x
,
ρ
(
x
)
)
)
(
y
)
d
y
∫
|
x
-
y
|
≤
t
‍
|
2
d
t
t
3
)
1
/
2
≤
C
∫
R
d
|
f
(
y
)
(
χ
Q
~
j
-
χ
B
(
x
,
ρ
(
x
)
)
)
(
y
)
|
000000000
×
|
K
j
Δ
(
x
,
y
)
|
(
∫
|
x
-
y
|
≤
t
d
t
t
3
)
1
/
2
≤
C
∫
Q
~
j
∖
B
(
x
,
ρ
(
x
)
)
|
f
(
y
)
|
|
x
j
-
y
j
|
|
x
-
y
|
d
|
x
-
y
|
-
1
d
y
≤
C
∫
Q
~
j
∖
B
(
x
,
ρ
(
x
)
)
|
f
(
y
)
|
|
x
-
y
|
d
d
y
≤
C
1
|
Q
~
j
|
∫
Q
~
j
|
f
(
y
)
|
d
y
,
where we have used |
x
-
y
|
≥
ρ
(
x
)
≃
ρ
(
x
j
) when x
∈
Q
j and y
∉
B
(
x
,
ρ
(
x
)
).
Therefore, for1
<
p
<
∞,
(40)
∥
(
μ
j
Δ
)
loc
⁡
(
f
)
∥
L
p
p
≤
C
∑
j
‍
∫
Q
j
‍
(
1
|
Q
~
j
|
∫
Q
~
j
|
f
(
y
)
|
d
y
)
p
d
x
+
∥
(
μ
j
Δ
)
0
(
f
)
∥
L
p
p
≤
C
∑
j
∫
Q
j
1
|
Q
~
j
|
∫
Q
~
j
|
f
(
y
)
|
p
d
y
d
x
+
∥
(
μ
j
Δ
)
0
(
f
)
∥
L
p
p
≤
C
∑
j
∫
Q
~
j
|
f
(
y
)
|
p
d
y
+
∥
(
μ
j
Δ
)
0
(
f
)
∥
L
p
p
≤
C
∫
R
d
|
f
(
y
)
|
p
d
y
+
∥
(
μ
j
Δ
)
0
(
f
)
∥
L
p
p
.
Hence from Proposition13 we have done it 1
<
p
<
∞.
For the casep
=
1, using the estimate of |
(
μ
j
Δ
)
loc
⁡
(
f
)
(
x
)
-
μ
j
Δ
(
χ
Q
~
j
f
)
(
x
)
| again, we have, for each j,
(41)
|
{
x
∈
Q
j
:
|
(
μ
j
Δ
)
loc
⁡
(
f
)
(
x
)
-
μ
j
Δ
(
χ
Q
~
j
f
)
(
x
)
|
>
λ
}
|
≤
C
1
λ
∫
Q
~
j
|
f
(
y
)
|
d
y
.
Besides, from Proposition 13,
(42)
|
{
x
∈
Q
j
:
|
(
μ
j
Δ
)
0
(
χ
Q
~
j
f
)
(
x
)
|
>
λ
}
|
≤
C
1
λ
∫
Q
~
j
|
f
(
y
)
|
d
y
.Therefore, summing overj we have the weak type (
1,1
).Lemma 15.
Letb
∈
B
M
O
θ
(
ρ
), and (
k
0
+
1
)
≤
η
<
∞. Set [
b
,
μ
j
Δ
]
loc
⁡
(
f
)
(
x
)
=
[
b
,
μ
j
Δ
]
(
f
χ
B
(
x
,
ρ
(
x
)
)
)
(
x
). Let 0
<
2
δ
<
ϵ
<
1. Then,
(43)
M
δ
,
η
♯
(
[
b
,
μ
j
Δ
]
loc
⁡
(
f
)
)
(
x
)
≤
C
[
b
]
θ
(
M
ϵ
,
η
Δ
(
(
μ
j
Δ
)
loc
⁡
(
f
)
)
(
x
)
0000000000
+
M
L
log
⁡
L
,
V
,
η
(
f
)
(
x
)
(
(
μ
j
Δ
)
loc
⁡
(
f
)
)
)
,
a
.
e
.
x
∈
R
d
,
holds for any f
∈
C
0
∞
(
R
d
).Proof.
We fixx
∈
R
d and let x
∈
Q
=
Q
(
x
0
,
r
) (dyadic cube). We consider two cases about r; that is, r
<
ρ
(
x
0
) and r
≥
ρ
(
x
0
).
Case  1. When r
<
ρ
(
x
0
), decomposing f
=
f
1
+
f
2, where f
1
=
f
χ
Q
*, where Q
*
=
Q
(
x
0
,
4
d
r
). Let C
Q a constant be fixed along the proof. Since 0
<
δ
<
1, we then have
(44)
(
1
|
Q
|
∫
Q
|
|
[
b
,
μ
j
Δ
]
loc
⁡
(
f
)
(
y
)
|
δ
-
|
C
Q
|
δ
|
d
y
)
1
/
δ
≤
(
1
|
Q
|
∫
Q
|
[
b
,
μ
j
Δ
]
loc
⁡
(
f
)
(
y
)
-
C
Q
|
δ
d
y
)
1
/
δ
≤
C
(
1
|
Q
|
∫
Q
|
(
b
(
y
)
-
b
Q
*
)
(
μ
j
Δ
)
loc
⁡
(
f
)
(
y
)
|
δ
d
y
)
1
/
δ
+
C
(
1
|
Q
|
∫
Q
|
(
μ
j
Δ
)
loc
⁡
(
(
b
-
b
Q
*
)
f
1
)
(
y
)
|
δ
d
y
)
1
/
δ
+
C
(
1
|
Q
|
∫
Q
|
(
μ
j
Δ
)
loc
⁡
(
(
b
-
b
Q
*
)
f
2
)
(
y
)
-
C
Q
|
δ
d
y
)
1
/
δ
=
I
+
II
+
III
,
where b
Q
*
=
(
1
/
|
Q
*
|
)
∫
Q
*
b
(
y
)
d
y.
We start withI. For any 1
<
γ
<
ϵ
/
δ, note that M
V
(
x
)
~
M
V
(
x
0
) for any x
∈
Q
* and Ψ
(
Q
*
)
~
1; by Lemma 9, we obtain
(45)
I
≤
(
1
|
Q
*
|
∫
Q
*
|
b
(
y
)
-
b
Q
*
|
δ
γ
′
d
y
)
1
/
(
δ
γ
′
)
×
(
1
|
Q
|
∫
Q
|
(
μ
j
Δ
)
loc
⁡
(
f
)
(
y
)
|
δ
γ
d
y
)
1
/
(
δ
γ
)
≤
C
[
b
]
θ
M
ϵ
,
η
Δ
(
(
μ
j
Δ
)
loc
⁡
(
f
)
)
(
x
)
,
where 1
/
γ
+
1
/
γ
′
=
1.
ForII, note that M
V
(
x
)
~
M
V
(
x
0
) for any x
∈
Q
* and Ψ
(
Q
*
)
~
1; by Kolmogorov’s inequality, Proposition 11, and Lemma 14, we have
(46)
II
≤
C
|
Q
|
∥
(
μ
j
Δ
)
loc
⁡
(
(
b
-
b
Q
*
)
f
1
)
∥
L
1
,
∞
≤
C
|
Q
*
|
∫
Q
*
|
(
b
(
y
)
-
b
Q
*
)
f
(
y
)
|
d
y
≤
C
[
b
]
θ
M
L
log
⁡
L
,
V
,
η
(
f
)
(
x
)
.
To deal withIII, we first fix the value of C
Q by taking C
Q
=
(
μ
j
Δ
)
loc
⁡
(
(
b
-
b
Q
*
)
f
2
)
(
y
0
) with y
0
∈
Q; we have
(47)
|
(
μ
j
Δ
)
loc
⁡
(
(
b
-
b
Q
*
)
f
2
)
(
y
)
-
(
μ
j
Δ
)
loc
⁡
(
(
b
-
b
Q
*
)
f
2
)
(
y
0
)
|
≤
(
×
f
2
(
z
)
χ
B
(
y
,
ρ
(
y
)
)
K
j
Δ
|
d
z
∫
|
z
-
y
|
≤
t
<
|
z
-
y
0
|
|
K
j
Δ
(
y
,
z
)
(
b
(
z
)
-
b
Q
*
)
|
2
∫
0
∞
|
∫
|
z
-
y
|
≤
t
<
|
z
-
y
0
|
|
K
j
Δ
(
y
,
z
)
(
b
(
z
)
-
b
Q
*
)
00000000000000000.000
×
f
2
(
z
)
χ
B
(
y
,
ρ
(
y
)
)
K
j
Δ
|
d
z
∫
|
z
-
y
|
≤
t
<
|
z
-
y
0
|
|
K
j
Δ
(
y
,
z
)
(
b
(
z
)
-
b
Q
*
)
|
2
d
t
t
3
)
1
/
2
+
(
×
f
2
(
z
)
χ
B
(
y
0
,
ρ
(
y
0
)
)
K
j
Δ
|
d
z
∫
|
z
-
y
0
|
≤
t
<
|
z
-
y
|
‍
|
2
∫
0
∞
‍
|
∫
|
z
-
y
0
|
≤
t
<
|
z
-
y
|
‍
|
K
j
Δ
(
y
0
,
z
)
(
b
(
z
)
-
b
Q
*
)
00000000000000.0000000
×
f
2
(
z
)
χ
B
(
y
0
,
ρ
(
y
0
)
)
K
j
Δ
|
d
z
∫
|
z
-
y
0
|
≤
t
<
|
z
-
y
|
‍
|
2
d
t
t
3
)
1
/
2
+
(
×
χ
B
(
y
0
,
ρ
(
y
0
)
)
|
d
z
∫
{
|
z
-
y
0
|
≤
t
,
|
z
-
y
|
≤
t
}
‍
|
2
∫
0
∞
‍
|
∫
{
|
z
-
y
0
|
≤
t
,
|
z
-
y
|
≤
t
}
‍
|
K
j
Δ
(
y
,
z
)
-
K
j
Δ
(
y
0
,
z
)
|
000000000000000000000
×
|
χ
B
(
y
0
,
ρ
(
y
0
)
)
(
b
(
z
)
-
b
Q
*
)
f
2
(
z
)
000000000000000000000000
×
χ
B
(
y
0
,
ρ
(
y
0
)
)
|
d
z
∫
{
|
z
-
y
0
|
≤
t
,
|
z
-
y
|
≤
t
}
‍
|
2
d
t
t
3
)
1
/
2
+
(
×
f
2
(
z
)
χ
B
(
y
0
,
ρ
(
y
0
)
)
K
j
Δ
|
d
z
∫
{
|
z
-
y
0
|
≤
t
,
|
z
-
y
|
≤
t
}
‍
|
2
∫
0
∞
‍
|
∫
{
|
z
-
y
0
|
≤
t
,
|
z
-
y
|
≤
t
}
‍
|
K
j
Δ
(
y
,
z
)
(
b
(
z
)
-
b
Q
*
)
000000000000000000000000
×
f
2
(
z
)
χ
B
(
y
0
,
ρ
(
y
0
)
)
K
j
Δ
|
d
z
∫
{
|
z
-
y
0
|
≤
t
,
|
z
-
y
|
≤
t
}
‍
|
2
d
t
t
3
)
1
/
2
+
(
×
f
2
(
z
)
χ
B
(
y
,
ρ
(
y
)
)
|
d
z
∫
{
|
z
-
y
0
|
≤
t
,
|
z
-
y
|
≤
t
}
‍
|
2
∫
0
∞
‍
|
∫
{
|
z
-
y
0
|
≤
t
,
|
z
-
y
|
≤
t
}
‍
|
χ
B
(
y
,
ρ
(
y
)
)
K
j
Δ
(
y
,
z
)
(
b
(
z
)
-
b
Q
*
)
00000000000.000000000000
×
f
2
(
z
)
χ
B
(
y
,
ρ
(
y
)
)
|
d
z
∫
{
|
z
-
y
0
|
≤
t
,
|
z
-
y
|
≤
t
}
‍
|
2
d
t
t
3
)
1
/
2
=
E
1
+
E
2
+
E
3
+
E
4
+
E
5
.
ForE
1, since ρ
(
y
)
~
ρ
(
x
0
) and |
z
-
y
|
~
|
z
-
y
0
|
~
|
z
-
x
0
|, using Minkowski’s inequality and Proposition 11, we obtain
(48)
E
1
≤
C
∫
R
d
|
K
j
Δ
(
y
,
z
)
(
b
(
z
)
-
b
Q
*
)
f
2
(
z
)
χ
B
(
y
,
ρ
(
y
)
)
|
×
(
∫
|
z
-
y
|
≤
t
<
|
z
-
y
0
|
d
t
t
3
)
1
/
2
d
z
≤
C
r
1
/
2
∫
(
Q
*
)
c
⋂
B
(
y
,
ρ
(
y
)
)
|
(
b
(
z
)
-
b
Q
*
)
f
(
z
)
|
×
|
z
j
-
y
j
|
/
|
z
-
y
|
|
z
-
y
|
d
-
1
1
|
z
-
y
|
3
/
2
d
z
≤
C
r
1
/
2
∫
(
Q
*
)
c
⋂
B
(
y
,
ρ
(
y
)
)
|
(
b
(
z
)
-
b
Q
*
)
f
(
z
)
|
×
1
|
z
-
y
|
d
+
1
/
2
d
z
≤
C
r
1
/
2
∫
4
d
r
<
|
z
-
x
0
|
≤
ρ
(
x
0
)
|
(
b
(
z
)
-
b
Q
*
)
f
(
z
)
|
×
1
|
z
-
x
0
|
d
+
1
/
2
d
z
≤
C
∑
k
=
2
N
0
2
-
k
/
2
(
2
k
d
r
)
d
∫
|
z
-
x
0
|
≤
2
k
+
1
d
r
|
(
b
(
z
)
-
b
Q
*
)
f
(
z
)
|
d
z
≤
C
[
b
]
θ
M
L
log
⁡
L
,
V
,
η
(
f
)
(
x
)
,
where the integer k
0 satisfies 2
N
0
d
r
≤
ρ
(
x
0
)
≤
2
N
0
+
1
d
r.
Similarly,(49)
E
2
≤
C
[
b
]
θ
M
L
log
⁡
L
,
V
,
η
(
f
)
(
x
)
.
ForE
3, since ρ
(
y
0
)
~
ρ
(
x
0
) and |
z
-
y
|
~
|
z
-
y
0
|
~
|
z
-
x
0
|, by Minkowski’s inequality and Proposition 11, we have
(50)
E
3
≤
C
∫
R
d
‍
|
K
j
Δ
(
y
,
z
)
-
K
j
Δ
(
y
0
,
z
)
|
×
|
(
b
(
z
)
-
b
Q
*
)
f
2
(
z
)
χ
B
(
y
0
,
ρ
(
y
0
)
)
|
×
(
∫
{
|
z
-
y
0
|
≤
t
,
|
z
-
y
|
≤
t
}
‍
d
t
t
3
)
1
/
2
d
z
≤
C
∫
(
Q
*
)
c
⋂
‍
B
(
y
0
,
ρ
(
y
0
)
)
|
(
b
(
z
)
-
b
Q
*
)
f
(
z
)
|
×
|
y
j
-
y
0
j
|
|
z
-
y
0
|
d
1
|
z
-
y
0
|
d
z
≤
C
∫
4
d
r
<
|
z
-
x
0
|
≤
ρ
(
x
0
)
|
(
b
(
z
)
-
b
Q
*
)
f
(
z
)
|
|
y
-
y
0
|
|
z
-
x
0
|
d
+
1
d
z
≤
C
∑
k
=
2
N
0
‍
r
(
2
k
d
r
)
d
+
1
∫
|
z
-
x
0
|
≤
2
k
+
1
d
r
|
(
b
(
z
)
-
b
Q
*
)
f
(
z
)
|
d
z
≤
C
∑
k
=
2
N
0
‍
2
-
k
(
2
k
d
r
)
d
∫
|
z
-
x
0
|
≤
2
k
+
1
d
r
|
(
b
(
z
)
-
b
Q
*
)
f
(
z
)
|
d
z
≤
C
[
b
]
θ
M
L
log
⁡
L
,
V
,
η
(
f
)
(
x
)
,
where the integer N
0 is the same as above.
ForE
4, since ρ
(
y
0
)
~
ρ
(
x
0
) and |
z
-
y
|
~
|
z
-
y
0
|
~
|
z
-
x
0
|, using Minkowski’s inequality and Proposition 11, we obtain
(51)
E
4
≤
C
∫
R
d
‍
|
K
j
Δ
(
y
,
z
)
(
b
(
z
)
-
b
Q
*
)
f
2
(
z
)
χ
B
(
y
0
,
ρ
(
y
0
)
)
|
×
(
∫
{
|
z
-
y
0
|
≤
t
,
|
z
-
y
|
≤
t
}
‍
d
t
t
3
)
1
/
2
d
z
≤
C
∫
(
Q
*
)
c
⋂
‍
B
(
y
0
,
ρ
(
y
0
)
)
‍
|
(
b
(
z
)
-
b
Q
*
)
f
(
z
)
|
000000000000000
×
|
z
j
-
y
j
|
/
|
z
-
y
|
|
z
-
y
|
d
-
1
1
|
z
-
y
|
d
z
≤
C
∫
4
d
r
<
|
z
-
x
0
|
≤
ρ
(
x
0
)
‍
|
(
b
(
z
)
-
b
Q
*
)
f
(
z
)
|
1
|
z
-
x
0
|
d
d
z
≤
C
∑
k
=
2
N
0
‍
1
(
2
k
d
r
)
d
∫
|
z
-
x
0
|
≤
2
k
+
1
d
r
‍
|
(
b
(
z
)
-
b
Q
*
)
f
(
z
)
|
d
z
≤
C
[
b
]
θ
M
L
log
⁡
L
,
V
,
η
(
f
)
(
x
)
,
where the integer N
0 is the same as above, and we know it is finite.
Similarly,(52)
E
5
≤
C
[
b
]
θ
M
L
log
⁡
L
,
V
,
η
(
f
)
(
x
)
.
So,(53)
III
≤
C
|
Q
|
∫
Q
‍
|
(
μ
j
Δ
)
loc
⁡
(
(
b
-
b
Q
*
)
f
2
)
(
y
)
00000000
-
(
μ
j
Δ
)
loc
⁡
(
(
b
-
b
Q
*
)
f
2
)
(
y
0
)
|
d
y
≤
C
[
b
]
θ
M
L
log
⁡
L
,
V
,
η
(
f
)
(
x
)
.
Case  2. When r
≥
ρ
(
x
0
), decomposing f
=
f
1
+
f
2, where f
1
=
f
χ
Q
*, where Q
*
=
Q
(
x
0
,
C
0
2
k
0
+
4
d
r
). Since 0
<
2
δ
<
ϵ
<
1, so a
=
η
/
δ and ϵ
/
δ
>
2; then,
(54)
1
Ψ
(
Q
)
a
(
1
|
Q
|
∫
Q
|
[
b
,
μ
j
Δ
]
loc
⁡
(
f
)
(
y
)
|
δ
d
y
)
1
/
δ
≤
1
Ψ
(
Q
)
a
(
1
|
Q
|
∫
Q
|
(
b
(
y
)
-
b
Q
*
)
(
μ
j
Δ
)
loc
⁡
(
f
)
(
y
)
00000000000000000
+
(
μ
j
Δ
)
loc
⁡
(
(
b
-
b
Q
*
)
f
)
(
y
)
|
δ
d
y
1
|
Q
|
∫
Q
|
(
b
(
y
)
-
b
Q
*
)
(
μ
j
Δ
)
loc
⁡
(
f
)
(
y
)
)
1
/
δ
≤
C
1
Ψ
(
Q
)
a
(
1
|
Q
|
∫
Q
‍
|
(
b
(
y
)
-
b
Q
*
)
(
μ
j
Δ
)
loc
⁡
(
f
)
(
y
)
|
δ
d
y
)
1
/
δ
+
C
1
Ψ
(
Q
)
a
(
1
|
Q
|
∫
Q
‍
|
(
μ
j
Δ
)
loc
⁡
(
(
b
-
b
Q
*
)
f
1
)
(
y
)
|
δ
d
y
)
1
/
δ
+
C
1
Ψ
(
Q
)
a
(
1
|
Q
|
∫
Q
‍
|
(
μ
j
Δ
)
loc
⁡
(
(
b
-
b
Q
*
)
f
2
)
(
y
)
|
δ
d
y
)
1
/
δ
=
I
+
II
+
III
,
where b
Q
*
=
(
1
/
|
Q
*
|
)
∫
Q
*
b
(
y
)
d
y.
To deal withI, for any 2
≤
γ
<
ϵ
/
δ, note that k
0
+
1
≤
η; by Lemma 9, we then have
(55)
I
≤
C
1
Ψ
θ
′
(
Q
)
(
1
|
Q
*
|
∫
Q
*
‍
|
b
(
y
)
-
b
Q
*
|
δ
γ
′
d
y
)
1
/
(
δ
γ
′
)
×
Ψ
θ
′
(
Q
)
Ψ
(
Q
)
a
-
η
/
(
2
δ
)
×
(
1
Ψ
(
Q
)
η
|
Q
|
∫
Q
|
(
μ
j
Δ
)
loc
⁡
(
f
)
(
y
)
|
δ
γ
d
y
)
1
/
(
δ
γ
)
≤
C
[
b
]
θ
M
ϵ
,
η
Δ
(
(
μ
j
Δ
)
loc
⁡
(
f
)
)
(
x
)
,
where 1
/
γ
+
1
/
γ
′
=
1.
ForII, using Kolmogorov’s inequality, Proposition 11, and Lemma 14, we have
(56)
II
≤
C
Ψ
(
Q
)
a
|
Q
|
∥
(
μ
j
Δ
)
loc
⁡
(
(
b
-
b
Q
*
)
f
1
)
∥
L
1
,
∞
≤
C
Ψ
(
Q
)
a
|
Q
*
|
∫
Q
*
|
(
b
(
y
)
-
b
Q
*
)
f
(
y
)
|
d
y
≤
C
[
b
]
θ
M
L
log
⁡
L
,
V
,
η
(
f
)
(
x
)
.
Finally, forIII, notice that B
(
y
,
ρ
(
y
)
)
⊂
Q
(
x
0
,
C
0
2
k
0
+
4
d
r
) for any y
∈
Q; then III
=
0.
Hence the proof is finished.The following information can be founded in [4]. Define the following maximal functions:
(57)
M
V
,
η
f
(
x
)
=
sup
⁡
x
∈
B
1
(
Ψ
(
B
)
)
η
|
B
|
∫
B
|
f
(
y
)
|
d
y
,
M
~
V
,
η
f
(
x
)
=
sup
⁡
ϵ
>
0
1
(
1
+
ϵ
ψ
(
B
(
x
,
ϵ
)
)
)
θ
η
×
∫
R
d
ϵ
-
n
φ
(
x
-
y
ϵ
)
|
f
(
y
)
|
d
y
,
and their commutators:
(58)
M
V
,
η
b
f
(
x
)
=
sup
⁡
x
∈
B
1
(
Ψ
(
B
)
)
η
|
B
|
∫
B
‍
|
b
(
x
)
-
b
(
y
)
|
|
f
(
y
)
|
d
y
,
M
~
V
,
η
b
f
(
x
)
=
sup
⁡
ϵ
>
0
1
(
1
+
ϵ
ψ
(
B
(
x
,
ϵ
)
)
)
θ
η
0000000
×
∫
R
d
‍
ϵ
-
n
φ
(
x
-
y
ϵ
)
|
b
(
x
)
-
b
(
y
)
|
|
f
(
y
)
|
d
y
,
where ψ
(
B
(
x
,
ϵ
)
)
=
(
1
/
|
B
(
x
,
ϵ
)
|
)
∫
B
(
x
,
ϵ
)
ρ
(
y
)
-
1
d
y.Obviously, we have(59)
M
V
,
η
′
b
f
(
x
)
≤
C
M
~
V
,
η
b
f
(
x
)
,
where η
′
=
(
k
0
+
1
)
η and η
>
0.Lemma 16 (see [1]).
Letb
∈
B
M
O
θ
(
ρ
), and (
k
0
+
1
)
(
1
+
1
/
θ
)
≤
η
<
∞, η
1
=
(
k
0
+
1
)
η, and η
2
=
(
k
0
+
1
)
η
1
(
1
+
1
/
θ
). Let 0
<
2
δ
<
ϵ
<
1; then,
(60)
M
δ
,
η
♯
(
M
~
V
,
η
2
b
(
f
)
)
(
x
)
≤
C
[
b
]
θ
(
M
ϵ
,
η
Δ
(
M
~
V
,
η
2
(
f
)
)
(
x
)
000000000
+
M
L
log
⁡
L
,
V
,
η
(
f
)
(
x
)
)
,
a
.
e
x
∈
R
d
,
holds for any f
∈
C
0
∞
(
R
d
).In the following Lemma, we setω
=
1.Lemma 17 (see [4]).
Let2
≤
η
<
∞, ω
∈
A
1
ρ, and A
(
t
)
=
t
log
⁡
(
e
+
t
). Then there exists a constant C
>
0 such that for all t
>
0
(61)
ω
(
{
x
∈
R
d
:
M
A
,
V
,
η
f
(
x
)
>
t
}
)
≤
C
∫
R
d
‍
A
(
|
f
(
x
)
|
t
)
ω
(
x
)
d
x
.Lemma 18 (see [15]).
Let0
<
η
<
∞ and let M
V
,
η
/
2
f be locally integral. Then there exist positive constants C
1 and C
2 independent of f and x such that
(62)
C
1
M
V
,
η
M
V
,
η
+
1
f
(
x
)
≤
M
L
log
⁡
L
,
V
,
η
+
1
f
(
x
)
≤
C
2
M
V
,
η
/
2
M
V
,
η
/
2
f
(
x
)
.


## 3. Proof of the Main Theorems

Proof of Theorem6.
(i) Without loss of generality, we may assume thatα
<
0. Pick any x
0
∈
R
d and r
>
0, and write
(63)
f
(
x
)
=
f
0
(
x
)
+
∑
i
=
1
∞
‍
f
i
(
x
)
,
where f
0
=
χ
B
(
x
0
,
2
r
) and f
i
=
χ
B
(
x
0
,
2
i
+
1
r
)
∖
B
(
x
0
,
2
i
r
), for i
≥
1. Hence, we have
(64)
(
∫
B
(
x
0
,
r
)
|
μ
j
L
f
(
x
)
|
p
d
x
)
1
/
p
≤
C
(
∫
B
(
x
0
,
r
)
|
μ
j
L
f
0
(
x
)
|
p
d
x
)
1
/
p
+
∑
i
=
1
∞
(
∫
B
(
x
0
,
r
)
|
μ
j
L
f
i
(
x
)
|
p
d
x
)
1
/
p
.
By theL
p boundedness of μ
j
L, we obtain
(65)
(
∫
B
(
x
0
,
r
)
|
μ
j
L
f
0
(
x
)
|
p
d
x
)
1
/
p
≤
C
(
1
+
r
ρ
(
x
0
)
)
-
α
|
B
|
1
/
p
-
1
/
q
∥
f
∥
M
p
,
V
q
,
α
(
R
d
)
.
By Proposition2, Lemma 8, and Minkowski’s inequality, we have
(66)
(
∫
B
(
x
0
,
r
)
|
μ
j
L
f
i
(
x
)
|
p
d
x
)
1
/
p
≤
C
(
f
i
(
y
)
d
y
∫
|
x
-
y
|
≤
t
K
j
L
|
2
d
t
t
3
)
p
/
2
∫
B
(
x
0
,
r
)
(
f
i
(
y
)
d
y
∫
|
x
-
y
|
≤
t
K
j
L
|
2
d
t
t
3
∫
0
∞
|
∫
|
x
-
y
|
≤
t
K
j
L
(
x
,
y
)
2222222222222222222222
×
f
i
(
y
)
d
y
∫
|
x
-
y
|
≤
t
K
j
L
|
2
d
t
t
3
)
p
/
2
d
x
)
1
/
p
≤
C
k
(
(
∫
|
x
-
y
|
≤
t
‍
d
t
t
3
)
1
/
2
d
y
)
p
∫
B
(
x
0
,
r
)
‍
(
(
∫
|
x
-
y
|
≤
t
‍
d
t
t
3
)
1
/
2
(
∫
|
x
-
y
|
≤
t
‍
d
t
t
3
)
1
/
2
∫
R
d
‍
|
f
i
(
y
)
|
(
1
+
|
x
-
y
|
/
ρ
(
x
)
)
k
1
|
x
-
y
|
d
-
1
2222222222222222
×
(
∫
|
x
-
y
|
≤
t
‍
d
t
t
3
)
1
/
2
d
y
)
p
d
x
)
1
/
p
≤
C
k
(
1
|
x
-
y
|
d
d
y
∫
B
(
x
0
,
2
i
+
1
r
)
∖
B
(
x
0
,
2
i
r
)
‍
)
p
∫
B
(
x
0
,
r
)
‍
(
∫
B
(
x
0
,
2
i
+
1
r
)
∖
B
(
x
0
,
2
i
r
)
‍
|
f
(
y
)
|
(
1
+
|
x
-
y
|
/
ρ
(
x
)
)
k
2222222222222222222222222
×
1
|
x
-
y
|
d
d
y
∫
B
(
x
0
,
2
i
+
1
r
)
∖
B
(
x
0
,
2
i
r
)
‍
)
p
d
x
)
1
/
p
≤
C
k
(
∫
B
(
x
0
,
r
)
‍
(
2
i
r
)
-
d
(
1
+
2
i
r
/
ρ
(
x
)
)
k
p
d
x
22222
z
z
z
222
×
∫
B
(
x
0
,
2
i
+
1
r
)
‍
|
f
(
y
)
|
p
d
y
∫
B
(
x
0
,
r
)
‍
(
2
i
r
)
-
d
(
1
+
2
i
r
/
ρ
(
x
)
)
k
p
)
1
/
p
≤
C
k
(
∫
B
(
x
0
,
r
)
‍
(
2
i
r
)
-
d
(
1
+
2
i
r
/
ρ
(
x
)
)
k
p
d
x
)
1
/
p
×
(
1
+
2
i
r
ρ
(
x
0
)
)
-
α
|
2
i
B
|
1
/
p
-
1
/
q
∥
f
∥
M
p
,
V
q
,
α
(
R
d
)
≤
C
k
(

∫
B
(
x
0
,
r
)

(
(
1
+
(
r
ρ
(
x
0
)
)
)
-
k
0
/
(
k
0
+
1
)
)
-
k
p

(
2
i
r
)
-
d
2222,222222222
×
(
(
1
+
(
r
ρ
(
x
0
)
)
)
-
k
0
/
(
k
0
+
1
)
1
+
2
i
r
ρ
(
x
0
)
22222
z
.
2222222222
×
(
1

+

r
ρ
(
x
0
)
)
-
k
0
/
(
k
0
+
1
)

)
-
k
p

)
d
x

)
1
/
p
×
(
1
+
2
i
r
ρ
(
x
0
)
)
-
α
|
2
i
B
|
1
/
p
-
1
/
q
∥
f
∥
M
p
,
V
q
,
α
(
R
d
)
≤
C
k
(
2
-
i
)
d
/
p
(
1
+
2
i
r
/
ρ
(
x
0
)
)
-
α
(
1
+
2
i
r
/
ρ
(
x
0
)
)
k
/
(
k
0
+
1
)
|
2
i
B
|
1
/
p
-
1
/
q
×
∥
f
∥
M
p
,
V
q
,
α
(
R
d
)
.
Let k
=
(
-
[
α
]
+
1
)
(
k
0
+
1
), and we obtain
(67)
∥
μ
j
L
f
∥
M
p
,
V
q
,
α
(
R
d
)
≤
C
∥
f
∥
M
p
,
V
q
,
α
(
R
d
)
.
(ii) When p
=
1 and noting that  μ
j
L are bounded from L
1
(
R
d
) to weak L
1
(
R
d
), we have
(68)
λ
|
{
y
∈
B
(
x
,
r
)
:
|
μ
j
L
f
(
y
)
|
>
λ
}
|
≤
∫
B
(
x
,
r
)
‍
|
f
(
y
)
|
d
y
≤
C
(
1
+
r
ρ
(
x
0
)
)
-
α
|
B
(
x
,
r
)
|
1
-
1
/
q
∥
f
∥
M
1
,
V
q
,
α
(
R
d
)
.
We have finished the proof of Theorem 6.Proof of Theorem7.
(i) Pick anyx
0
∈
R
d and r
>
0, as in Theorem 6; we write
(69)
f
(
x
)
=
f
0
(
x
)
+
∑
i
=
1
∞
‍
f
i
(
x
)
,
and, by the L
p boundedness of [
b
,
μ
j
L
], we obtain
(70)
(
∫
B
(
x
0
,
r
)
‍
|
[
b
,
μ
j
L
]
f
0
(
x
)
|
p
d
x
)
1
/
p
≤
C
(
1
+
r
ρ
(
x
0
)
)
-
α
|
B
|
1
/
p
-
1
/
q
∥
f
∥
M
p
,
V
q
,
α
(
R
d
)
.
Set b
r
=
(
1
/
|
B
(
x
0
,
r
)
|
)
∫
B
(
x
0
,
r
)
‍
b
(
x
)
d
x.
We write(71)
|
[
b
,
μ
j
L
]
f
i
|
≤
|
b
-
b
r
|
|
μ
j
L
f
i
|
+
|
μ
j
L
(
(
b
-
b
r
)
f
i
)
|
.
Wheni
≥
1, by Lemmas 8 and 9, Proposition 2, Corollary 10, and Minkowski’s inequality, we have
(72)
(
∫
B
(
x
0
,
r
)
‍
|
[
b
,
μ
j
L
]
f
i
(
x
)
|
p
d
x
)
1
/
p
≤
(
∫
B
(
x
0
,
r
)
‍
|
b
(
x
)
-
b
r
|
p
|
μ
j
L
f
i
(
x
)
|
p
d
x
)
1
/
p
+
(
∫
B
(
x
0
,
r
)
‍
|
μ
j
L
(
(
b
-
b
r
)
f
i
)
(
x
)
|
p
d
x
)
1
/
p
≤
(
(
∫
0
∞
‍
|
∫
|
x
-
y
|
≤
t
‍
K
j
L
(
x
,
y
)
f
i
(
y
)
d
y
|
2
d
t
t
3
)
p
/
2
∫
B
(
x
0
,
r
)
‍
|
b
(
x
)
-
b
r
|
p
×
(
∫
0
∞
‍
|
∫
|
x
-
y
|
≤
t
‍
K
j
L
(
x
,
y
)
f
i
(
y
)
d
y
|
2
d
t
t
3
)
p
/
2
d
x
)
1
/
p
+
(
f
i
(
y
)
d
y
∫
|
x
-
y
|
≤
t
‍
|
2
d
t
t
3
)
p
/
2
∫
B
(
x
0
,
r
)
‍
(
f
i
(
y
)
d
y
∫
|
x
-
y
|
≤
t
‍
|
2
d
t
t
3
∫
0
∞
‍
|
∫
|
x
-
y
|
≤
t
‍
K
j
L
(
x
,
y
)
(
b
(
y
)
-
b
r
)
000000000000000000000000
×
f
i
(
y
)
d
y
∫
|
x
-
y
|
≤
t
‍
|
2
d
t
t
3
)
p
/
2
d
x
)
1
/
p
≤
C
k
(
2
i
r
)
-
d
(
1
+
2
i
r
/
ρ
(
x
0
)
)
k
/
(
k
0
+
1
)
×
{
(
∫
B
(
x
0
,
r
)
‍
|
b
(
x
)
-
b
r
|
p
d
x
)
1
/
p
∫
B
(
x
0
,
2
i
+
1
r
)
‍
|
f
(
y
)
|
d
y
+
r
d
/
p
∫
B
(
x
0
,
2
i
+
1
r
)
‍
|
b
(
x
)
-
b
r
|
|
f
(
y
)
|
d
y
}
≤
C
k
(
2
i
r
)
-
d
(
1
+
2
i
r
/
ρ
(
x
0
)
)
k
/
(
k
0
+
1
)
×
{
(
∫
B
(
x
0
,
r
)
‍
|
b
(
x
)
-
b
r
|
p
d
x
)
1
/
p
×
(
∫
B
(
x
0
,
2
i
+
1
r
)
‍
|
f
(
y
)
|
p
d
y
)
1
/
p
(
2
i
r
)
d
(
1
-
1
/
p
)
+
r
d
/
p
(
∫
B
(
x
0
,
2
i
+
1
r
)
‍
|
b
(
y
)
-
b
r
|
p
′
d
y
)
1
/
p
′
×
(
∫
B
(
x
0
,
2
i
+
1
r
)
‍
|
f
(
y
)
|
p
d
y
)
1
/
p
}
≤
C
k
(
2
i
r
)
-
d
(
1
+
2
i
r
/
ρ
(
x
0
)
)
k
/
(
k
0
+
1
)
×
{
[
b
]
θ
r
d
/
p
(
1
+
r
ρ
(
x
0
)
)
θ
′
(
2
i
r
)
d
(
1
-
1
/
p
)
0000000
×
(
1
+
2
i
r
ρ
(
x
0
)
)
-
α
|
2
i
B
|
1
/
p
-
1
/
q
∥
f
∥
M
p
,
V
q
,
α
(
R
d
)
0000000
+
r
d
/
p
i
[
b
]
θ
(
1
+
r
ρ
(
x
0
)
)
θ
′
(
2
i
r
)
d
(
1
-
1
/
p
)
0000000
×
(
1
+
2
i
r
ρ
(
x
0
)
)
-
α
|
2
i
B
|
1
/
p
-
1
/
q
∥
f
∥
M
p
,
V
q
,
α
(
R
d
)
[
b
]
θ
r
d
/
p
(
1
+
r
ρ
(
x
0
)
)
θ
′
}
≤
C
k
i
[
b
]
θ
(
2
i
)
-
d
/
p
(
1
+
2
i
r
/
ρ
(
x
0
)
)
-
α
(
1
+
2
i
r
/
ρ
(
x
0
)
)
k
/
(
k
0
+
1
)
-
θ
′
×
|
2
i
B
|
1
/
p
-
1
/
q
∥
f
∥
M
p
,
V
q
,
α
(
R
d
)
,
where 1
/
p
+
1
/
p
′
=
1. Choosing k large enough such that k
/
(
k
0
+
1
)
-
θ
′
>
0, we obtain
(73)
∥
[
b
,
μ
j
L
]
f
∥
M
p
,
V
q
,
α
(
R
d
)
≤
C
[
b
]
θ
∥
f
∥
M
p
,
V
q
,
α
(
R
d
)
.
(ii) We adapt a similar argument to that of Theorem 1.2 in [5].
Define(74)
[
b
,
μ
j
L
]
loc
⁡
(
f
)
(
x
)
=
[
b
,
μ
j
L
]
(
f
χ
B
(
x
,
ρ
(
x
)
)
)
(
x
)
,
[
b
,
μ
j
L
]
glob
(
f
)
(
x
)
=
[
b
,
μ
j
L
]
(
f
χ
B
c
(
x
,
ρ
(
x
)
)
)
(
x
)
.
We start with[
b
,
μ
j
L
]
glob. Using Lemma 8 with N
=
1, we have
(75)
[
b
,
μ
j
L
]
glob
(
f
)
(
x
)
=
(
f
(
y
)
χ
B
c
(
x
,
ρ
(
x
)
)
d
y
∫
|
x
-
y
|
≤
t
‍
|
2
d
t
t
3
∫
0
∞
‍
|
∫
|
x
-
y
|
≤
t
‍
K
j
L
(
x
,
y
)
(
b
(
x
)
-
b
(
y
)
)
0000000000000
×
f
(
y
)
χ
B
c
(
x
,
ρ
(
x
)
)
d
y
∫
|
x
-
y
|
≤
t
‍
|
2
d
t
t
3
)
1
/
2
≤
(
f
(
y
)
d
y
∫
ρ
(
x
)
<
|
x
-
y
|
≤
t
‍
|
2
d
t
t
3
∫
ρ
(
x
)
∞
‍
|
∫
ρ
(
x
)
<
|
x
-
y
|
≤
t
‍
K
j
L
(
x
,
y
)
(
b
(
x
)
-
b
(
y
)
)
000000000000000
×
f
(
y
)
d
y
∫
ρ
(
x
)
<
|
x
-
y
|
≤
t
‍
|
2
d
t
t
3
)
1
/
2
≤
C
(
∫
ρ
(
x
)
∞
‍
|
∫
ρ
(
x
)
<
|
x
-
y
|
≤
t
‍
(
1
+
|
x
-
y
|
/
ρ
(
x
)
)
-
1
|
x
-
y
|
d
-
1
0000000000000000
×
|
(
b
(
x
)
-
b
(
y
)
)
f
(
y
)
|
d
y
∫
ρ
(
x
)
<
|
x
-
y
|
≤
t
‍
|
2
d
t
t
3
)
1
/
2
≤
C
ρ
(
x
)
(
f
(
y
)
(
b
(
x
)
-
b
(
y
)
)
|
d
y
∑
k
=
1
[
log
⁡
2
t
/
ρ
(
x
)
]
+
1
‍
|
2
∫
ρ
(
x
)
∞
‍
|
∑
k
=
1
[
log
⁡
2
t
/
ρ
(
x
)
]
+
1
‍
1
(
2
k
ρ
(
x
)
)
n
0000000000000000
×
∫
|
x
-
y
|
≤
2
k
ρ
(
x
)
‍
|
(
b
(
x
)
-
b
(
y
)
)
0000000000000000000000000000
×
f
(
y
)
(
b
(
x
)
-
b
(
y
)
)
|
d
y
∑
k
=
1
[
log
⁡
2
t
/
ρ
(
x
)
]
+
1
‍
|
2
d
t
t
3
)
1
/
2
≤
C
ρ
(
x
)
M
V
,
η
b
f
(
x
)
×
(
∫
ρ
(
x
)
∞
‍
(
[
log
⁡
2
t
ρ
(
x
)
]
+
1
)
2
d
t
t
3
)
1
/
2
≤
C
M
V
,
η
b
f
(
x
)
.
Therefore, the estimates for[
b
,
μ
j
L
]
glob follow from those for M
V
,
η
b
f
(
x
) by Lemmas 16 and 18.
To deal with[
b
,
μ
j
L
]
loc
⁡ we write
(76)
[
b
,
μ
j
L
]
loc
⁡
(
f
)
(
x
)
≤
I
(
x
)
+
[
b
,
μ
j
Δ
]
loc
⁡
(
f
)
(
x
)
+
II
(
x
)
,
where [
b
,
μ
j
Δ
]
loc
⁡
(
f
)
(
x
) is defined in Lemma 15 and
(77)
I
(
x
)
=
(
(
b
(
x
)
-
b
(
y
)
)
f
(
y
)
d
y
∫
|
x
-
y
|
≤
t
[
K
j
L
(
x
,
y
)
-
K
j
Δ
(
x
,
y
)
]
∫
|
x
-
y
|
≤
t
‍
|
2
∫
0
ρ
(
x
)
|
∫
|
x
-
y
|
≤
t
[
K
j
L
(
x
,
y
)
-
K
j
Δ
(
x
,
y
)
]
0000000000
×
(
b
(
x
)
-
b
(
y
)
)
f
(
y
)
d
y
∫
|
x
-
y
|
≤
t
[
K
j
L
(
x
,
y
)
-
K
j
Δ
(
x
,
y
)
]
∫
|
x
-
y
|
≤
t
‍
|
2
d
t
t
3
∫
0
ρ
(
x
)
‍
)
1
/
2
,
II
(
x
)
≤
(
(
b
(
x
)
-
b
(
y
)
)
f
(
y
)
d
y
∫
|
x
-
y
|
≤
ρ
(
x
)
K
j
L
(
x
,
y
)
∫
|
x
-
y
|
≤
ρ
(
x
)
‍
|
2
∫
ρ
(
x
)
∞
|
∫
|
x
-
y
|
≤
ρ
(
x
)
K
j
L
(
x
,
y
)
0000000000
×
(
b
(
x
)
-
b
(
y
)
)
f
(
y
)
d
y
∫
|
x
-
y
|
≤
ρ
(
x
)
K
j
L
(
x
,
y
)
∫
|
x
-
y
|
≤
ρ
(
x
)
‍
|
2
d
t
t
3
)
1
/
2
.
ForII
(
x
), by Lemma 8 with N
=
0,
(78)
II
(
x
)
≤
C
(
∫
ρ
(
x
)
∞
|
∫
|
x
-
y
|
≤
ρ
(
x
)
1
|
x
-
y
|
d
-
1
000000000000
×
|
(
b
(
x
)
-
b
(
y
)
)
f
(
y
)
|
d
y
∫
ρ
(
x
)
∞
|
∫
|
x
-
y
|
≤
ρ
(
x
)
1
|
x
-
y
|
d
-
1
|
2
d
t
t
3
∫
ρ
(
x
)
∞
|
∫
|
x
-
y
|
≤
ρ
(
x
)
1
|
x
-
y
|
d
-
1
)
1
/
2
≤
C
(
∫
ρ
(
x
)
∞
‍
|
∑
k
=
-
∞
0
1
(
2
k
-
1
ρ
(
x
)
)
d
-
1
0000000000
×
∫
|
x
-
y
|
≤
2
k
ρ
(
x
)
‍
|
(
b
(
x
)
-
b
(
y
)
)
f
(
y
)
|
d
y
∑
k
=
-
∞
0
1
(
2
k
-
1
ρ
(
x
)
)
d
-
1
|
2
d
t
t
3
)
1
/
2
≤
C
(
∫
ρ
(
x
)
∞
|
∑
k
=
-
∞
0
2
k
ρ
(
x
)
(
2
k
ρ
(
x
)
)
d
000000000000
×
∫
|
x
-
y
|
≤
2
k
ρ
(
x
)
|
(
b
(
x
)
-
b
(
y
)
)
f
(
y
)
|
d
y
∑
k
=
-
∞
0
2
k
ρ
(
x
)
(
2
k
ρ
(
x
)
)
d
|
2
d
t
t
3
)
1
/
2
≤
C
ρ
(
x
)
M
V
,
η
b
f
(
x
)
(
∫
ρ
(
x
)
∞
d
t
t
3
)
1
/
2
≤
C
M
V
,
η
b
f
(
x
)
.
ForI
(
x
), using Lemma 8 again, we have
(79)
I
(
x
)
≤
C
(
∫
0
ρ
(
x
)
‍
|
∫
|
x
-
y
|
≤
t
‍
1
|
x
-
y
|
d
-
1
(
|
x
-
y
|
ρ
(
x
)
)
σ
00000000000000
×
|
(
b
(
x
)
-
b
(
y
)
)
f
(
y
)
|
d
y
(
|
x
-
y
|
ρ
(
x
)
)
σ
|
2
d
t
t
3
)
1
/
2
≤
C
ρ
(
x
)
-
σ
(
∫
0
ρ
(
x
)
‍
|
∑
k
=
-
∞
0
‍
1
(
2
k
-
1
t
)
d
-
σ
-
1
00000000000000000
×
∫
|
x
-
y
|
≤
2
k
t
‍
|
(
b
(
x
)
-
b
(
y
)
)
00000000000000000000000000
×
(
b
(
x
)
-
b
(
y
)
)
f
(
y
)
(
b
(
x
)
-
b
(
y
)
)
|
d
y
∑
k
=
-
∞
0
‍
|
2
d
t
t
3
)
1
/
2
≤
C
ρ
(
x
)
-
σ
(
f
(
y
)
|
d
y
∑
k
=
-
∞
0
‍
|
2
∫
0
ρ
(
x
)
‍
|
∑
k
=
-
∞
0
‍
(
2
k
)
σ
+
1
t
σ
+
1
(
2
k
t
)
d
00000000000000000
×
∫
|
x
-
y
|
≤
2
k
t
‍
|
(
b
(
x
)
-
b
(
y
)
)
0000000000000000000000000000
×
f
(
y
)
|
d
y
∑
k
=
-
∞
0
‍
(
2
k
)
σ
+
1
t
σ
+
1
(
2
k
t
)
d
|
2
d
t
t
3
)
1
/
2
≤
C
ρ
(
x
)
-
σ
(
∫
0
ρ
(
x
)
‍
|
∑
k
=
-
∞
0
‍
(
2
k
)
σ
+
1
t
σ
+
1
|
2
d
t
t
3
)
1
/
2
M
V
,
η
b
f
(
x
)
≤
C
ρ
(
x
)
-
σ
(
∫
0
ρ
(
x
)
‍
t
2
σ
-
1
d
t
)
1
/
2
M
V
,
η
b
f
(
x
)
≤
C
M
V
,
η
b
f
(
x
)
.
Using Lemmas 15, 16, 17, and 18 and Proposition 12, by adapting an argument in [2], we can obtain the desired result.



---

*Source: 901267-2014-08-20.xml*