# The Commutators of Fractional Integrals on Generalized Herz Spaces

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

---

## Abstract

LetI
l be the fractional integral, 0
<
l
<
n and let b
∈
BMO. We will obtain the weighted estimates for the commutator b
,
I
l on the generalized Herz spaces.


---

## Body

## 1. Introduction and Results

Let0
<
l
<
n, the fractional integral operator I
l is defined by
(1)
I
l
f
(
x
)
=
∫
ℝ
n
‍
f
(
y
)
|
x
-
y
|
n
-
l
d
y
.
Consider the commutator associated with the fractional integrals I
l and the locally integrable function b,
(2)
[
b
,
I
l
]
f
(
x
)
=
b
(
x
)
I
l
f
(
x
)
-
I
l
(
b
f
)
(
x
)
.
The area of the commutators of fractional integrals has been under intensive research. When b
∈
BMO
(
ℝ
n
), 1
/
p
2
=
1
/
p
1
-
l
/
n and 1
<
p
1
<
n
/
l, and Chanillo proved in [1] that [
b
,
I
l
] is bounded from L
p
1
(
ℝ
n
) to L
p
2
(
ℝ
n
). Segovia and Torrea in [2] proved that [
b
,
I
l
] is also bounded from L
p
1
(
ω
p
1
,
ℝ
n
) to L
p
2
(
ω
p
2
,
ℝ
n
) if ω
∈
A
p
1
,
p
2
(
ℝ
n
). Lu and Yang in [3] studied [
b
,
I
l
] Herz spaces; they proved that if 1
/
p
2
=
1
/
p
1
-
l
/
n,1
<
p
1
<
n
/
l,-
n
/
p
1
+
l
<
α
<
n
(
1
-
1
/
p
1
) and 0
<
q
1
≤
q
2
<
∞, then [
b
,
I
l
] maps K
˙
p
1
α
,
q
1
(
ℝ
n
) continuously into K
˙
p
2
α
,
q
2
(
ℝ
n
).In [4], Komori and Matsuoka considered the boundedness of singular integral operators and fractional integral operators on weighted Herz spaces; they introduced generalized Herz spaces, and, as corollaries of their general theory, they obtained the boundedness of these operators on weighted Herz spaces.For a sequenceφ
=
{
φ
(
k
)
}
-
∞
∞
,
φ
(
k
)
>
0, we suppose that φ satisfies doubling condition of order (
s
,
t
) and write φ
∈
D
(
s
,
t
) if there exists C
≥
1 such that
(3)
C
-
1
2
s
(
k
-
j
)
≤
φ
(
k
)
φ
(
j
)
≤
C
2
t
(
k
-
j
)
,
for k
>
j.LetB
k
=
{
x
∈
ℝ
n
:
|
x
|
≤
2
k
}
,
C
k
=
B
k
∖
B
k
-
1 and let χ
k
=
χ
C
k be the characteristic function of the set C
k for any k
∈
ℤ. Suppose that ω is a weight function on ℝ
n. For 1
<
p
<
∞, 0
<
q
<
∞, the generalized Herz space is defined by
(4)
K
p
q
(
φ
,
ω
)
(
ℝ
n
)
=
{
∥
f
∥
K
p
q
(
φ
,
ω
)
f
:
f
is
a
measurable
function
on
⁡
ℝ
n
,
∥
f
∥
K
p
q
(
φ
,
ω
)
<
∞
}
,
where
(5)
∥
f
∥
K
p
q
(
φ
,
ω
)
=
(
∑
k
=
-
∞
+
∞
‍
φ
(
k
)
q
∥
f
χ
k
∥
L
p
(
ω
)
q
)
1
/
q
,
(6)
∥
f
∥
L
p
(
ω
)
=
(
∫
ℝ
n
‍
|
f
(
x
)
|
q
ω
(
x
)
d
x
)
1
/
q
.Letφ
(
k
)
=
ω
1
(
B
k
)
α
/
n, ω
=
ω
2, and ω
1, ω
2 be weight functions in (5); then, K
p
q
(
φ
,
ω
)
(
ℝ
n
) is the weighted Herz spaces K
˙
p
α
(
ω
1
,
ω
2
)
(
ℝ
n
) (see [5]).Letδ
>
0. We say ω
∈
R
D
(
δ
) (centered reverse doubling) if there is a positive number C such that
(7)
ω
(
B
k
)
ω
(
B
j
)
≥
C
2
δ
(
k
-
j
)
for
k
>
j
.In 2000, Lu et al. in [6] proved boundedness results for sublinear operators on weighted Herz spaces with general Muckenhoupt weights. Recently, many authors considered the boundedness of operators on weighted Herz spaces. In [7], Wang proved the boundedness properties of intrinsic square function on weighted Herz spaces. Tomita in [8] studied the Strang-Fix theory for approximation order in weighted Herz spaces with general Muckenhoupt weights. In [9], Hu and Wang consider the boundedness properties of commutator operators generated by BMO
(
ℝ
n
) function and intrinsic square function on weighted Herz spaces.In this paper, we investigate the boundedness of commutators generated by fractional integrals andBMO
(
ℝ
n
) on generalized Herz spaces K
p
q
(
φ
,
ω
)
(
ℝ
n
) with general Muckenhoupt weights. Our main results in this paper are formulated as follows.Theorem 1.
Letb
∈
B
M
O
(
ℝ
n
), 1
<
p
1
<
n
/
l, 0
<
q
<
∞, δ
>
0 and let 1
/
p
2
=
1
/
p
1
-
l
/
n. One assumes that (1)
φ
∈
D
(
a
,
b
), where -
(
δ
/
p
2
)
<
a
≤
b
<
n
(
1
-
(
1
/
p
1
)
);
(2)
ω
p
2
∈
A
r, where r
=
min
⁡
(
1
+
(
p
2
/
p
1
′
)
,
1
+
(
p
2
/
p
1
′
)
-
(
b
p
2
/
n
)
);
(3)
ω
p
2
∈
R
D
(
δ
).Then, [
b
,
I
l
] is bounded from K
p
1
q
(
φ
,
ω
p
1
)
(
ℝ
n
) to K
p
2
q
(
φ
,
ω
p
2
)
(
ℝ
n
).Remark 2.
Whenb
≤
0, condition (2) is equivalent to the condition that ω
∈
A
p
1
,
p
2
(
ℝ
n
), but when b
>
0, (2) is stronger than A
p
1
,
p
2-condition. Komori and Matsuoka in [4] showed that the condition (2) is the best possible by a counterexample.Letφ
∈
D
(
0
,
α
) in Theorem 1; we haveCorollary 3.
Let1
<
p
1
<
n
/
l, 0
<
q
<
∞, 0
≤
α
<
n
(
1
-
1
/
p
1
) and let 1
/
p
2
=
1
/
p
1
-
l
/
n. If b
∈
BMO
(
ℝ
n
), ω
p
2
∈
A
1
+
p
2
/
p
1
′
-
α
p
2
/
n
(
ℝ
n
), then [
b
,
I
l
] is bounded from K
˙
p
1
α
,
q
(
1
,
ω
p
1
)
(
ℝ
n
) to K
˙
p
2
α
,
q
(
1
,
ω
p
2
)
(
ℝ
n
).The paper is organized as follows. In Section2, we give some basic notation and definitions and recall some preliminaries. In Section 3, we will prove Theorem 1.


## 2. Definitions and Preliminaries

We begin with some properties ofA
p weights which play a great role in the proofs of our main results.A weightω 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 ω and a measurable set E, we also denote the Lebesgue measure of E by |
E
| and set the weighted measure ω
(
E
)
=
∫
E
‍
ω
(
x
)
d
x.A weightω is said to belong to A
p
(
ℝ
n
) for 1
<
p
<
∞, if there exists a constant C such that for every ball B
⊂
ℝ
n,
(8)
(
1
|
B
|
∫
B
‍
ω
(
x
)
d
x
)
(
1
|
B
|
∫
B
‍
ω
(
x
)
1
-
p
′
d
x
)
p
-
1
≤
C
,
where s
′ is the dual of s such that 1
/
s
+
1
/
s
′
=
1. The class A
1
(
ℝ
n
) is defined by replacing the above inequality with
(9)
1
|
B
|
∫
B
‍
w
(
y
)
d
y
≤
C
·
ess
inf
⁡
x
∈
B
w
(
x
)
for
every
ball
B
⊂
ℝ
n
.A weightω is said to belong to A
∞
(
ℝ
n
) if there are positive numbers C and δ so that
(10)
ω
(
E
)
ω
(
B
)
≤
C
(
|
E
|
|
B
|
)
δ
,
for all balls B and all measurable E
⊂
B. It is well known that
(11)
A
∞
(
ℝ
n
)
=
⋃
1
≤
p
<
∞
‍
A
p
(
ℝ
n
)
.The classicalA
p
(
ℝ
n
) weight theory was first introduced by Muckenhoupt in the study of weighted L
p boundedness of Hardy-Littlewood maximal function in [10].Lemma 4 (see [10, 12]).
Supposeω
∈
A
p
(
ℝ
n
) and the following statements hold. (i)
For any1
≤
p
<
∞, there are positive numbers C and δ such that
(12)
ω
(
B
k
)
ω
(
B
j
)
≤
C
2
n
p
(
k
-
j
)
f
o
r
k
>
j
,
ω
(
B
k
)
ω
(
B
j
)
≤
C
2
n
p
δ
(
k
-
j
)
f
o
r
k
<
j
;
(ii)
A
p
1
(
ℝ
n
)
⊂
A
p
2
(
ℝ
n
) for any 1
≤
p
1
<
p
2
≤
∞;
(iii)
For any1
<
p
<
∞, one has ω
1
-
p
′
∈
A
p
′
(
ℝ
n
).We also need another weight classA
p
,
q
(
ℝ
n
) introduced by Muckenhoupt and Wheeden in [11] to study weighted boundedness of fractional integral operators.Given1
≤
p
≤
q
<
∞, we say that ω
∈
A
p
,
q
(
ℝ
n
) if there exists a constant C such that for every ball B
⊂
ℝ
n, the inequality
(13)
(
1
|
B
|
∫
B
‍
ω
(
y
)
-
p
′
d
y
)
1
/
p
′
(
1
|
B
|
∫
B
‍
ω
(
y
)
q
d
y
)
1
/
q
≤
C
holds when 1
<
p
<
∞, and the inequality
(14)
(
1
|
B
|
∫
B
‍
ω
(
y
)
q
d
y
)
1
/
q
≤
C
·
ess
inf
⁡
x
∈
B
w
(
x
)
for
every
ball
B
⊂
ℝ
n
holds when p
=
1.By (13), we have
(15)
(
∫
B
‍
ω
(
y
)
-
p
′
d
y
)
1
/
p
′
(
∫
B
‍
ω
(
y
)
q
d
y
)
1
/
q
≤
C
|
B
|
1
/
p
′
+
1
/
q
.We summarize some properties about weightsA
p
,
q
(
ℝ
n
) (see [11, 12]).Lemma 5.
Given1
≤
p
≤
q
<
∞, (i)
ω
∈
A
p
,
q
(
ℝ
n
) if and only if ω
q
∈
A
1
+
q
/
p
′
(
ℝ
n
);
(ii)
ω
∈
A
p
,
q
(
ℝ
n
) if and only if ω
-
p
′
∈
A
1
+
p
′
/
q
(
ℝ
n
);
(iii)
ω
∈
A
p
,
p
(
ℝ
n
) if and only if ω
p
∈
A
p
(
ℝ
n
);
(iv)
Ifp
1
<
p
2 and q
2
>
q
1, then A
p
1
,
q
1
(
ℝ
n
)
⊂
A
p
2
,
q
2
(
ℝ
n
).A locally integrable functionb is said to be in BMO
(
ℝ
n
) if
(16)
sup
⁡
B
⊂
ℝ
n
1
|
B
|
∫
B
‍
|
b
(
x
)
-
b
B
|
d
x
=
∥
b
∥
*
<
∞
,
where b
B
=
|
B
|
-
1
∫
B
‍
b
(
y
)
d
y.Lemma 6 (John-Nirenberg inequality; see [13]).
Letb
∈
B
M
O
(
ℝ
n
). Then, for any ball B
⊂
ℝ
n, there exist positive constants C
1 and C
2 such that, for all λ
>
0,
(17)
|
{
x
∈
B
:
|
b
(
x
)
-
b
B
|
>
λ
}
|
≤
C
1
|
B
|
exp
⁡
(
-
C
2
λ
∥
b
∥
*
)
.Lemma 7 (see [14]).
Letω
∈
A
∞
(
ℝ
n
). Then, the norm of B
M
O
(
ω
,
ℝ
n
) is equivalent to the norm of B
M
O
(
ℝ
n
), where
(18)
B
M
O
(
ω
,
ℝ
n
)
=
{
1
ω
(
B
)
∫
B
‍
|
b
(
x
)
-
b
B
,
ω
|
b
:
∥
b
∥
*
,
ω
=
sup
⁡
B
⊂
ℝ
n
1
ω
(
B
)
×
∫
B
‍
|
b
(
x
)
-
b
B
,
ω
|
ω
(
x
)
d
x
<
∞
}
,
b
B
,
ω
=
1
ω
(
B
)
∫
B
‍
b
(
z
)
ω
(
z
)
d
z
.Lemma 8.
Supposeω
∈
A
∞
(
ℝ
n
) and b
∈
B
M
O
(
ℝ
n
). Then, for any p
≥
1, one has
(19)
(
1
ω
(
B
)
∫
B
‍
|
b
(
x
)
-
b
B
,
ω
|
p
ω
(
x
)
d
x
)
1
/
p
≤
C
∥
b
∥
*
.Proof.
Sinceω
(
x
)
∈
A
∞
(
ℝ
n
), (10) and Lemma 6 imply that
(20)
ω
(
{
x
∈
B
:
|
b
(
x
)
-
b
B
|
>
λ
}
)
≤
C
ω
(
B
)
exp
⁡
(
-
C
2
δ
λ
∥
b
∥
*
)
.
Therefore,
(21)
∫
B
‍
|
b
(
x
)
-
b
B
|
p
ω
(
x
)
d
x
≤
p
∫
0
∞
‍
λ
p
-
1
ω
(
{
x
∈
B
:
|
b
(
x
)
-
b
B
|
>
λ
}
)
d
λ
≤
C
ω
(
B
)
∫
0
∞
‍
λ
p
-
1
exp
⁡
(
-
C
2
δ
λ
∥
b
∥
*
)
d
λ
≤
C
ω
(
B
)
∥
b
∥
*
p
.
Hence,
(22)
1
ω
(
B
)
∫
B
‍
|
b
(
x
)
-
b
B
,
ω
|
p
ω
(
x
)
d
x
≤
C
ω
(
B
)
∫
B
‍
|
b
(
x
)
-
b
B
|
p
ω
(
x
)
d
x
≤
C
∥
b
∥
*
p
.


## 3. Proof of Theorem1

Letf
∈
K
p
1
q
(
φ
,
ω
p
1
)
(
ℝ
n
), b
∈
BMO
(
ℝ
n
). Then,
(23)
∥
[
b
,
I
l
]
f
∥
K
p
2
q
(
φ
,
ω
p
2
)
q
≤
C
∑
j
=
-
∞
∞
‍
φ
(
j
)
q
(
∑
k
=
-
∞
j
-
2
‍
∥
[
b
,
I
l
]
(
f
χ
k
)
χ
j
(
x
)
∥
L
p
2
(
ω
p
2
)
)
q
+
C
∑
j
=
-
∞
∞
‍
φ
(
j
)
q
(
∑
k
=
j
-
1
j
+
1
‍
∥
[
b
,
I
l
]
(
f
χ
k
)
χ
j
(
x
)
∥
L
p
2
(
ω
p
2
)
)
q
+
C
∑
j
=
-
∞
∞
‍
φ
(
j
)
q
(
∑
k
=
j
+
2
∞
‍
∥
[
b
,
I
l
]
(
f
χ
k
)
χ
j
(
x
)
∥
L
p
2
(
ω
p
2
)
)
q
=
L
1
+
L
2
+
L
3
.
Since ω
p
2
∈
A
r, where r
=
min
⁡
{
1
+
(
p
2
/
p
1
′
)
,
1
+
(
p
2
/
p
1
′
)
-
(
b
p
2
/
n
)
}, then, by (i) of Lemma 5, we know thatω
∈
A
p
1
,
p
2
(
ℝ
n
). Using the fact that [
b
,
I
l
] is a bounded operator from L
p
1
(
ω
p
1
,
ℝ
n
) to L
p
2
(
ω
p
2
,
ℝ
n
) (see [2]), we obtain
(24)
L
2
=
C
∑
j
=
-
∞
∞
‍
φ
(
j
)
q
(
∑
k
=
j
-
1
j
+
1
‍
∥
[
b
,
I
l
]
(
f
χ
k
)
χ
j
(
x
)
∥
L
p
2
(
ω
p
2
)
)
q
≤
C
∥
b
∥
*
q
∑
j
=
-
∞
∞
‍
φ
(
j
)
q
∑
k
=
j
-
1
j
+
1
‍
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
q
≤
C
∥
b
∥
*
q
∑
j
=
-
∞
∞
‍
φ
(
j
)
q
∥
f
χ
j
∥
L
p
1
(
ω
p
1
)
q
≤
C
∥
b
∥
*
q
∥
f
∥
K
p
1
q
(
φ
,
ω
p
1
)
q
.
We now estimate L
1. Obviously,
(25)
∥
[
b
,
I
l
]
(
f
χ
k
)
χ
j
(
x
)
∥
L
p
2
(
ω
p
2
)
≤
C
(
∫
C
j
‍
|
(
b
(
x
)
-
b
B
k
)
∫
C
k
‍
1
|
x
-
y
|
n
-
l
×
(
f
χ
k
)
(
y
)
d
y
(
b
(
x
)
-
b
B
k
)
∫
C
k
‍
1
|
x
-
y
|
n
-
l
|
p
2
ω
(
x
)
p
2
d
x
)
1
/
p
2
+
C
(
∫
C
j
‍
|
∫
C
k
‍
1
|
x
-
y
|
n
-
l
(
b
(
y
)
-
b
B
k
)
×
(
f
χ
k
)
(
y
)
d
y
∫
C
k
‍
1
|
x
-
y
|
n
-
l
(
b
(
y
)
-
b
B
k
)
|
p
2
ω
(
x
)
p
2
d
x
)
1
/
p
2
=
M
1
+
M
2
.
First, we estimate M
1. From j
≥
k
+
2, one can see then that
(26)
M
1
=
(
∫
C
j
‍
|
(
b
(
x
)
-
b
B
k
)
∫
C
k
‍
1
|
x
-
y
|
n
-
l
×
(
f
χ
k
)
(
y
)
d
y
(
b
(
x
)
-
b
B
k
)
∫
C
k
‍
1
|
x
-
y
|
n
-
l
|
p
2
ω
(
x
)
p
2
d
x
)
1
/
p
2
≤
C
2
-
j
(
n
-
l
)
∥
f
χ
k
∥
L
1
×
(
(
∫
B
j
‍
|
b
(
x
)
-
b
B
j
,
ω
p
2
|
p
2
ω
(
x
)
p
2
d
x
)
1
/
p
2
+
(
|
b
B
j
-
b
B
j
,
ω
p
2
|
+
|
b
B
j
-
b
B
k
|
)
×
(
∫
B
j
‍
ω
(
x
)
p
2
d
x
)
1
/
p
2
)
.Since ω
p
2
∈
A
r, then, by Lemma 8,
(27)
(
∫
B
j
‍
|
b
(
x
)
-
b
B
j
,
ω
p
2
|
p
2
ω
(
x
)
p
2
d
x
)
1
/
p
2
≤
C
∥
b
∥
*
(
∫
B
j
‍
ω
(
x
)
p
2
d
x
)
1
/
p
2
.
Similar to the proof of Lemma 8, we can deduce that
(28)
|
b
B
j
-
b
B
j
,
ω
p
2
|
≤
C
∥
b
∥
*
.
From the definition of BMO
(
ℝ
n
), it is easy to see that
(29)
|
b
B
j
-
b
B
k
|
≤
C
(
j
-
k
)
∥
b
∥
*
.
By Hölder's inequality, we get
(30)
∥
f
χ
k
∥
L
1
≤
C
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
(
∫
B
k
‍
ω
-
p
1
′
d
x
)
1
/
p
1
′
.
Then,
(31)
M
1
≤
C
(
j
-
k
)
2
-
j
(
n
-
l
)
∥
b
∥
*
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
×
(
∫
B
k
‍
ω
-
p
1
′
d
x
)
1
/
p
1
′
(
∫
B
j
‍
ω
(
x
)
p
2
d
x
)
1
/
p
2
.We now turn to estimateM
2. Since ω
∈
A
p
1
,
p
2
(
ℝ
n
), then, by (ii) of Lemma 5, we know ω
-
p
1
′
∈
A
∞
(
ℝ
n
). Similar to the proof of Lemma 8, we have
(32)
(
∫
B
k
‍
|
b
(
x
)
-
b
B
k
|
p
1
′
ω
-
p
1
′
d
x
)
1
/
p
2
≤
C
∥
b
∥
*
(
∫
B
k
‍
ω
(
x
)
-
p
1
′
d
x
)
1
/
p
1
′
.
Using Hölder's inequality and (32),
(33)
M
2
≤
C
2
-
j
(
n
-
l
)
∫
B
k
‍
|
b
(
y
)
-
b
B
k
|
|
f
χ
k
(
y
)
|
d
y
×
(
∫
B
j
‍
ω
(
x
)
p
2
d
x
)
1
/
p
2
≤
C
2
-
j
(
n
-
l
)
(
∫
B
k
‍
|
b
(
y
)
-
b
B
k
|
p
1
′
ω
(
y
)
-
p
1
′
d
y
)
1
/
p
1
′
×
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
(
∫
B
j
‍
ω
(
x
)
p
2
d
x
)
1
/
p
2
≤
C
2
-
j
(
n
-
l
)
∥
b
∥
*
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
×
(
∫
B
k
‍
ω
(
x
)
-
p
1
′
d
x
)
1
/
p
1
′
(
∫
B
j
‍
ω
(
x
)
p
2
d
x
)
1
/
p
2
.
Since ω
(
x
)
p
2
∈
A
r, by (iii) of Lemma 4, we have ω
(
x
)
p
2
∈
A
r
¯ for some r
¯
<
r. Then, from (15) and (i) of Lemma 4,
(34)
(
∫
B
k
‍
ω
(
x
)
-
p
1
′
d
x
)
1
/
p
1
′
(
∫
B
j
‍
ω
p
2
d
x
)
1
/
p
2
=
(
∫
B
k
‍
ω
-
p
1
′
d
x
)
1
/
p
1
′
(
∫
B
k
‍
ω
(
x
)
p
2
d
x
)
1
/
p
2
×
(
ω
p
2
(
B
j
)
ω
p
2
(
B
k
)
)
1
/
p
2
≤
C
2
k
n
(
1
/
p
1
′
+
1
/
p
2
)
+
(
j
-
k
)
n
r
¯
/
p
2
.
Hence,
(35)
∥
[
b
,
I
l
]
(
f
χ
k
)
χ
j
(
x
)
∥
L
p
2
(
ω
p
2
)
≤
C
∥
b
∥
*
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
(
j
-
k
)
2
-
j
(
n
-
l
)
×
(
∫
B
k
‍
ω
-
p
1
′
d
x
)
1
/
p
1
′
(
∫
B
j
‍
ω
(
x
)
p
2
d
x
)
1
/
p
2
≤
C
∥
b
∥
*
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
(
j
-
k
)
2
-
j
(
n
-
l
)
≤
C
∥
b
∥
*
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
×
(
j
-
k
)
2
j
(
-
n
+
l
+
n
r
¯
/
p
2
)
2
n
k
(
1
/
p
1
′
+
1
/
p
2
-
r
¯
/
p
2
)
=
C
∥
b
∥
*
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
(
j
-
k
)
2
(
j
-
k
)
(
-
n
+
l
+
n
r
¯
/
p
2
)
.
Using the condition that φ
∈
D
(
a
,
b
), we obtain
(36)
L
1
=
C
∑
j
=
-
∞
∞
‍
φ
(
j
)
q
(
∑
k
=
-
∞
j
-
2
‍
∥
[
b
,
I
l
]
(
f
χ
k
)
χ
j
(
x
)
∥
L
p
2
(
ω
p
2
)
)
q
≤
C
∥
b
∥
*
q
∑
j
=
-
∞
∞
‍
φ
(
j
)
q
×
(
∑
k
=
-
∞
j
-
2
‍
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
(
j
-
k
)
2
(
j
-
k
)
(
-
n
+
l
+
n
r
¯
/
p
2
)
)
q
≤
C
∥
b
∥
*
q
∑
j
=
-
∞
∞
‍
(
∑
k
=
-
∞
j
-
2
‍
φ
(
k
)
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
×
(
j
-
k
)
2
(
j
-
k
)
(
-
n
+
l
+
n
r
¯
/
p
2
+
b
)
∑
k
=
-
∞
j
-
2
‍
φ
(
k
)
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
)
q
.Sincer
≤
1
+
p
2
/
p
1
′
-
b
p
2
/
n, we have -
n
+
l
+
n
r
/
p
2
+
b
≤
0, and, then, -
n
+
l
+
n
r
¯
/
p
2
+
b
<
0. When 0
<
q
≤
1, we have
(37)
L
1
≤
C
∥
b
∥
*
q
∑
k
=
-
∞
∞
‍
φ
(
k
)
q
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
q
×
∑
j
=
k
+
2
∞
‍
(
j
-
k
)
q
2
(
j
-
k
)
q
(
-
n
+
l
+
n
r
¯
/
p
2
+
b
)
≤
C
∥
b
∥
*
q
∑
k
=
-
∞
∞
‍
φ
(
k
)
q
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
q
≤
C
∥
b
∥
*
q
∥
f
∥
K
p
q
(
ℝ
n
)
q
.
When q
>
1, by Hölder's inequality, we have
(38)
L
1
≤
C
∥
b
∥
*
q
∑
j
=
-
∞
∞
(
∑
k
=
-
∞
j
-
2
‍
φ
(
k
)
q
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
q
‍
×
(
j
-
k
)
2
(
j
-
k
)
(
-
n
+
l
+
n
r
¯
/
p
2
+
b
)
∑
k
=
-
∞
j
-
2
‍
φ
(
k
)
q
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
q
)
×
(
∑
k
=
-
∞
j
-
2
‍
(
j
-
k
)
2
(
j
-
k
)
(
-
n
+
l
+
n
r
¯
/
p
2
+
b
)
)
q
/
q
′
≤
C
∥
b
∥
*
q
∑
k
=
-
∞
∞
‍
φ
(
k
)
q
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
q
≤
C
∥
b
∥
*
q
∥
f
∥
K
p
q
(
ℝ
n
)
q
.Now, we estimateL
3. Since k
≥
j
+
2, similar to the estimation of L
1, we have
(39)
∥
[
b
,
I
l
]
(
f
χ
k
)
χ
j
(
x
)
∥
L
p
2
(
ω
p
2
)
≤
C
(
k
-
j
)
2
-
k
(
n
-
l
)
∥
b
∥
*
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
×
(
∫
B
k
‍
ω
(
x
)
-
p
1
′
d
x
)
1
/
p
1
′
(
∫
B
j
‍
ω
(
x
)
p
2
d
x
)
1
/
p
2
.
From the fact ω
∈
A
p
1
,
p
2
(
ℝ
n
) and (15), we get
(40)
(
∫
B
k
‍
ω
-
p
1
′
d
x
)
1
/
p
1
′
(
∫
B
k
‍
ω
(
x
)
p
2
d
x
)
1
/
p
2
≤
C
2
k
n
(
1
/
p
1
′
+
1
/
p
2
)
.
By ω
p
2
∈
R
D
(
δ
), we have
(41)
(
ω
p
2
(
B
j
)
ω
p
2
(
B
k
)
)
1
/
p
2
≤
C
2
(
j
-
k
)
δ
/
p
2
.
Then,
(42)
(
∫
B
k
‍
ω
(
x
)
-
p
1
′
d
x
)
1
/
p
1
′
(
∫
B
j
‍
ω
(
x
)
p
2
d
x
)
1
/
p
2
=
(
∫
B
k
‍
ω
-
p
1
′
d
x
)
1
/
p
1
′
(
∫
B
k
‍
ω
(
x
)
p
2
d
x
)
1
/
p
2
×
(
ω
p
2
(
B
j
)
ω
p
2
(
B
k
)
)
1
/
p
2
≤
C
2
k
n
(
1
/
p
1
′
+
1
/
p
2
)
+
(
j
-
k
)
δ
/
p
2
.
Hence,
(43)
∥
[
b
,
I
l
]
(
f
χ
k
)
χ
j
(
x
)
∥
L
p
2
(
ω
p
2
)
≤
C
∥
b
∥
*
(
k
-
j
)
2
k
n
(
1
/
p
1
′
+
1
/
p
2
)
+
(
j
-
k
)
δ
/
p
2
-
k
(
n
-
l
)
=
C
∥
b
∥
*
(
k
-
j
)
2
(
j
-
k
)
δ
/
p
2
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
.
Using the condition that φ
∈
D
(
a
,
b
),
(44)
L
3
=
C
∑
j
=
-
∞
∞
‍
φ
(
j
)
q
(
∑
k
=
j
+
2
∞
‍
∥
[
b
,
I
l
]
(
f
χ
k
)
χ
j
(
x
)
∥
L
p
2
(
ω
p
2
)
)
q
≤
C
∥
b
∥
*
q
∑
j
=
-
∞
∞
‍
(
∑
k
=
j
+
2
∞
‍
φ
(
k
)
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
×
(
k
-
j
)
2
(
j
-
k
)
(
δ
/
p
2
+
a
)
∑
k
=
j
+
2
∞
‍
)
q
.
When 0
<
q
≤
1, we have
(45)
L
3
≤
C
∥
b
∥
*
q
∑
k
=
-
∞
∞
‍
φ
(
k
)
q
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
q
×
∑
j
=
-
∞
k
-
2
‍
(
k
-
j
)
q
2
(
j
-
k
)
(
δ
/
p
2
+
a
)
q
≤
C
∥
b
∥
*
q
∑
k
=
-
∞
∞
‍
φ
(
k
)
q
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
q
≤
C
∥
b
∥
*
∥
f
∥
K
p
q
(
φ
,
ω
p
1
)
q
.
When q
>
1, by the Hölder inequality, we get
(46)
L
3
≤
C
∥
b
∥
*
q
∑
j
=
-
∞
∞
‍
(
∑
k
=
j
+
2
∞
‍
φ
(
k
)
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
×
(
k
-
j
)
2
(
j
-
k
)
(
δ
/
p
2
+
a
)
∑
k
=
j
+
2
∞
‍
φ
(
k
)
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
)
q
.
≤
C
∥
b
∥
*
q
∑
j
=
-
∞
∞
‍
(
∑
k
=
j
+
2
∞
‍
φ
(
k
)
q
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
q
×
(
k
-
j
)
2
(
j
-
k
)
(
δ
/
p
2
+
a
)
∑
k
=
j
+
2
∞
‍
φ
(
k
)
q
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
q
)
×
(
∑
k
=
j
+
2
∞
‍
(
k
-
j
)
2
(
j
-
k
)
(
δ
/
p
2
+
a
)
)
q
/
q
′
≤
C
∥
b
∥
*
q
∑
k
=
-
∞
∞
‍
φ
(
k
)
q
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
q
×
(
∑
j
=
-
∞
k
-
2
‍
(
k
-
j
)
q
2
(
j
-
k
)
(
δ
/
p
2
+
a
)
q
)
≤
C
∥
b
∥
*
q
∑
k
=
-
∞
∞
‍
φ
(
k
)
q
∥
f
χ
k
∥
L
p
1
(
ω
p
1
)
q
≤
C
∥
b
∥
*
q
∥
f
∥
K
p
q
(
φ
,
ω
p
1
)
q
.
Combining the above estimates for L
1, L
2, and L
3, the proof of Theorem 1 is completed.



---

*Source: 428493-2014-04-03.xml*