# Estimates for Multilinear Commutators of Generalized Fractional Integral Operators on Weighted Morrey Spaces

Authors: Sha He; Taotao Zheng; Xiangxing Tao
Journal: Journal of Function Spaces
 (2015)

---

## Abstract

LetL be the infinitesimal generator of an analytic semigroup on L
2
(
R
n
) with Gaussian kernel bounds, and let L
-
α
/
2 be the fractional integrals of L for 0
<
α
<
n. Assume that b
→
=
(
b
1
,
b
2
,
…
,
b
m
) is a finite family of locally integrable functions; then the multilinear commutators generated by b
→ and L
-
α
/
2 are defined by L
b
→
-
α
/
2
f
=
[
b
m
,
…
,
[
b
2
,
[
b
1
,
L
-
α
/
2
]
]
,
…
]
f. Assume that b
j belongs to weighted BMO space, j
=
1,2
,
…
,
m; the authors obtain the boundedness of L
b
→
-
α
/
2 on weighted Morrey spaces. As a special case, when L
=
-
Δ is the Laplacian operator, the authors also obtain the boundedness of the multilinear fractional commutator I
α
b
→ on weighted Morrey spaces. The main results in this paper are substantial improvements and extensions of some known results.


---

## Body

## 1. Introduction and Main Results

Assume thatL is a linear operator on L
2
(
R
n
), which generates an analytic semigroup e
-
t
L with a kernel p
t
(
x
,
y
) satisfying a Gaussian upper bound; that is,(1)
p
t
x
,
y
≤
C
t
n
/
2
e
-
c
(
x
-
y
2
/
t
)
,for x
,
y
∈
R
n and all t
>
0.The property (1) is satisfied by a large amount of differential operators. One can see [1] for details and examples.For0
<
α
<
n, the fractional integral L
-
α
/
2 generated by the operator L is defined by (2)
L
-
α
/
2
f
(
x
)
=
1
Γ
(
α
/
2
)
∫
0
∞
‍
e
-
t
L
(
f
)
d
t
t
-
α
/
2
+
1
(
x
)
.Letb
→
=
(
b
1
,
b
2
,
…
,
b
m
) be a finite family of locally integrable functions; then the multilinear commutators generated by L
-
α
/
2 and b
→ are defined by(3)
L
b
→
-
α
/
2
f
=
[
b
m
,
…
,
[
b
2
,
[
b
1
,
L
-
α
/
2
]
]
,
…
]
f
,where m
∈
Z
+.Note that ifL
=
-
Δ, which is Laplacian on R
n, then L
-
α
/
2 is the classical fractional integral I
α: (4)
I
α
f
(
x
)
=
Γ
(
(
n
-
α
)
/
2
)
π
n
/
2
2
α
Γ
(
α
/
2
)
∫
R
n
‍
f
(
y
)
x
-
y
n
-
α
d
y
,while L
b
→
-
α
/
2 is the iterated commutator generated by b
→ and I
α: (5)
I
α
b
→
f
=
[
b
m
,
…
,
[
b
2
,
[
b
1
,
I
α
]
]
,
…
]
f
,where m
∈
Z
+.Whenm
=
1, it is easy to see that L
b
→
-
α
/
2
f
=
[
b
,
L
-
α
/
2
]
f is the commutator generated by L
-
α
/
2 and b, and when b
1
=
b
2
=
⋯
=
b
m, L
b
→
-
α
/
2 is the higher commutator.As we all know, ifb
∈
BMO, the commutator of fractional integral operator [
b
,
I
α
] is bounded from L
p
(
R
n
) to L
q
(
R
n
), where 1
<
p
<
n
/
α and 1
/
q
=
1
/
p
-
α
/
n (see [2]). In 2004, Duong and Yan [1] generalized the above classical result and obtained the (
L
p
,
L
q
) boundedness of the commutator [
b
,
L
-
α
/
2
] under the same conditions. Simultaneously, the theory on multilinear integral operators and multilinear commutators has attracted much attention as a rapid developing field in harmonic analysis. Mo and Lu [3] studied the (
L
p
,
L
q
) boundedness of the multilinear commutators L
b
→
-
α
/
2, where b
→
=
(
b
1
,
…
,
b
m
), b
j
∈
BMO, and j
=
1,2
,
…
,
m.On the other hand, Muckenhoupt and Wheeden [4] gave some definitions of weighted bounded mean oscillation and obtained some equivalent conditions for them.Definition 1 (see [4]).
Let1
≤
p
<
∞ and w be locally integral in R
n and let w
≥
0. A locally integrable function b is said to be in BM
O
p
(
w
) if (6)
b
BM
O
p
w
=
sup
⁡
Q
1
w
Q
∫
Q
‍
b
x
-
b
Q
p
w
x
1
-
p
d
x
1
/
p
≤
C
,where b
Q
=
(
1
/
|
Q
|
)
∫
Q
‍
b
(
y
)
d
y and the supremum is taken over all balls Q
∈
R
n.
We may note that other weighted definitions for the bounded mean oscillation also have been given by Muckenhoupt and Wheeden in [4].Definition 2 (see [4]).
Letw be locally integral in R
n and w
≥
0. A locally integrable function b is said to be in BMO
(
w
) if the norm of BMO
(
w
): ∥
·
∥
∗
,
w satisfies (7)
b
∗
,
w
=
sup
⁡
Q
1
w
(
Q
)
∫
Q
‍
b
x
-
b
Q
,
w
w
(
x
)
d
x
≤
C
,where b
Q
,
w
=
(
1
/
w
(
Q
)
)
∫
Q
‍
b
(
z
)
w
(
z
)
d
z and the supremum is taken over all balls Q
∈
R
n.
The above two definitions cannot contain each other; throughout this paper, we will make some investigations on the basis of Definition2.
Recently, Wang [5] obtained some estimates for the commutator [
b
,
I
α
] on weighted Morrey space (see Definitions 3 and 4), where b
∈
BM
O
1
(
w
). Furthermore, Wang and Si [6] obtained the necessary and sufficient conditions for the boundedness of [
b
,
L
-
α
/
2
] on weighted Morrey spaces when b
∈
BM
O
1
(
w
).Motivated by [1, 3, 5, 6], it is natural to raise the following question: how to establish corresponding boundedness of the multilinear commutator L
b
→
-
α
/
2
f on the weighted Morrey space, where b
→
=
(
b
1
,
…
,
b
m
), b
j
∈
BMO
(
w
)?The question is not motivated only by a mere quest to extend the multilinear commutatorL
b
→
-
α
/
2
f from the classical commutator [
b
,
I
α
] but rather by their natural appearance in analysis (see [3]).To state the main results, we now give some definitions and notations.A weight is a locally integrable function onR
n which takes values in (
0
,
∞
) almost everywhere. For a weight w and a measurable set E, we define w
(
E
)
=
∫
E
‍
w
(
x
)
d
x, the Lebesgue measure of E, by |
E
| and the characteristic function of E by χ
E. For a real number p, 1
<
p
<
∞; p
′ is the conjugate of p; that is, 1
/
p
+
1
/
p
′
=
1. The letter C denotes a positive constant that may vary at each occurrence but is independent of the essential variable.Definition 3 (see [7]).
Let1
≤
p
<
∞, 0
<
κ
<
1; let w be a weight; then weighted Morrey space is defined by (8)
L
p
,
κ
(
w
)
:
=
f
∈
L
loc
⁡
p
w
:
f
L
p
,
κ
w
<
∞
,where (9)
f
L
p
,
κ
(
w
)
=
sup
⁡
B
1
w
B
κ
∫
B
‍
f
x
p
w
x
d
x
1
/
p
,and the supremum is taken over all balls B in R
n.Definition 4 (see [7]).
Let1
≤
p
<
∞, 0
<
κ
<
1; let u
,
v be weight; then two weights weighted Morrey space are defined by (10)
L
p
,
κ
(
u
,
v
)
:
=
f
:
f
L
p
,
κ
u
,
v
<
∞
,where (11)
f
L
p
,
κ
(
u
,
v
)
=
sup
⁡
B
1
v
B
κ
∫
B
‍
f
x
p
u
x
d
x
1
/
p
,and the supremum is taken over all balls B in R
n. If u
=
v, then we denote L
p
,
κ
(
u
) for short.Remark 5.
(1) Ifw
=
1, κ
=
λ
/
n, and 0
<
λ
<
n, then L
p
,
κ
(
w
)
=
L
p
,
λ
(
R
n
), the classical Morrey space.
(2) Ifκ
=
0, then L
p
,
0
(
w
)
=
L
p
(
w
), the weighted Lebesgue space; if w
=
1, κ
=
0, then L
p
,
κ
(
w
)
=
L
p
(
R
n
), the Lebesgue space.Definition 6 (see [8]).
A weight functionw is in the Muckenhoupt class A
p with 1
<
p
<
∞ if for every ball B in R
n, there exists a positive constant C which is independent of B such that (12)
(
1
B
∫
B
‍
w
(
x
)
d
x
)
1
B
∫
B
‍
w
x
-
1
/
(
p
-
1
)
d
x
p
-
1
≤
C
.When p
=
1, w
∈
A
1, if (13)
1
B
∫
B
‍
w
(
x
)
d
x
≤
C
ess
inf
⁡
x
∈
B
w
(
x
)
.When p
=
∞, w
∈
A
∞, if there exist positive constants δ and C such that given a ball B and E is a measurable subset of B, then (14)
w
(
E
)
w
(
B
)
≤
C
E
B
δ
.Definition 7 (see [9]).
A weight functionw belongs to the reverse Hölder class R
H
r, if there exist two constants r
>
1 and C
>
0 such that the following reverse Hölder inequality (15)
1
B
∫
B
‍
w
x
r
d
x
1
/
r
≤
C
1
B
∫
B
‍
w
(
x
)
d
x
,holds for every ball B in R
n.It is well known that ifw
∈
A
p with 1
≤
p
<
∞, then there exists r
>
1 such that w
∈
R
H
r. It follows from Hölder’s inequality that w
∈
R
H
r implies w
∈
R
H
s for all 1
<
s
<
r. Moreover, if w
∈
R
H
r, r
>
1, then we have w
∈
R
H
r
+
ɛ for some ε
>
0. We thus write r
w
≡
sup
⁡
{
r
>
1
:
w
∈
R
H
r
} to denote the critical index of w for the reverse Hölder condition.Definition 8.
The Hardy-Littlewood maximal operatorM is defined by (16)
M
f
(
x
)
=
sup
⁡
x
∈
B
1
B
∫
B
‍
f
y
d
y
.Let w be a weight. The weighted maximal operator M
w is defined by (17)
M
w
f
(
x
)
=
sup
⁡
x
∈
B
1
w
B
∫
B
‍
f
y
w
(
y
)
d
y
.For 0
<
α
<
n, r
≥
1, the fractional maximal operator M
α
,
r is defined by (18)
M
α
,
r
f
(
x
)
=
sup
⁡
x
∈
B
1
B
1
-
α
r
/
n
∫
B
‍
f
y
r
d
y
1
/
r
.And the fractional weighted maximal operator M
α
,
r
,
w is defined by (19)
M
α
,
r
,
w
f
(
x
)
=
sup
⁡
x
∈
B
1
w
B
1
-
α
r
/
n
∫
B
‍
f
y
r
w
y
d
y
1
/
r
.If α
=
0, we denote M
r
,
w for short.Definition 9.
A family of operators{
A
t
:
t
>
0
} is said to be an “approximation to identity” if, for every t
>
0, A
t is represented by the kernel a
t
(
x
,
y
), which is a measurable function defined on R
n
×
R
n, in the following sense: for every f
∈
L
p
(
R
n
), p
≥
1, (20)
A
t
f
(
x
)
=
∫
R
n
‍
a
t
(
x
,
y
)
f
(
y
)
d
y
,
a
t
x
,
y
≤
h
t
(
x
,
y
)
=
t
-
n
/
2
g
x
-
y
2
t
,for (
x
,
y
)
∈
R
n
×
R
n, t
>
0. Here, g is a positive, bounded, decreasing function satisfying (21)
lim
r
→
∞
r
n
+
ɛ
g
(
r
2
)
=
0
,for some ε
>
0.Associated with an “approximation to identity”{
A
t
:
t
>
0
}, Martell [10] introduced the sharp maximal function as follows: (22)
M
A
♯
f
x
=
sup
⁡
x
∈
B
1
B
∫
B
‍
f
y
-
A
t
B
f
y
d
y
,where t
B
=
r
B
2, r
B is the radius of the ball B, and f
∈
L
p
(
R
n
) for some p
≥
1.Notice that our analytic semigroup{
e
-
t
L
:
t
>
0
} is an “approximation to identity.” In particular, denote (23)
M
L
♯
f
x
=
sup
⁡
x
∈
B
1
B
∫
B
‍
f
y
-
e
-
t
B
L
f
y
d
y
.Next, we make some conventions on notation. Given any positive integerm, for all 0
≤
j
≤
m, we denote by C
j
m the family of all finite subsets σ
=
{
σ
1
,
σ
2
,
…
,
σ
j
} of {
1,2
,
…
,
m
} of different elements, and, for any σ
∈
C
j
m, let σ
′
=
{
1,2
,
…
,
m
}
∖
σ. Let b
→
=
(
b
1
,
b
2
,
…
,
b
m
); then, for any σ
=
{
σ
1
,
σ
2
,
…
,
σ
j
}
∈
C
j
m, we denote b
→
σ
=
(
b
σ
1
,
b
σ
2
,
…
,
b
σ
j
), b
σ
(
x
)
=
∏
σ
j
∈
σ
b
σ
j
(
x
) and ∥
b
→
σ
∥
BMO
(
w
)
=
∏
σ
j
∈
σ
∥
b
σ
j
∥
BMO
(
w
), and ∥
b
→
∥
BMO
(
w
)
=
∏
σ
j
∈
{
1,2
,
…
,
m
}
∥
b
σ
j
∥
BMO
(
w
)
=
∏
j
=
1
m
∥
b
j
∥
BMO
(
w
).In this paper, our main results are stated as follows.Theorem 10.
Assume the condition (1) holds. Let 0
<
α
<
n, 1
<
p
<
n
/
α, 1
/
q
=
1
/
p
-
α
/
n, 0
≤
κ
<
p
/
q, w
q
/
p
∈
A
1, and r
w
>
(
1
-
κ
)
/
(
p
/
q
-
κ
), where r
w denotes the critical index of w for the reverse Hölder condition. If b
j
∈
B
M
O
(
w
), j
=
1,2
,
…
,
m, then (24)
L
b
→
-
α
/
2
f
L
q
,
κ
q
/
p
(
w
q
/
p
,
w
)
≤
C
b
→
B
M
O
(
w
)
f
L
p
,
κ
(
w
)
.Theorem 11.
Let0
<
α
<
n, 1
<
p
<
n
/
α, 1
/
q
=
1
/
p
-
α
/
n, w
q
/
p
∈
A
1, and r
w
>
q
/
p, where r
w denotes the critical index of w for the reverse Hölder condition. If b
j
∈
B
M
O
(
w
), j
=
1,2
,
…
,
m, then (25)
L
b
→
-
α
/
2
f
L
q
(
w
q
/
p
)
≤
C
b
→
B
M
O
(
w
)
f
L
p
(
w
)
.Moreover, if L
=
-
Δ is the Laplacian, then (26)
I
α
b
→
f
L
q
(
w
q
/
p
)
≤
C
b
→
B
M
O
(
w
)
f
L
p
(
w
)
.Remark 12.
We note that our results extend some results in [1, 3]. To be specific, if we take m
=
1, w
=
1, and κ
=
0 in Theorem 10, it is easy to see that our conclusion is the main result of Duong and Yan [1]. If we only take w
=
1 and κ
=
0 in Theorem 10, our result contains the corresponding conclusion of [3].The remaining part of this paper will be organized as follows. In Section2, we will give some known results and prove some requisite lemmas. Section 3 is devoted to proving the theorems of this paper.


## 2. Requisite Lemmas

In this section, we will prove some lemmas and state some known results about weights and weighted Morrey space.Lemma 13 (see [11]).
Lets
>
1, 1
≤
p
<
∞, and A
p
s
=
{
w
:
w
s
∈
A
p
}. Then(27)
A
p
s
=
A
1
+
(
p
-
1
)
/
s
∩
R
H
s
.In particular, A
1
s
=
A
1
∩
R
H
s.Lemma 14 (see [10]).
Assume that the semigroupe
-
t
L has a kernel p
t
(
x
,
y
) which satisfies the upper bound (1). Take λ
>
0, f
∈
L
0
1
(
R
n
) (the set of functions in L
1
(
R
n
) with bounded support) and a ball B
0 such that there exists x
0
∈
B
0 with M
f
(
x
0
)
≤
λ. Then, for every w
∈
A
∞, 0
<
η
<
1, we can find γ
>
0 (independent of λ
,
B
0
,
f
,
x
0) and constants C
,
r
>
0 (which only depend on w) such that (28)
w
x
∈
B
0
:
M
f
x
>
A
λ
,
M
L
♯
f
x
≤
γ
λ
≤
C
η
r
w
(
B
0
)
,where A
>
1 is a fixed constant which depends only on n.As a result, using the above good-λ inequality together with the standard arguments, we have the following estimates:For everyf
∈
L
p
,
κ
(
u
,
v
), 1
<
p
<
∞, 0
≤
κ
<
1; if u
,
v
∈
A
∞, then (29)
f
L
p
,
κ
(
u
,
v
)
≤
M
f
L
p
,
κ
(
u
,
v
)
≤
M
L
♯
f
L
p
,
κ
(
u
,
v
)
.In particular, when u
=
v
=
w, w
∈
A
∞, we have (30)
f
L
p
,
κ
(
w
)
≤
M
f
L
p
,
κ
(
w
)
≤
M
L
♯
f
L
p
,
κ
(
w
)
.Lemma 15 (Kolmogorov’s inequality; see [9, page 455]).
Let0
<
r
<
l
<
∞, for f
≥
0; define ∥
f
∥
L
l
,
∞
=
sup
⁡
t
>
0
t
|
{
x
∈
R
n
:
|
f
(
x
)
|
>
t
}
|
1
/
l, N
l
,
r
(
f
)
=
sup
⁡
E
(
∥
f
χ
E
∥
r
/
∥
χ
E
∥
h
), and 1
/
h
=
1
/
r
-
1
/
l; then (31)
f
L
l
,
∞
≤
N
l
,
r
(
f
)
≤
l
l
-
r
1
/
r
f
L
l
,
∞
.Lemma 16 (see [4]).
Letw
∈
A
∞. Then the norm of B
M
O
(
w
): ∥
·
∥
∗
,
w, is equivalent to the norm of B
M
O
(
R
n
): ∥
·
∥
∗; that is, if b is a locally integrable function, then (32)
b
∗
,
w
=
sup
⁡
Q
1
w
(
Q
)
∫
Q
‍
b
x
-
b
Q
,
w
w
(
x
)
d
x
≤
Cis equivalent to (33)
b
∗
=
sup
⁡
Q
1
Q
∫
Q
‍
b
x
-
b
Q
d
x
≤
C
.Lemma 17 (see [5]).
Let0
<
α
<
n, 1
<
p
<
n
/
α, 1
/
q
=
1
/
p
-
α
/
n, and w
q
/
p
∈
A
1; if  0
<
κ
<
p
/
q, r
w
>
(
1
-
κ
)
/
(
p
/
q
-
κ
), then (34)
M
α
,
1
f
L
q
,
κ
q
/
p
(
w
q
/
p
,
w
)
≤
C
f
L
p
,
κ
(
w
)
.It also holds for I
α.Lemma 18 (see [5]).
Let0
<
α
<
n, 1
<
p
<
n
/
α, 1
/
q
=
1
/
p
-
α
/
n, and w
q
/
p
∈
A
1; if 0
<
κ
<
p
/
q, 1
<
r
<
p, r
w
>
(
1
-
κ
)
/
(
p
/
q
-
κ
), then (35)
M
r
,
w
f
L
q
,
κ
q
/
p
(
w
q
/
p
,
w
)
≤
C
f
L
q
,
κ
q
/
p
(
w
q
/
p
,
w
)
.Lemma 19 (see [5]).
Let0
<
α
<
n, 1
<
p
<
n
/
α, 1
/
q
=
1
/
p
-
α
/
n, 0
<
κ
<
p
/
q, and w
∈
A
∞. Then, for 1
<
r
<
p, (36)
M
α
,
r
,
w
f
L
q
,
κ
q
/
p
(
w
)
≤
C
f
L
p
,
κ
(
w
)
.Remark 20.
It is easy to see that Lemmas17, 18, and 19 still hold for κ
=
0.Lemma 21.
Let0
<
α
<
n, 1
<
p
<
n
/
α, 1
/
q
=
1
/
p
-
α
/
n, and w
q
/
p
∈
A
1; if 0
≤
κ
<
p
/
q, r
w
>
(
1
-
κ
)
/
(
p
/
q
-
κ
), then (37)
L
-
α
/
2
f
L
q
,
κ
q
/
p
(
w
q
/
p
,
w
)
≤
C
f
L
p
,
κ
(
w
)
.Proof.
Since semigroupe
-
t
L has a kernel p
t
(
x
,
y
) that satisfies the upper bound (1), it is easy to see that, for x
∈
R
n, L
-
α
/
2
f
(
x
)
≤
I
α
(
|
f
|
)
(
x
). From the boundedness of I
α on weighted Morrey space (see Lemma 17), we get (38)
L
-
α
/
2
f
L
q
,
κ
q
/
p
(
w
q
/
p
,
w
)
≤
I
α
f
L
q
,
κ
q
/
p
(
w
q
/
p
,
w
)
≤
C
f
L
p
,
κ
(
w
)
.Remark 22.
SinceI
α is of weak (
1
,
n
/
(
n
-
α
)
) type, from the above proof, we can obtain that L
-
α
/
2 is of weak (
1
,
n
/
(
n
-
α
)
) type.Lemma 23 (see [1]).
Assume the semigroupe
-
t
L has a kernel p
t
(
x
,
y
) which satisfies the upper bound (1). Then, for 0
<
α
<
n, the differential operator L
-
α
/
2
-
e
-
t
L
L
-
α
/
2 has an associated kernel K
~
α
,
t
(
x
,
y
) which satisfies (39)
K
~
α
,
t
(
x
,
y
)
≤
C
x
-
y
n
-
α
t
x
-
y
2
.Lemma 24.
Assume the semigroupe
-
t
L has a kernel p
t
(
x
,
y
) which satisfies the upper bound (1), b
∈
B
M
O
(
w
), and w
∈
A
1. Then, for f
∈
L
p
(
R
n
), p
>
1, σ
∈
C
j
m
(
j
=
1,2
,
…
,
m
), 1
<
τ
<
∞, and(40)
sup
⁡
x
∈
B
1
B
∫
B
‍
e
-
t
B
L
b
-
b
B
σ
f
y
d
y
≤
C
b
→
σ
B
M
O
w
M
τ
,
w
f
x
,where t
B
=
r
B
2 and r
B is the radius of B.Proof.
For anyf
∈
L
p
(
R
n
), x
∈
B, we have(41)
1
B
∫
B
‍
e
-
t
B
L
b
-
b
B
σ
f
y
d
y
≤
1
B
∫
B
‍
∫
R
n
p
t
B
y
,
z
‍
b
z
-
b
B
σ
f
z
d
z
d
y
≤
1
B
∫
B
‍
∫
2
B
p
t
B
(
y
,
z
)
b
z
-
b
B
σ
f
(
z
)
‍
d
z
d
y
+
1
B
∫
B
‍
∑
k
=
1
∞
‍
∫
2
k
+
1
B
∖
2
k
B
p
t
B
y
,
z
‍
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
×
b
z
-
b
B
σ
f
z
d
z
d
y
∶
=
I
+
I
I
.Noticing that y
∈
B, z
∈
2
B, from (1), we get (42)
p
t
B
y
,
z
≤
C
t
B
-
n
/
2
≤
C
2
B
-
1
.Thus,(43)
I
≤
C
2
B
∫
2
B
‍
b
z
-
b
B
σ
f
z
d
z
=
C
2
B
∫
2
B
‍
∏
σ
j
∈
σ
‍
b
σ
j
(
z
)
-
b
σ
j
B
f
(
z
)
d
z
.For simplicity, we only consider the case of j
=
2. We also want to point out that although we state our results on the case of j
=
2, all results are valid on the multilinear case (
j
>
2
) without any essential difference and difficulty in the proof. So it follows that (44)
I
≤
C
2
B
∫
2
B
‍
b
σ
1
z
-
b
σ
1
2
B
+
b
σ
1
2
B
-
b
σ
1
B
×
b
σ
2
z
-
b
σ
2
2
B
+
b
σ
2
2
B
-
b
σ
2
B
f
z
d
z
≤
C
2
B
∫
2
B
‍
b
σ
1
z
-
b
σ
1
2
B
b
σ
2
(
z
)
-
b
σ
2
2
B
f
(
z
)
d
z
+
C
2
B
∫
2
B
‍
b
σ
1
z
-
b
σ
1
2
B
b
σ
2
2
B
-
b
σ
2
B
f
(
z
)
d
z
+
C
2
B
∫
2
B
‍
b
σ
1
2
B
-
b
σ
1
B
b
σ
2
(
z
)
-
b
σ
2
2
B
f
(
z
)
d
z
+
C
2
B
∫
2
B
‍
b
σ
1
2
B
-
b
σ
1
B
b
σ
2
2
B
-
b
σ
2
B
f
(
z
)
d
z
∶
=
I
1
+
I
2
+
I
3
+
I
4
.
We splitI
1 as follows: (45)
I
1
≤
C
2
B
∫
2
B
‍
b
σ
1
z
-
b
σ
1
2
B
,
w
+
b
σ
1
2
B
,
w
-
b
σ
1
2
B
hhhhhhi
×
b
σ
2
z
-
b
σ
2
2
B
,
w
hhhhhhhhh
+
b
σ
2
2
B
,
w
-
b
σ
2
2
B
f
z
d
z
=
C
2
B
∫
2
B
‍
b
σ
1
z
-
b
σ
1
2
B
,
w
h
h
h
h
h
h
h
×
b
σ
2
z
-
b
σ
2
2
B
,
w
f
z
d
z
+
C
2
B
∫
2
B
‍
b
σ
1
z
-
b
σ
1
2
B
,
w
h
h
h
h
h
h
h
h
×
b
σ
2
2
B
,
w
-
b
σ
2
2
B
f
z
d
z
+
C
2
B
∫
2
B
‍
b
σ
1
2
B
,
w
-
b
σ
1
2
B
h
h
h
h
h
h
h
h
×
b
σ
2
z
-
b
σ
2
2
B
,
w
f
z
d
z
+
C
2
B
∫
2
B
‍
b
σ
1
2
B
,
w
-
b
σ
1
2
B
h
h
h
h
h
h
h
h
×
b
σ
2
2
B
,
w
-
b
σ
2
2
B
f
(
z
)
d
z
∶
=
I
11
+
I
12
+
I
13
+
I
14
.We now consider the four terms, respectively. Choose τ
1
,
τ
2
,
τ
,
s
>
1 satisfing 1
/
τ
1
+
1
/
τ
2
+
1
/
τ
+
1
/
s
=
1. According to Hölder’s inequality and w
∈
A
1, we have (46)
I
11
=
C
2
B
∫
2
B
‍
b
σ
1
z
-
b
σ
1
2
B
,
w
w
z
1
/
τ
1
h
h
h
h
h
h
×
b
σ
2
z
-
b
σ
2
2
B
,
w
w
z
1
/
τ
2
f
z
h
h
h
h
h
h
×
w
z
1
/
τ
w
z
-
1
+
1
/
s
d
z
≤
C
2
B
∫
2
B
‍
b
σ
1
z
-
b
σ
1
2
B
,
w
τ
1
w
z
d
z
1
/
τ
1
×
∫
2
B
‍
b
σ
2
z
-
b
σ
2
2
B
,
w
τ
2
w
z
d
z
1
/
τ
2
×
∫
2
B
‍
f
z
τ
w
z
d
z
1
/
τ
∫
2
B
‍
w
z
-
s
+
1
d
z
1
/
s
≤
C
2
B
b
→
σ
∗
,
w
M
τ
,
w
f
(
x
)
w
2
B
1
-
1
/
s
w
x
-
1
+
1
/
s
2
B
1
/
s
=
C
b
→
σ
∗
M
τ
,
w
f
(
x
)
w
2
B
2
B
1
-
1
/
s
w
x
-
1
+
1
/
s
≤
C
b
→
σ
∗
M
τ
,
w
f
(
x
)
.
In the above inequalities, we use the fact that ifw
∈
A
1, then w
∈
A
∞. Thus, the norm of BMO
(
w
) is equivalent to the norm of BMO
(
R
n
) (see Lemma 16).
ForI
12, we first estimate the term that contains (
b
σ
2
)
2
B. In fact, it follows from the John-Nirenberg lemma that there exist C
1
>
0 and C
2
>
0 such that, for any ball B and α
>
0, (47)
z
∈
2
B
:
b
σ
2
z
-
b
σ
2
2
B
>
α
≤
C
1
2
B
e
-
C
2
α
/
b
σ
2
∗
,since b
∈
BMO
(
R
n
). Using the definition of A
∞, we get(48)
w
z
∈
2
B
:
b
σ
2
z
-
b
σ
2
2
B
>
α
≤
C
w
2
B
e
-
C
2
α
δ
/
b
σ
2
∗
,for some δ
>
0. We can see that (48) yields (49)
∫
2
B
‍
b
σ
2
z
-
b
σ
2
2
B
w
(
z
)
d
z
=
∫
0
∞
‍
w
z
∈
2
B
:
b
σ
2
z
-
b
σ
2
2
B
>
α
d
α
≤
C
w
(
2
B
)
∫
0
∞
‍
e
-
C
2
α
δ
/
b
σ
2
∗
d
α
=
C
w
(
2
B
)
b
σ
2
∗
.Thus,(50)
b
σ
2
2
B
,
w
-
b
σ
2
2
B
≤
1
w
2
B
∫
2
B
‍
b
σ
2
z
-
b
σ
2
2
B
w
(
z
)
d
z
≤
C
w
(
2
B
)
w
(
2
B
)
b
σ
2
∗
=
C
b
σ
2
∗
.On the basis of (50), we now estimate I
12. For the above τ, select u, v such that 1
/
u
+
1
/
v
+
1
/
τ
=
1; by virtue of Hölder’s inequality and w
∈
A
1, we have (51)
I
12
≤
C
b
σ
2
∗
1
2
B
∫
2
B
‍
b
σ
1
z
-
b
σ
1
2
B
,
w
f
(
z
)
d
z
=
C
b
σ
2
∗
1
2
B
∫
2
B
‍
b
σ
1
z
-
b
σ
1
2
B
,
w
h
h
h
h
h
h
h
h
h
h
h
h
h
×
w
z
1
/
u
f
z
w
z
1
/
τ
w
z
-
1
+
1
/
v
d
z
≤
C
b
σ
2
∗
1
2
B
∫
2
B
‍
b
σ
1
z
-
b
σ
1
2
B
,
w
u
w
z
d
z
1
/
u
×
∫
2
B
‍
f
z
τ
w
z
d
z
1
/
τ
∫
2
B
‍
w
z
-
v
+
1
d
z
1
/
v
≤
C
b
→
σ
∗
M
τ
,
w
f
(
x
)
1
2
B
w
2
B
1
/
u
+
1
/
τ
w
x
-
1
+
1
/
v
2
B
1
/
v
=
C
b
→
σ
∗
M
τ
,
w
f
(
x
)
w
2
B
2
B
1
-
1
/
v
w
x
-
1
+
1
/
v
=
C
b
→
σ
∗
M
τ
,
w
f
x
.Analogous to the estimate of I
12, we also have I
13
≤
C
∥
b
→
σ
∥
∗
M
τ
,
w
f
(
x
).
As forI
14, taking advantage of Hölder’s inequality and (50), we get (52)
I
14
≤
C
b
→
σ
∗
1
2
B
∫
2
B
‍
f
z
d
z
=
C
b
→
σ
∗
1
2
B
∫
2
B
‍
f
z
w
z
1
/
τ
w
z
1
/
τ
′
-
1
d
z
≤
C
b
→
σ
∗
1
2
B
∫
2
B
‍
f
z
τ
w
z
d
z
1
/
τ
×
∫
2
B
‍
w
z
1
-
τ
′
d
z
1
/
τ
′
=
C
b
→
σ
∗
1
2
B
M
τ
,
w
f
(
x
)
w
2
B
1
/
τ
w
x
1
/
τ
′
-
1
2
B
1
/
τ
′
=
C
b
→
σ
∗
M
τ
,
w
f
(
x
)
w
2
B
2
B
1
/
τ
w
x
1
/
τ
′
-
1
≤
C
b
→
σ
∗
M
τ
,
w
f
(
x
)
.Collecting the estimates of I
11, I
12, I
13, and I
14, it is easy to see that (53)
I
1
≤
C
b
→
σ
∗
M
τ
,
w
f
(
x
)
.
For termI
2, using the fact that |
b
2
B
-
b
B
|
≤
C
∥
b
∥
∗ (see [12]), we now get (54)
I
2
≤
b
σ
2
∗
C
2
B
∫
2
B
‍
b
σ
1
z
-
b
σ
1
2
B
,
w
h
h
h
h
h
h
h
h
h
h
i
h
h
h
h
+
b
σ
1
2
B
,
w
-
b
σ
1
2
B
hhhhhhhhhhhi
×
f
z
d
z
=
C
b
σ
2
∗
2
B
∫
2
B
‍
b
σ
1
(
z
)
-
b
σ
1
2
B
,
w
f
z
d
z
+
C
b
σ
2
∗
2
B
∫
2
B
‍
b
σ
1
2
B
,
w
-
b
σ
1
2
B
f
z
d
z
∶
=
I
21
+
I
22
.By some estimates similar to those used in the estimate for I
12, we conclude that (55)
I
21
≤
C
b
σ
∗
M
τ
,
w
f
(
x
)
.For I
22, using the same method as in dealing with I
14, we get (56)
I
22
≤
C
b
σ
∗
M
τ
,
w
f
(
x
)
.Therefore, (57)
I
2
≤
C
b
σ
∗
M
τ
,
w
f
(
x
)
.Analogously, (58)
I
3
≤
C
b
σ
∗
M
τ
,
w
f
(
x
)
.An argument similar to that used in the estimate for I
14 leads to (59)
I
4
≤
C
b
σ
∗
M
τ
,
w
f
(
x
)
.Hence, (60)
I
≤
I
1
+
I
2
+
I
3
+
I
4
≤
C
b
σ
∗
M
τ
,
w
f
(
x
)
.
Next, we will consider the second termI
I in (41). For any y
∈
B, z
∈
2
k
+
1
B
∖
2
k
B, it is easy to get that |
y
-
z
|
≥
2
k
-
1
r
B and (61)
p
t
B
y
,
z
≤
C
e
-
C
2
2
(
k
-
1
)
2
(
k
+
1
)
n
2
k
+
1
B
.Thus, (62)
I
I
≤
C
∑
k
=
1
∞
‍
e
-
C
2
2
(
k
-
1
)
2
(
k
+
1
)
n
2
k
+
1
B
∫
2
k
+
1
B
‍
b
z
-
b
B
σ
f
(
z
)
d
z
=
C
∑
k
=
1
∞
‍
e
-
C
2
2
k
-
1
2
k
+
1
n
2
k
+
1
B
×
∫
2
k
+
1
B
‍
b
σ
1
z
-
b
σ
1
B

⋯
b
σ
j
z
-
b
σ
j
B
f
z
d
z
.
For simplicity, we also consider the case of j
=
2. (63)
I
I
≤
C
∑
k
=
1
∞
‍
e
-
C
2
2
k
-
1
2
k
+
1
n
2
k
+
1
B
h
h
h
h
×
∫
2
k
+
1
B
‍
b
σ
1
z
-
b
σ
1
2
k
+
1
B
hhhh
×
∫
2
k
+
1
B
‍
+
b
σ
1
2
k
+
1
B
-
b
σ
1
B
h
h
h
h
h
h
×
b
σ
2
z
-
b
σ
2
2
k
+
1
B
hhhhhhhh
+
b
σ
2
2
k
+
1
B
-
b
σ
2
B
f
z
d
z
=
C
∑
k
=
1
∞
‍
e
-
C
2
2
k
-
1
2
k
+
1
n
2
k
+
1
B
×
∫
2
k
+
1
B
‍
b
σ
1
z
-
b
σ
1
2
k
+
1
B
h
h
h
h
h
×
b
σ
2
z
-
b
σ
2
2
k
+
1
B
f
z
d
z
+
C
∑
k
=
1
∞
‍
e
-
C
2
2
k
-
1
2
k
+
1
n
2
k
+
1
B
×
∫
2
k
+
1
B
b
σ
1
2
k
+
1
B
-
b
σ
1
B
‍
h
h
h
h
h
×
b
σ
2
z
-
b
σ
2
2
k
+
1
B
f
z
d
z
+
C
∑
k
=
1
∞
‍
e
-
C
2
2
k
-
1
2
k
+
1
n
2
k
+
1
B
×
∫
2
k
+
1
B
‍
b
σ
1
z
-
b
σ
1
2
k
+
1
B
h
h
h
h
h
×
b
σ
2
2
k
+
1
B
-
b
σ
2
B
f
z
d
z
+
C
∑
k
=
1
∞
‍
e
-
C
2
2
k
-
1
2
k
+
1
n
2
k
+
1
B
×
∫
2
k
+
1
B
‍
b
σ
1
2
k
+
1
B
-
b
σ
1
B
h
h
h
h
h
×
b
σ
2
2
k
+
1
B
-
b
σ
2
B
f
z
d
z
∶
=
I
I
1
+
I
I
2
+
I
I
3
+
I
I
4
.For I
I
1, similar to the estimate of I
1, we have (64)
I
I
1
≤
C
b
→
σ
∗
M
τ
,
w
f
(
x
)
∑
k
=
1
∞
‍
e
-
C
2
2
(
k
-
1
)
2
(
k
+
1
)
n
≤
C
b
→
σ
∗
M
τ
,
w
f
(
x
)
.Noticing that we could use similar methods of the estimates of I
2, I
3, and I
4 in the estimates of I
I
2, I
I
3, and I
I
4, respectively. From this together with the fact that |
b
2
j
+
1
B
-
b
B
|
≤
2
n
(
j
+
1
)
∥
b
∥
∗ (see [12]), it is easy to get (65)
I
I
2
+
I
I
3
+
I
I
4
≤
C
b
→
σ
∗
M
τ
,
w
f
(
x
)
.Therefore, (66)
I
I
≤
C
∥
b
→
σ
∥
∗
M
τ
,
w
f
(
x
)
.According to the estimates of I and I
I, the lemma has been proved.Now, we will establish a lemma which plays an important role in the proof of Theorem10.Lemma 25.
Let0
<
α
<
n, w
∈
A
1, and b
∈
B
M
O
(
w
); then, for all r
>
1, τ
>
1, and x
∈
R
n, we have (67)
M
L
♯
(
L
b
→
-
α
/
2
f
)
(
x
)
≤
C
b
→
∗
M
r
,
w
(
L
-
α
/
2
f
)
(
x
)
hhhihh
+
∑
j
=
1
m
-
1
‍
∑
σ
∈
C
j
m
‍
C
j
,
m
b
→
σ
∗
M
τ
,
w
(
L
b
→
σ
′
-
α
/
2
f
)
(
x
)
hhhihh
+
b
→
∗
w
x
-
α
/
n
M
α
,
r
,
w
f
(
x
)
hhhihh
+
b
→
∗
M
α
,
1
f
x
.Proof.
For any givenx
∈
R
n, take a ball B
=
B
(
x
0
,
r
B
) which contains x. For f
∈
L
p
(
R
n
), let f
1
=
f
χ
2
B, f
2
=
f
-
f
1. Denote the kernel of L
-
α
/
2 by K
α
(
x
,
y
), λ
→
=
(
λ
1
,
λ
2
,
…
,
λ
m
), where λ
j
∈
R
n, j
=
1,2
,
…
,
m. Then L
b
→
-
α
/
2
f can be written in the following form:(68)
L
b
→
-
α
/
2
f
y
=
∫
R
n
‍
∏
j
=
1
m
‍
(
b
j
(
y
)
-
b
j
(
z
)
)
K
α
(
y
,
z
)
f
(
z
)
d
z
=
∫
R
n
‍
∏
j
=
1
m
‍
(
(
b
j
(
y
)
-
λ
j
)
-
(
b
j
(
z
)
-
λ
j
)
)
K
α
(
y
,
z
)
f
(
z
)
d
z
=
∑
i
=
0
m
‍
∑
σ
∈
C
i
m
‍
-
1
m
-
i
b
y
-
λ
σ
h
h
h
h
h
h
h
×
∫
R
n
‍
b
z
-
λ
σ
′
K
α
(
y
,
z
)
f
(
z
)
d
z
.Now expanding (
b
(
z
)
-
λ
)
σ
′ as(69)
b
z
-
λ
σ
′
=
b
z
-
b
y
+
b
y
-
λ
σ
′
,and combining (68) with (69), it is easy to see that (70)
e
-
t
B
L
L
b
→
-
α
/
2
f
y
=
e
-
t
B
L
∏
j
=
1
m
‍
b
j
-
λ
j
L
-
α
/
2
f
(
y
)
+
∑
j
=
1
m
-
1
‍
∑
σ
∈
C
j
m
‍
C
j
,
m
e
-
t
B
L
b
-
λ
σ
L
b
→
σ
′
-
α
/
2
f
(
y
)
+
-
1
m
e
-
t
B
L
L
-
α
/
2
∏
j
=
1
m
‍
b
j
-
λ
j
f
1
(
y
)
+
-
1
m
e
-
t
B
L
L
-
α
/
2
∏
j
=
1
m
‍
b
j
-
λ
j
f
2
y
,where t
B
=
r
B
2 and r
B is the radius of ball B.
Takeλ
j
=
(
b
j
)
B, j
=
1,2
,
…
,
m, and denote b
→
B
=
(
(
b
1
)
B
,
(
b
2
)
B
,
…
,
(
b
m
)
B
); then (71)
1
B
∫
B
‍
L
b
→
-
α
/
2
f
y
-
e
-
t
B
L
L
b
→
-
α
/
2
f
y
d
y
≤
1
B
∫
B
‍
∏
j
=
1
m
‍
b
j
y
-
b
j
B
L
-
α
/
2
f
y
d
y
+
1
B
∫
B
‍
∑
j
=
1
m
-
1
‍
∑
σ
∈
C
j
m
‍
C
j
,
m
b
y
-
b
B
σ
L
b
→
σ
′
-
α
/
2
f
y
d
y
+
1
B
∫
B
‍
L
-
α
/
2
∏
j
=
1
m
‍
b
j
-
b
j
B
f
1
y
d
y
+
1
B
∫
B
‍
e
-
t
B
L
∏
j
=
1
m
‍
b
j
-
b
j
B
L
-
α
/
2
f
y
d
y
+
1
B
∫
B
‍
∑
j
=
1
m
-
1
‍
∑
σ
∈
C
j
m
‍
C
j
,
m
e
-
t
B
L
b
-
b
B
σ
L
b
→
σ
′
-
α
/
2
f
y
d
y
+
1
B
∫
B
‍
e
-
t
B
L
L
-
α
/
2
∏
j
=
1
m
‍
b
j
-
b
j
B
f
1
y
d
y
+
1
B
∫
B
‍
L
-
α
/
2
∏
j
=
1
m
‍
b
j
-
b
j
B
f
2
(
y
)
hhhhhhhhhh
-
e
-
t
B
L
L
-
α
/
2
∏
j
=
1
m
‍
b
j
-
b
j
B
f
2
y
d
y



∶
=
∑
i
=
1
7
‍
G
i
.We now estimate the above seven terms, respectively. We take m
=
2 as an example; the estimate for the case m
>
2 is the same. For the first term G
1, we split it as follows: (72)
G
1
=
1
B
∫
B
‍
b
1
(
y
)
-
b
1
B
,
w

b
2
(
y
)
-
b
2
B
,
w

L
-
α
/
2
f
(
y
)
d
y

+
1
B
∫
B
‍
b
1
(
y
)
-
b
1
B
,
w

b
2
B
,
w
-
b
2
B

L
-
α
/
2
f
y
d
y

+
1
B
∫
B
‍
b
1
B
,
w
-
b
1
B

b
2
(
y
)
-
b
2
B
,
w

L
-
α
/
2
f
y
d
y

+
1
B
∫
B
‍
b
1
B
,
w
-
b
1
B

b
2
B
,
w
-
b
2
B

L
-
α
/
2
f
y
d
y

∶
=
G
11
+
G
12
+
G
13
+
G
14
.Choose r
1
,
r
2
,
r
,
q
>
1 such that 1
/
r
1
+
1
/
r
2
+
1
/
r
+
1
/
q
=
1. Noticing that w
∈
A
1, by Hölder’s inequality and the similar estimate of I
1, we have (73)
G
1
≤
C
b
→
∗
M
r
,
w
(
L
-
α
/
2
f
)
(
x
)
.
ForG
2, take τ
1
,
…
,
τ
j
,
τ
,
ν
>
1 that satisfy 1
/
τ
1
+
⋯
+
1
/
τ
j
+
1
/
τ
+
1
/
ν
=
1; then following Hölder’s inequality and the same idea as that of G
1 yields(74)
G
2
≤
C
∑
j
=
1
m
-
1
‍
∑
σ
∈
C
j
m
‍
C
j
,
m
b
→
σ
∗
M
τ
,
w
(
L
b
→
σ
′
-
α
/
2
f
)
(
x
)
.
To estimateG
3, applying Lemma 15 (Kolmogorov’s inequality), weak (
1
,
n
/
(
n
-
α
)
) boundedness of L
-
α
/
2 (see Remark 22), and Hölder’s inequality, we have (75)
G
3
=
1
B
∫
B
‍
L
-
α
/
2
∏
j
=
1
m
‍
b
j
-
b
j
B
f
1
y
d
y
≤
C
B
1
-
α
/
n
L
-
α
/
2
∏
j
=
1
m
‍
b
j
-
b
j
B
f
1
L
n
/
(
n
-
α
)
,
∞
≤
C
B
1
-
α
/
n
∫
2
B
‍
∏
j
=
1
m
‍
b
j
y
-
b
j
B
f
y
d
y
.We consider the case of m
=
2, for example, and we split G
3 as follows: (76)
G
3
≤
C
B
1
-
α
/
n
∫
2
B
‍
b
1
y
-
b
1
2
B
+
b
1
2
B
-
b
1
B
hhhhhhhhh
×
b
2
y
-
b
2
2
B
hhhhhhhhhhh
+
b
2
2
B
-
b
2
B
f
y
d
y
≤
C
B
1
-
α
/
n
∫
2
B
b
1
(
y
)
-
b
1
2
B
b
2
(
y
)
-
b
2
2
B
‍
f
y
d
y
+
C
B
1
-
α
/
n
∫
2
B
‍
b
1
(
y
)
-
b
1
2
B
b
2
2
B
-
b
2
B
f
y
d
y
+
C
B
1
-
α
/
n
∫
2
B
b
1
2
B
-
b
1
B
‍
b
2
(
y
)
-
b
2
2
B
f
y
d
y
+
C
B
1
-
α
/
n
∫
2
B
‍
b
1
2
B
-
b
1
B
b
2
2
B
-
b
2
B
f
y
d
y
∶
=
G
31
+
G
32
+
G
33
+
G
34
.Take r
1
,
r
2
,
r
,
q
>
1 such that 1
/
r
1
+
1
/
r
2
+
1
/
r
+
1
/
q
=
1; by virtue of Hölder’s inequality and the same manner as that used in dealing with I
1
,
I
2
,
I
3 in Lemma 24, we get (77)
G
31
+
G
32
+
G
33
≤
C
b
→
∗
M
α
,
r
,
w
f
(
x
)
.For G
34, (78)
G
34
≤
C
b
→
∗
1
B
1
-
α
/
n
∫
2
B
‍
f
y
d
y
≤
C
b
→
∗
M
α
,
1
f
(
x
)
.Therefore, (79)
G
3
≤
C
b
→
∗
M
α
,
r
,
w
f
(
x
)
+
C
b
→
∗
M
α
,
1
f
(
x
)
.
By Lemma24, we have (80)
G
4
≤
C
b
→
∗
M
r
,
w
L
-
α
/
2
f
x
,
G
5
≤
C
∑
j
=
1
m
-
1
‍
∑
σ
∈
C
j
m
‍
C
j
,
m
b
→
σ
∗
M
τ
,
w
(
L
b
→
σ
′
-
α
/
2
f
)
(
x
)
.Next, we consider the term G
6: (81)
G
6
=
1
B
∫
B
‍
∏
j
=
1
m
‍
b
j
-
b
j
B
f
1
∫
R
n
‍
p
t
B
y
,
z
h
h
h
h
h
h
h
h
×
L
-
α
/
2
∏
j
=
1
m
‍
b
j
-
b
j
B
f
1
z
d
z
d
y
≤
1
B
∫
B
‍
∫
2
B
‍
p
t
B
y
,
z
h
h
h
h
h
h
h
h
×
L
-
α
2
∏
j
=
1
m
‍
b
j
-
b
j
B
f
1
z
d
z
d
y
≤
1
B
∫
B
‍
∫
R
n
∖
2
B
‍
p
t
B
y
,
z
h
h
h
h
h
h
h
h
×
L
-
α
/
2
∏
j
=
1
m
‍
b
j
-
b
j
B
f
1
z
d
z
d
y
∶
=
G
61
+
G
62
.For G
61, since y
∈
B, z
∈
2
B, and |
p
t
B
(
y
,
z
)
|
≤
C
|
2
B
|
-
1, it follows that (82)
G
61
≤
C
2
B
∫
2
B
‍
L
-
α
/
2
∏
j
=
1
m
‍
b
j
-
b
j
B
f
1
z
d
z
.Analogous to the estimate of G
3, we have (83)
G
61
≤
C
b
→
∗
M
α
,
r
,
w
f
(
x
)
+
C
b
→
∗
M
α
,
1
f
(
x
)
.For G
62, note that y
∈
B, z
∈
2
k
+
1
B
∖
2
k
B, and |
p
t
B
(
y
,
z
)
|
≤
C
(
e
-
C
2
2
k
-
1
2
k
+
1
n
/
|
2
k
+
1
B
|
). Hence, the estimate of G
62 runs as that of G
61 yields that (84)
G
62
≤
C
b
→
∗
M
α
,
r
,
w
f
(
x
)
+
C
b
→
∗
M
α
,
1
f
(
x
)
.
For the last termG
7, applying Lemma 23, we have (85)
G
7
≤
1
B
∫
B
‍
∏
j
=
1
m
‍
b
j
-
b
j
B
f
2
L
-
α
/
2
-
e
-
t
B
L
L
-
α
/
2
h
h
h
h
h
h
h
h
×
∏
j
=
1
m
‍
b
j
-
b
j
B
f
2
y
d
y
≤
1
B
∫
B
‍
∫
R
n
∖
2
B
‍
K
~
α
,
t
B
y
,
z
hhhhhhhh
×
∏
j
=
1
m
‍
b
j
z
-
b
j
B
f
z
d
z
d
y
≤
C
∑
k
=
1
∞
‍
∫
2
k
+
1
B
∖
2
k
B
‍
t
B
x
0
-
z
n
-
α
+
2
hhhhhhh
×
∏
j
=
1
m
‍
b
j
z
-
b
j
B
f
z
d
z
≤
C
∑
k
=
1
∞
‍
2
-
2
k
2
k
B
1
-
α
/
n
∫
2
k
+
1
B
‍
∏
j
=
1
m
‍
b
j
z
-
b
j
B
f
z
d
z
.The same manner as that of G
3 gives us that (86)
G
7
≤
C
∑
k
=
1
∞
‍
2
-
2
k
b
→
∗
M
α
,
r
,
w
f
x
+
b
→
∗
M
α
,
1
f
x
≤
C
b
→
∗
M
α
,
r
,
w
f
x
+
M
α
,
1
f
x
.According to the above estimates, we have completed the proof of Lemma 25.


## 3. Proof of Theorems

At first, we give the proof of Theorem10.Proof.
It follows from Lemmas13, 14, 25, and 17–21 that (87)
L
b
→
-
α
/
2
f
L
q
,
κ
q
/
p
(
w
q
/
p
,
w
)
≤
M
L
b
→
-
α
/
2
f
L
q
,
κ
q
/
p
(
w
q
/
p
,
w
)
≤
M
L
♯
(
L
b
→
-
α
/
2
f
)
L
q
,
κ
q
/
p
(
w
q
/
p
,
w
)
≤
C
b
→
∗
M
r
,
w
(
L
-
α
/
2
f
)
L
q
,
κ
q
/
p
(
w
q
/
p
,
w
)
hhhhhh
+
∑
j
=
1
m
-
1
‍
∑
σ
∈
C
j
m
‍
C
j
,
m
b
→
σ
∗
hhhhhh
×
M
τ
,
w
(
L
b
→
σ
′
-
α
/
2
f
)
L
q
,
κ
q
/
p
(
w
q
/
p
,
w
)
hhhhhh
+
b
→
∗
w
-
α
/
n
M
α
,
r
,
w
(
f
)
L
q
,
κ
q
/
p
(
w
q
/
p
,
w
)
hhhhhh
+
b
→
∗
M
α
,
1
(
f
)
L
q
,
κ
q
/
p
(
w
q
/
p
,
w
)
≤
C
b
→
∗
L
-
α
/
2
f
L
q
,
κ
q
/
p
(
w
q
/
p
,
w
)
+
∑
j
=
1
m
-
1
‍
∑
σ
∈
C
j
m
‍
C
j
,
m
b
→
σ
∗
L
b
→
σ
′
-
α
/
2
f
L
q
,
κ
q
/
p
(
w
q
/
p
,
w
)
+
C
b
→
∗
M
α
,
r
,
w
f
L
q
,
κ
q
/
p
(
w
)
+
C
b
→
∗
f
L
p
,
κ
(
w
)
≤
C
b
→
∗
f
L
p
,
κ
(
w
)
+
∑
j
=
1
m
-
1
‍
∑
σ
∈
C
j
m
‍
C
j
,
m
b
→
σ
∗
(
L
b
→
σ
′
-
α
/
2
f
)
L
q
,
κ
q
/
p
(
w
q
/
p
,
w
)
.Then, we can make use of an induction on σ
⊆
{
1,2
,
…
,
m
} to get that (88)
L
b
→
-
α
/
2
f
L
q
,
κ
q
/
p
(
w
q
/
p
,
w
)
≤
C
b
→
∗
f
L
p
,
κ
(
w
)
.This completes the proof of Theorem 10.Now, we are in the position of proving Theorem11. If we take κ
=
0 in Theorem 10, we will immediately get our desired results.



---

*Source: 670649-2015-04-06.xml*