# The Boundedness of the Hardy-Littlewood Maximal Operator and Multilinear Maximal Operator in Weighted Morrey Type Spaces

Authors: Takeshi Iida
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

The aim of this paper is to prove the boundedness of the Hardy-Littlewood maximal operator on weighted Morrey spaces and multilinear maximal operator on multiple weighted Morrey spaces. In particular, the result includes the Komori-Shirai theorem and the Iida-Sato-Sawano-Tanaka theorem for the Hardy-Littlewood maximal operator and multilinear maximal function.


---

## Body

## 1. Introduction

The aim of this paper is to investigate the boundedness of the Hardy-Littlewood maximal operator on weighted Morrey spaces and multilinear maximal operator on weighted Morrey type spaces. The Hardy-Littlewood maximal functionM
f is given by
(1)
M
f
(
x
)
:
=
sup
⁡
Q
∋
x
Q
:
cube
⨏
Q
|
f
(
y
)
|
d
y
,
where x
∈
ℝ
n, f is locally integrable function, and the description ⨏
Q is an integral mean on Q:
(2)
⨏
Q
F
(
y
)
d
y
:
=
1
|
Q
|
∫
Q
‍
F
(
y
)
d
y
.
In this paper, for every cube Q
⊂
ℝ
n, we assume that the sides of Q are parallel to the coordinate axes and |
Q
| describes the volume of Q. A weight is a nonnegative locally integrable function on ℝ
n that takes values in (
0
,
∞
) almost everywhere. Given a weight w and a measurable set E, we use the notation
(3)
w
(
E
)
:
=
∫
E
‍
w
(
x
)
d
x
to denote the w-measure of the set E. We recall weighted Lebesgue spaces, Morrey spaces, weighted Morrey spaces, and multi-Morrey spaces.Definition 1 (weighted Lebesgue spaces).
Let0
<
p
<
∞ and let w be a weight. The weighted Lebesgue space is defined by the norm (or quasinorm)
(4)
∥
f
∥
L
p
(
w
)
:
=
(
∫
ℝ
n
‍
|
f
(
x
)
|
p
w
(
x
)
d
x
)
1
/
p
.Definition 2 (Morrey spaces).
For0
<
p
≤
p
0
<
∞, the Morrey space ℳ
p
p
0
(
ℝ
n
) is defined by the norm (or quasi-norm)
(5)
∥
f
∥
ℳ
p
p
0
(
ℝ
n
)
=
sup
⁡
Q
⊂
ℝ
n
;
Q
:
cube
|
Q
|
1
/
p
0
(
⨏
Q
|
f
(
x
)
|
p
d
x
)
1
/
p
.Definition 3 (weighted Morrey spaces).
Letw
1 and w
2 be weights. For 0
<
p
≤
p
0
<
∞, the weighted Morrey space is defined by the norm (or quasinorm)
(6)
∥
f
∥
ℳ
p
p
0
(
w
1
,
w
2
)
:
=
sup
⁡
Q
⊂
ℝ
n
;
Q
:
cube
w
1
(
Q
)
1
/
p
0
(
1
w
1
(
Q
)
∫
Q
‍
|
f
(
x
)
|
p
w
2
(
x
)
d
x
)
1
/
p
.Definition 4 (multi-Morrey space).
Let0
<
p
1
,
…
,
p
m
<
∞ and 1
/
p
0
≤
1
/
p
1
+
⋯
+
1
/
p
m. The multi-Morrey space is defined by the quantity for vector valued function f
→
=
(
f
1
,
…
,
f
m
):
(7)
∥
f
→
∥
ℳ
P
→
p
0
(
ℝ
n
)
:
=
sup
⁡
Q
⊂
ℝ
n
;
Q
:
cube
|
Q
|
1
/
p
0
∏
j
=
1
m
‍
(
⨏
Q
|
f
j
(
y
j
)
|
p
j
d
y
j
)
1
/
p
j
.Hence it is natural to consider multiple weighted Morrey spaces.Definition 5 (multiple weighted Morrey spaces).
Let0
<
p
1
,
…
,
p
m
<
∞ and 1
/
p
0
≤
1
/
p
1
+
⋯
+
1
/
p
m. Let w
→
=
(
w
1
,
…
,
w
m
) be a multiple weight. Let W be a weight. The multiple weighted Morrey space is defined by the quantity for vector valued function f
→
=
(
f
1
,
…
,
f
m
):
(8)
∥
f
→
∥
ℳ
P
→
p
0
(
W
,
w
→
)
:
=
sup
⁡
Q
⊂
ℝ
n
;
Q
:
cube
W
(
Q
)
1
/
p
0
∏
j
=
1
m
‍
(
1
W
(
Q
)
∫
Q
‍
|
f
j
(
y
j
)
|
p
j
w
j
(
y
j
)
d
y
j
)
1
/
p
j
.In this framework, we investigate the well-known results. The following theorem which is elementary result was discovered by Muckenhoupt [1].Theorem A.
Letw be a weight and 1
<
p
<
∞:
(9)
∥
M
f
∥
L
p
(
w
)
≤
C
∥
f
∥
L
p
(
w
)
if and only if
(10)
w
∈
A
p
(
ℝ
n
)
,
where
(11)
[
w
]
A
p
:
=
sup
⁡
Q
⊂
ℝ
n
;
Q
:
c
u
b
e
(
⨏
Q
w
(
x
)
d
x
)
(
⨏
Q
w
(
x
)
1
/
(
1
-
p
)
d
x
)
p
-
1
<
∞
.Komori and Shirai [2] introduced the weighted Morrey spaces and proved the following theorem. The following theorem gives us the boundedness of the Hardy-Littlewood maximal operator on weighted Morrey spaces.Theorem B.
Letw be a weight and 1
<
p
<
p
0
<
∞. If w
∈
A
p
(
ℝ
n
), then one has
(12)
∥
M
f
∥
ℳ
p
p
0
(
w
,
w
)
≤
C
∥
f
∥
ℳ
p
p
0
(
w
,
w
)
.On the other hand, in [3], the following theorem was proved. The following result gives also the boundedness of the Hardy-Littlewood maximal operator on weighted Morrey spaces.Theorem C.
Letw be a weight and 1
<
p
≤
p
0
<
∞. If
(13)
sup
⁡
Q
⊂
Q
′
;
Q
,
Q
′
:
c
u
b
e
s
(
|
Q
|
|
Q
′
|
)
1
/
p
0
(
⨏
Q
w
(
x
)
p
d
x
)
1
/
p
(
⨏
Q
′
w
(
x
)
-
p
′
d
x
)
1
/
p
′
<
∞
,
then one has
(14)
∥
M
f
∥
ℳ
p
p
0
(
1
,
w
p
)
≤
C
∥
f
∥
ℳ
p
p
0
(
1
,
w
p
)
.Moreover we can extend Theorem C to the multilinear version (see [3, 4]).Theorem D.
Let1
<
p
1
,
…
,
p
m
<
∞ and 1
/
p
0
≤
1
/
p
1
+
⋯
+
1
/
p
m. Let w
→
=
(
w
1
,
…
,
w
m
) be a multiple weight. If
(15)
sup
⁡
Q
⊂
Q
′
;
Q
,
Q
′
:
c
u
b
e
s
(
|
Q
|
|
Q
′
|
)
1
/
p
0
(
⨏
Q
(
w
1
⋯
w
m
)
(
x
)
p
d
x
)
1
/
p
×
∏
j
=
1
m
‍
(
⨏
Q
′
w
j
(
y
j
)
-
p
′
d
y
j
)
1
/
p
′
<
∞
.
Then one has
(16)
∥
ℳ
f
→
∥
ℳ
p
p
0
(
1
,
(
w
1
⋯
w
m
)
p
)
≤
C
∥
f
∥
ℳ
p
p
0
(
1
,
w
→
P
→
)
,
where w
→
P
→
=
(
w
1
p
1
,
…
,
w
m
p
m
) and
(17)
ℳ
f
→
(
x
)
:
=
sup
⁡
Q
∋
x
∏
j
=
1
m
‍
⨏
Q
|
f
j
(
y
j
)
|
d
y
j
.Because the ways of the evaluation by the weighted Morrey norm are different, Theorems B and C are independent. Therefore, it is natural to consider unifying Theorems B and C. The question is not settled yet. In this paper, we unify Theorems B–D.


## 2. Main Results

Theorem 6.
Let1
<
p
≤
p
0
<
∞ and let w
1 and w
2 be weights. Additionally assume that w
1 satisfies the doubling condition: there exists C
>
0 such that
(18)
w
1
(
2
Q
)
≤
C
w
1
(
Q
)
.
Assume that the following condition holds:
(19)
[
w
1
,
w
2
]
p
0
,
p
=
sup
⁡
Q
⊂
Q
′
Q
,
Q
′
:
c
u
b
e
s
(
w
1
(
Q
)
w
1
(
Q
′
)
)
1
/
p
0
-
1
/
p
(
|
Q
|
|
Q
′
|
)
1
/
p
×
(
⨏
Q
w
2
(
x
)
p
d
x
)
1
/
p
(
⨏
Q
′
w
2
(
x
)
-
p
′
d
x
)
1
/
p
′
•
<
∞
.
Then, one has
(20)
w
1
(
Q
)
1
/
p
0
(
1
w
1
(
Q
)
∫
Q
‍
M
f
(
x
)
p
w
2
(
x
)
p
d
x
)
1
/
p
≤
C
[
w
1
,
w
2
]
p
0
,
p
sup
⁡
Q
⊂
Q
′
Q
,
Q
′
:
c
u
b
e
s
w
1
(
Q
′
)
1
/
p
0
×
(
1
w
1
(
Q
′
)
∫
Q
′
‍
|
f
(
x
)
|
p
w
2
(
x
)
p
d
x
)
1
/
p
.Remark 7.
If condition (19) holds, then w
2
p
∈
A
p
(
ℝ
n
). In fact, for every cube Q
⊂
ℝ
n,
(21)
(
⨏
Q
w
2
(
x
)
p
d
x
)
1
/
p
(
⨏
Q
w
2
(
x
)
-
p
′
d
x
)
1
/
p
′
=
(
w
1
(
Q
)
w
1
(
Q
)
)
1
/
p
0
-
1
/
p
(
|
Q
|
|
Q
|
)
1
/
p
(
⨏
Q
w
2
(
x
)
p
d
x
)
1
/
p
×
(
⨏
Q
w
2
(
x
)
-
p
′
d
x
)
1
/
p
′
≤
[
w
1
,
w
2
]
p
0
,
p
<
∞
.
Therefore, we obtain w
2
p
∈
A
p
(
ℝ
n
).Theorem 8.
Letw
1 and w
2 be weights. Let 1
<
p
≤
p
0
<
∞. Additionally assume that w
1 satisfies the doubling condition. If [
w
1
,
w
2
]
p
0
,
p
<
∞, then one has
(22)
∥
M
f
∥
ℳ
p
p
0
(
w
1
,
w
2
p
)
≤
C
[
w
1
,
w
2
]
p
0
,
p
∥
f
∥
ℳ
p
p
0
(
w
1
,
w
2
p
)
.Remark 9.
Theorem8 includes Theorems B and C. In Theorem 8, if w
1
=
1, then Theorem 8 corresponds with Theorem C. In Theorem 8, if w
1
=
w
2
p
=
w, then Theorem 8 corresponds with Theorem B. In fact, condition (19) corresponds with w
∈
A
p
(
ℝ
n
). We state the detail:
(23)
[
w
,
w
1
/
p
]
p
0
,
p
=
sup
⁡
Q
⊂
Q
′
Q
,
Q
′
:
cubes
(
w
(
Q
)
w
(
Q
′
)
)
1
/
p
0
-
1
/
p
(
|
Q
|
|
Q
′
|
)
1
/
p
×
(
⨏
Q
w
(
x
)
d
x
)
1
/
p
(
⨏
Q
′
w
(
x
)
-
p
′
/
p
d
x
)
1
/
p
′
=
sup
⁡
Q
⊂
Q
′
Q
,
Q
′
:
cubes
(
w
(
Q
)
w
(
Q
′
)
)
1
/
p
0
(
w
(
Q
′
)
|
Q
′
|
)
1
/
p
×
(
⨏
Q
′
w
(
x
)
-
p
′
/
p
d
x
)
1
/
p
′
=
sup
⁡
Q
⊂
Q
′
Q
,
Q
′
:
cubes
(
w
(
Q
)
w
(
Q
′
)
)
1
/
p
0
×
[
(
⨏
Q
′
w
(
x
)
d
x
)
(
⨏
Q
′
w
(
x
)
1
/
(
1
-
p
)
d
x
)
p
-
1
]
1
/
p
≤
[
w
]
A
p
1
/
p
·
sup
⁡
Q
⊂
Q
′
Q
,
Q
′
:
cubes
(
w
(
Q
)
w
(
Q
′
)
)
1
/
p
0
≤
[
w
]
A
p
1
/
p
<
∞
.Theorem 10.
Letw
1
,
w
2, and v be weights. Additionally assume that w
1 satisfies the doubling condition. Let 1
<
p
≤
p
0
<
∞, a
>
1, and
(24)
[
w
1
,
w
2
,
v
]
p
0
,
p
,
p
/
a
≔
sup
⁡
Q
⊂
Q
′
Q
,
Q
′
:
c
u
b
e
s
(
w
1
(
Q
)
w
1
(
Q
′
)
)
1
/
p
0
-
1
/
p
(
|
Q
|
|
Q
′
|
)
1
/
p
×
(
⨏
Q
v
(
y
)
p
d
y
)
1
/
p
(
⨏
Q
′
w
2
(
y
)
-
(
p
/
a
)
′
d
y
)
1
/
(
p
/
a
)
′
<
∞
.
Then one has
(25)
∥
M
f
∥
ℳ
p
p
0
(
w
1
,
v
p
)
≤
C
[
w
1
,
w
2
,
v
]
p
0
,
p
,
p
/
a
∥
f
∥
ℳ
p
p
0
(
w
1
,
w
2
p
)
.We can extend Theorem10 as follows.Theorem 11.
Letw
1
,
w
2, and v be weights. Additionally assume that w
1 satisfies the doubling condition. Let 1
<
p
≤
p
0
≤
q
0
<
∞, a
>
1, and
(26)
[
w
1
,
w
2
,
v
]
p
0
,
q
0
,
p
,
p
/
a
≔
sup
⁡
Q
⊂
Q
′
Q
,
Q
′
:
c
u
b
e
s
w
1
(
Q
)
1
/
q
0
w
1
(
Q
′
)
1
/
p
0
(
w
1
(
Q
′
)
w
1
(
Q
)
)
1
/
p
(
|
Q
|
|
Q
′
|
)
1
/
p
×
(
⨏
Q
v
(
y
)
p
d
y
)
1
/
p
(
⨏
Q
′
w
2
(
y
)
-
(
p
/
a
)
′
d
y
)
1
/
(
p
/
a
)
′
.
<
∞
.
Then one has
(27)
∥
M
f
∥
ℳ
p
q
0
(
w
1
,
v
p
)
≤
C
[
w
1
,
w
2
,
v
]
p
0
,
q
0
,
p
,
p
/
a
∥
f
∥
ℳ
p
p
0
(
w
1
,
w
2
p
)
.By considering the multilinear version, we obtain the following theorems.Theorem 12.
Let1
<
p
1
,
…
,
p
m
<
∞, 1
/
p
=
1
/
p
1
+
⋯
+
1
/
p
m, and a
>
1. Let v and W be weights. Let w
→
=
(
w
1
,
…
,
w
m
) be a multiple weight. Additionally assume that W satisfies the doubling condition and
(28)
[
v
,
W
,
w
→
]
p
0
,
p
,
P
→
/
a
:
=
sup
⁡
Q
⊂
Q
′
Q
,
Q
′
:
c
u
b
e
s
(
W
(
Q
)
W
(
Q
′
)
)
1
/
p
0
-
1
/
p
(
|
Q
|
|
Q
′
|
)
1
/
p
×
(
⨏
Q
v
(
x
)
p
d
x
)
1
/
p
×
∏
j
=
1
m
‍
(
⨏
Q
′
w
j
(
y
j
)
-
(
p
j
/
a
)
′
d
y
j
)
1
/
(
p
j
/
a
)
′
<
∞
.
Then one has
(29)
∥
ℳ
f
→
∥
ℳ
p
p
0
(
W
,
v
p
)
≤
C
∥
f
→
∥
ℳ
P
→
p
0
(
W
,
w
→
P
→
)
,
where w
→
P
→
=
(
w
1
p
1
,
…
,
w
m
p
m
).We can extend Theorem12 as follows.Theorem 13.
Let1
<
p
1
,
…
,
p
m
<
∞, 1
/
p
=
1
/
p
1
+
⋯
+
1
/
p
m, 0
<
p
≤
p
0
≤
q
0
<
∞, and a
>
1. Let v and W be weights. Let w
→
=
(
w
1
,
…
,
w
m
) be a multiple weight. Additionally assume that W satisfies the doubling condition and
(30)
[
v
,
W
,
w
→
]
p
0
,
q
0
,
p
,
P
→
/
a
:
=
sup
⁡
Q
⊂
Q
′
Q
,
Q
′
:
c
u
b
e
s
W
(
Q
)
1
/
q
0
W
(
Q
′
)
1
/
p
0
(
W
(
Q
′
)
W
(
Q
)
)
1
/
p
×
(
|
Q
|
|
Q
′
|
)
1
/
p
(
⨏
Q
v
(
x
)
p
d
x
)
1
/
p
×
∏
j
=
1
m
‍
(
⨏
Q
′
w
j
(
y
j
)
-
(
p
j
/
a
)
′
d
y
j
)
1
/
(
p
j
/
a
)
′
<
∞
.
Then one has
(31)
∥
ℳ
f
→
∥
ℳ
p
q
0
(
W
,
v
p
)
≤
C
∥
f
→
∥
ℳ
P
→
p
0
(
W
,
w
→
P
→
)
,
where w
→
P
→
=
(
w
1
p
1
,
…
,
w
m
p
m
).


## 3. Some Lemmas

We use the following lemma (see [3, 5–9]).Lemma 14.
Let0
<
p
<
∞. Let v be a weight. For every cube Q
0
⊂
ℝ
n, one has
(32)
∥
M
(
f
χ
3
Q
0
)
v
∥
L
p
(
Q
0
)
≤
C
∥
M
~
p
Q
0
(
f
,
v
)
∥
L
p
(
Q
0
)
,
where
(33)
M
~
p
Q
0
(
f
,
v
)
(
x
)
:
=
sup
⁡
Q
0
⊃
Q
∋
x
(
⨏
3
Q
|
f
(
y
)
|
d
y
)
(
⨏
Q
v
(
y
)
p
d
y
)
1
/
p
.Moreover we can extend Lemma14 to the multilinear maximal function.Lemma 15.
Let0
<
p
<
∞. Let v be a weight. For every cube Q
0
⊂
ℝ
n, one has
(34)
∥
ℳ
(
f
1
χ
3
Q
0
,
…
,
f
m
χ
3
Q
0
)
v
∥
L
p
(
Q
0
)
≤
C
∥
M
~
p
Q
0
(
f
→
,
v
)
∥
L
p
(
Q
0
)
,
where
(35)
M
~
p
Q
0
(
f
→
,
v
)
(
x
)
:
=
sup
⁡
Q
0
⊃
Q
∋
x
∏
j
=
1
m
‍
(
⨏
3
Q
|
f
j
(
y
j
)
|
d
y
j
)
(
⨏
Q
v
(
y
)
p
d
y
)
1
/
p
.Remark 16.
In [3], there is not the restriction Q
⊂
Q
0 in the maximal function M
~
p
Q
0
(
f
→
,
v
). However, reexamining the argument of the proof in [3], we can remove the restriction. Hence the norm inequalities in Lemmas 14 and 15 are sharper than the norm inequalities in [3].


## 4. Proof of Theorems

### 4.1. Onelinear Version

We prove Theorem11. We omit the proof of Theorems 6, 8, and 10.Proof of Theorem11.
For every cubeQ
0
⊂
ℝ
n, by sublinearity, M
f
(
x
)
≤
M
(
f
χ
3
Q
0
)
(
x
)
+
M
(
f
χ
(
3
Q
0
)
c
)
(
x
). Let f
0
=
f
χ
3
Q
0 and f
∞
=
f
χ
(
3
Q
0
)
c. By Lemma 14,
(36)
∥
(
M
f
0
)
v
∥
L
p
(
Q
0
)
≤
C
∥
M
~
p
Q
0
(
f
,
v
)
∥
L
p
(
Q
0
)
.
By Hölder’s inequality, we have
(37)
M
~
p
Q
0
(
f
,
v
)
(
x
)
≤
sup
⁡
x
∈
Q
⊂
Q
0
(
⨏
3
Q
|
f
(
y
)
w
2
(
y
)
|
p
/
a
d
y
)
a
/
p
×
(
⨏
3
Q
w
2
(
y
)
-
(
p
/
a
)
′
d
y
)
1
/
(
p
/
a
)
′
(
⨏
Q
v
(
y
)
p
d
y
)
1
/
p
=
sup
⁡
x
∈
Q
⊂
Q
0
w
1
(
Q
)
1
/
q
0
w
1
(
3
Q
)
1
/
p
0
(
w
1
(
3
Q
)
w
1
(
Q
)
)
1
/
p
(
|
Q
|
|
3
Q
|
)
1
/
p
×
(
⨏
Q
v
(
y
)
p
d
y
)
1
/
p
(
⨏
3
Q
w
2
(
y
)
-
(
p
/
a
)
′
d
y
)
1
/
(
p
/
a
)
′
×
w
1
(
Q
)
-
1
/
q
0
w
1
(
3
Q
)
-
1
/
p
0
(
w
1
(
3
Q
)
w
1
(
Q
)
)
-
1
/
p
(
|
3
Q
|
|
Q
|
)
1
/
p
×
(
⨏
3
Q
|
f
(
y
)
w
2
(
y
)
|
p
/
a
d
y
)
a
/
p
.
By condition (26), we have
(38)
M
~
p
Q
0
(
f
,
v
)
(
x
)
≤
C
[
w
1
,
w
2
,
v
]
p
0
,
q
0
,
p
,
p
/
a
sup
⁡
x
∈
Q
⊂
Q
0
w
1
(
Q
)
-
1
/
q
0
w
1
(
3
Q
)
-
1
/
p
0
×
(
w
1
(
3
Q
)
w
1
(
Q
)
)
-
1
/
p
(
⨏
3
Q
|
f
(
y
)
w
2
(
y
)
|
p
/
a
d
y
)
a
/
p
.
Since w
1
(
Q
)
≤
w
1
(
3
Q
), we have
(39)
sup
⁡
x
∈
Q
⊂
Q
0
w
1
(
Q
)
-
1
/
q
0
w
1
(
3
Q
)
-
1
/
p
0
(
w
1
(
3
Q
)
w
1
(
Q
)
)
-
1
/
p
×
(
⨏
3
Q
|
f
(
y
)
w
2
(
y
)
|
p
/
a
d
y
)
a
/
p
=
sup
⁡
x
∈
Q
⊂
Q
0
w
1
(
3
Q
)
1
/
p
0
w
1
(
Q
)
1
/
q
0
(
w
1
(
Q
)
w
1
(
3
Q
)
)
1
/
p
×
(
⨏
3
Q
|
f
(
y
)
w
2
(
y
)
|
p
/
a
d
y
)
a
/
p
≤
sup
⁡
x
∈
Q
⊂
Q
0
w
1
(
3
Q
)
1
/
p
0
w
1
(
Q
)
1
/
q
0
(
⨏
3
Q
|
f
(
y
)
w
2
(
y
)
|
p
/
a
d
y
)
a
/
p
.
Since w
1 satisfies the doubling condition, we obtain the following inequality; for every cube Q
⊂
ℝ
n,
(40)
w
1
(
3
Q
)
1
/
p
0
w
1
(
Q
)
1
/
q
0
≤
C
1
/
q
0
w
1
(
3
Q
)
1
/
p
0
w
1
(
3
Q
)
1
/
q
0
=
C
1
/
q
0
w
1
(
3
Q
)
1
/
p
0
-
1
/
q
0
.
Since p
0
≤
q
0, for pair of cubes Q
⊂
Q
0, we have
(41)
w
1
(
3
Q
)
(
1
/
p
0
)
-
(
1
/
q
0
)
≤
w
1
(
3
Q
0
)
(
1
/
p
0
)
-
(
1
/
q
0
)
.
Hence we have
(42)
sup
⁡
x
∈
Q
⊂
Q
0
w
1
(
Q
)
-
1
/
q
0
w
1
(
3
Q
)
-
1
/
p
0
(
w
1
(
3
Q
)
w
1
(
Q
)
)
-
1
/
p
×
(
⨏
3
Q
|
f
(
y
)
w
2
(
y
)
|
p
/
a
d
y
)
a
/
p
≤
C
1
/
q
0
w
1
(
3
Q
0
)
1
/
p
0
-
1
/
q
0
M
[
|
f
w
2
|
p
/
a
χ
3
Q
0
]
(
x
)
a
/
p
.
This implies that
(43)
(
∫
Q
0
‍
M
f
0
(
x
)
p
v
(
x
)
p
d
x
)
1
/
p
≤
C
[
w
1
,
w
2
,
v
]
p
0
,
q
0
,
p
,
p
/
a
·
w
1
(
3
Q
0
)
1
/
p
0
-
1
/
q
0
×
(
∫
Q
0
‍
M
[
|
f
w
2
|
p
/
a
·
χ
3
Q
0
]
(
x
)
(
a
/
p
)
·
p
d
x
)
1
/
p
=
C
[
w
1
,
w
2
,
v
]
p
0
,
q
0
,
p
,
p
/
a
·
w
1
(
3
Q
0
)
1
/
p
0
-
1
/
q
0
×
(
∫
Q
0
‍
M
[
|
f
w
2
|
p
/
a
χ
3
Q
0
]
(
x
)
a
d
x
)
1
/
p
.
By the boundedness of M
:
L
a
(
ℝ
n
)
→
L
a
(
ℝ
n
), we obtain
(44)
(
∫
Q
0
‍
M
[
|
f
w
2
|
p
/
a
χ
3
Q
0
]
(
x
)
a
d
x
)
1
/
p
≤
C
(
∫
3
Q
0
‍
|
f
(
x
)
w
2
(
x
)
|
p
d
x
)
1
/
p
.
Since p
≤
q
0, by the doubling condition of w
1, we have
(45)
w
1
(
Q
0
)
1
/
q
0
-
1
/
p
(
∫
Q
0
‍
M
f
0
(
x
)
p
v
(
x
)
p
d
x
)
1
/
p
≤
C
[
w
1
,
w
2
,
v
]
p
0
,
q
0
,
p
,
p
/
a
·
w
1
(
Q
0
)
1
/
q
0
-
1
/
p
w
1
(
3
Q
0
)
1
/
p
0
-
1
/
q
0
×
(
∫
3
Q
0
‍
|
f
(
x
)
w
2
(
x
)
|
p
d
x
)
1
/
p
≤
C
[
w
1
,
w
2
,
v
]
p
0
,
q
0
,
p
,
p
/
a
·
w
1
(
3
Q
0
)
1
/
p
0
-
1
/
p
(
∫
3
Q
0
‍
|
f
(
x
)
w
2
(
x
)
|
p
d
x
)
1
/
p
≤
C
[
w
1
,
w
2
,
v
]
p
0
,
q
0
,
p
,
p
/
a
·
w
1
(
3
Q
0
)
1
/
p
0
-
1
/
p
(
∫
3
Q
0
‍
|
f
(
x
)
w
2
(
x
)
|
p
d
x
)
1
/
p
.
Taking the weighted Morrey norm, we obtain the following inequality:
(46)
∥
M
f
0
∥
ℳ
p
q
0
(
w
1
,
v
p
)
≤
C
∥
f
∥
ℳ
p
p
0
(
w
1
,
w
2
p
)
.
Next, we estimateM
f
∞
(
x
). By routine geometric observation, for x
∈
Q
0, we have
(47)
M
f
∞
(
x
)
≤
C
sup
⁡
Q
⊃
Q
0
⨏
Q
|
f
(
y
)
|
d
y
.
For pair of cubes Q
⊃
Q
0, by Hölder’s inequality, we obtain
(48)
M
f
∞
(
x
)
≤
C
sup
⁡
Q
⊃
Q
0
(
⨏
Q
|
f
(
y
)
|
p
/
a
w
2
(
y
)
p
/
a
d
y
)
a
/
p
×
(
⨏
Q
w
2
(
y
)
-
(
p
/
a
)
′
d
y
)
1
/
(
p
/
a
)
′
≤
C
sup
⁡
Q
⊃
Q
0
w
1
(
Q
)
1
/
p
0
-
1
/
p
×
(
∫
Q
‍
|
f
(
y
)
w
2
(
y
)
|
p
d
y
)
1
/
p
1
|
Q
|
1
/
p
×
(
⨏
Q
w
2
(
y
)
-
(
p
/
a
)
′
d
y
)
1
/
(
p
/
a
)
′
w
1
(
Q
)
1
/
p
-
1
/
p
0
≤
C
∥
f
∥
ℳ
p
p
0
(
w
1
,
w
2
p
)
·
sup
⁡
Q
⊃
Q
0
1
|
Q
|
1
/
p
×
(
⨏
Q
w
2
(
y
)
-
(
p
/
a
)
′
d
y
)
1
/
(
p
/
a
)
′
w
1
(
Q
)
1
/
p
-
1
/
p
0
.
Therefore we have
(49)
w
1
(
Q
0
)
1
/
q
0
-
1
/
p
(
∫
Q
0
‍
M
f
∞
(
x
)
p
v
(
x
)
p
d
x
)
1
/
p
≤
C
∥
f
∥
ℳ
p
p
0
(
w
1
,
w
2
p
)
sup
⁡
Q
⊃
Q
0
w
1
(
Q
0
)
1
/
q
0
w
1
(
Q
)
1
/
p
0
(
w
1
(
Q
)
w
1
(
Q
0
)
)
1
/
p
×
(
|
Q
0
|
|
Q
|
)
1
/
p
(
⨏
Q
0
v
(
x
)
p
d
x
)
1
/
p
×
(
⨏
Q
w
2
(
y
)
-
(
p
/
a
)
′
d
y
)
1
/
(
p
/
a
)
′
.
By condition (26), we have
(50)
sup
⁡
Q
⊃
Q
0
Q
,
Q
0
:
cubes
w
1
(
Q
0
)
1
/
q
0
w
1
(
Q
)
1
/
p
0
(
w
1
(
Q
)
w
1
(
Q
0
)
)
1
/
p
(
|
Q
0
|
|
Q
|
)
1
/
p
×
(
⨏
Q
0
v
(
x
)
p
d
x
)
1
/
p
(
⨏
Q
w
2
(
y
)
-
(
p
/
a
)
′
d
y
)
1
/
(
p
/
a
)
′
≤
[
v
,
w
1
,
w
2
]
p
0
,
q
0
,
p
,
p
/
a
.
This implies that
(51)
∥
M
f
∞
∥
ℳ
p
q
0
(
w
1
,
v
p
)
≤
C
[
v
,
w
1
,
w
2
]
p
0
,
q
0
,
p
,
p
/
a
∥
f
∥
ℳ
p
p
0
(
w
1
,
w
2
p
)
.
Therefore we obtain the desired result.


### 4.2. Multilinear Version

We prove Theorem13.Proof of Theorem13.
For every cubeQ
⊂
ℝ
n, we have f
j
(
y
j
)
=
f
j
(
y
j
)
χ
3
Q
0
(
y
j
)
+
f
j
(
y
j
)
χ
(
3
Q
0
)
c
(
y
j
)
=
f
j
0
(
y
j
)
+
f
j
∞
(
y
j
). Let f
→
0
=
(
f
1
0
,
…
,
f
m
0
), f
→
l
→
=
(
f
1
l
1
,
…
,
f
m
l
m
), and l
→
=
(
l
1
,
…
,
l
m
):
(52)
ℳ
(
f
→
)
(
x
)
≤
ℳ
(
f
→
0
)
(
x
)
+
∑
l
→
≠
0
→
‍
ℳ
(
f
→
l
→
)
(
x
)
.
Firstly, we estimate ℳ
(
f
→
0
)
(
x
). By Lemma 15, we have
(53)
∥
ℳ
(
f
→
0
)
v
∥
L
p
(
Q
0
)
≤
C
∥
M
~
p
Q
0
(
f
→
,
v
)
∥
L
p
(
Q
0
)
,
where
(54)
M
~
p
Q
0
(
f
→
,
v
)
(
x
)
:
=
sup
⁡
Q
0
⊃
Q
∋
x
∏
j
=
1
m
‍
(
⨏
3
Q
|
f
j
(
y
j
)
|
d
y
j
)
(
⨏
Q
v
(
y
)
p
d
y
)
1
/
p
.
By Hölder’s inequality, we obtain
(55)
M
~
p
Q
0
(
f
→
,
v
)
(
x
)
≤
sup
⁡
Q
0
⊃
Q
∋
x
∏
j
=
1
m
‍
(
⨏
3
Q
|
f
j
(
y
j
)
|
p
j
/
a
w
j
(
y
j
)
p
j
/
a
d
y
j
)
a
/
p
j
×
(
⨏
3
Q
w
j
(
y
j
)
-
(
p
j
/
a
)
′
d
y
j
)
1
/
(
p
j
/
a
)
′
(
⨏
Q
v
(
y
)
p
d
y
)
1
/
p
.
By condition (30), we obtain
(56)
M
~
p
Q
0
(
f
→
,
v
)
(
x
)
≤
C
[
v
,
W
,
w
→
]
p
0
,
q
0
,
p
,
P
→
/
a
×
sup
⁡
x
∈
Q
⊂
Q
0
∏
j
=
1
m
‍
(
⨏
3
Q
|
f
j
(
y
j
)
|
p
j
/
a
w
j
(
y
j
)
p
j
/
a
d
y
j
)
a
/
p
j
×
W
(
3
Q
)
1
/
p
0
W
(
Q
)
1
/
q
0
(
W
(
Q
)
W
(
3
Q
)
)
1
/
p
.
Since W
(
Q
)
≤
W
(
3
Q
), we have
(57)
(
W
(
Q
)
W
(
3
Q
)
)
1
/
p
≤
1
.
Since W satisfies the doubling condition, we have
(58)
W
(
3
Q
)
1
/
p
0
W
(
Q
)
1
/
q
0
≤
C
W
(
3
Q
)
1
/
p
0
-
1
/
q
0
.
Since p
0
≤
q
0, for pair of cubes Q
⊂
Q
0, we have
(59)
W
(
3
Q
)
1
/
p
0
-
1
/
q
0
≤
W
(
3
Q
0
)
1
/
p
0
-
1
/
q
0
.
Therefore we obtain
(60)
M
~
p
Q
0
(
f
→
,
v
)
(
x
)
≤
C
[
v
,
W
,
w
→
]
p
0
,
q
0
p
,
P
→
/
a
·
W
(
3
Q
0
)
1
/
p
0
-
1
/
q
0
·
sup
⁡
x
∈
Q
⊂
Q
0
∏
j
=
1
m
‍
(
⨏
3
Q
|
f
j
(
y
j
)
|
p
j
/
a
w
j
(
y
j
)
p
j
/
a
d
y
j
)
a
/
p
j
≤
C
[
v
,
W
,
w
→
]
p
0
,
q
0
,
p
,
P
→
/
a
·
W
(
3
Q
0
)
1
/
p
0
-
1
/
q
0
·
∏
j
=
1
m
‍
M
[
|
f
j
w
j
|
p
j
/
a
χ
3
Q
0
]
(
x
)
a
/
p
j
.
This implies that
(61)
(
∫
Q
0
‍
M
~
p
Q
0
(
f
→
,
v
)
(
x
)
p
d
x
)
1
/
p
≤
C
[
v
,
W
,
w
→
]
p
0
,
q
0
,
p
,
P
→
/
a
·
W
(
3
Q
0
)
1
/
p
0
-
1
/
q
0
×
(
∫
Q
0
‍
∏
j
=
1
m
‍
M
[
|
f
j
w
j
|
p
j
/
a
χ
3
Q
0
]
(
x
)
(
a
/
p
j
)
p
d
x
)
1
/
p
.
Since 1
/
p
=
1
/
p
1
+
⋯
+
1
/
p
m, by Hölder’s inequality, we have
(62)
(
∫
Q
0
‍
M
~
p
Q
0
(
f
→
,
v
)
(
x
)
p
d
x
)
1
/
p
≤
C
[
v
,
W
,
w
→
]
p
0
,
q
0
,
p
,
P
→
/
a
·
W
(
3
Q
0
)
1
/
p
0
-
1
/
q
0
×
∏
j
=
1
m
‍
(
∫
Q
0
‍
M
[
|
f
j
w
j
|
p
j
/
a
χ
3
Q
0
]
(
x
)
a
d
x
)
1
/
p
j
.
By the boundedness of M on L
a
(
ℝ
n
)
→
L
a
(
ℝ
n
), we have
(63)
(
∫
Q
0
‍
M
~
p
Q
0
(
f
→
,
v
)
(
x
)
p
d
x
)
1
/
p
≤
C
[
v
,
W
,
w
→
]
p
0
,
q
0
,
p
,
P
→
/
a
·
W
(
3
Q
0
)
1
/
p
0
-
1
/
q
0
×
∏
j
=
1
m
‍
(
∫
3
Q
0
‍
|
f
j
(
x
)
|
p
j
w
j
(
x
)
p
j
d
x
)
1
/
p
j
.
Since p
≤
q
0, by the doubling condition of W, we have
(64)
W
(
Q
0
)
1
/
q
0
(
1
W
(
Q
0
)
∫
Q
0
‍
ℳ
(
f
→
0
)
(
x
)
p
v
(
x
)
p
d
x
)
1
/
p
≤
C
[
v
,
W
,
w
→
]
p
0
,
q
0
,
p
,
P
→
/
a
·
W
(
3
Q
0
)
1
/
p
0
-
1
/
p
×
∏
j
=
1
m
‍
(
∫
3
Q
0
‍
|
f
j
(
x
)
|
p
j
w
j
(
x
)
p
j
d
x
)
1
/
p
j
.
Taking the multiple weighted Morrey quantity, we have
(65)
W
(
Q
0
)
1
/
p
0
(
1
W
(
Q
0
)
∫
Q
0
‍
ℳ
(
f
→
0
)
(
x
)
p
v
(
x
)
p
d
x
)
1
/
p
≤
C
[
v
,
W
,
w
→
]
p
0
,
q
0
,
p
,
P
→
/
a
·
∥
f
→
∥
ℳ
P
→
p
0
(
W
,
w
→
P
→
)
.
Next, we estimateℳ
(
f
→
l
→
)
(
x
). By routine geometric observation, for x
∈
Q
0, we have
(66)
ℳ
(
f
→
l
→
)
(
x
)
≤
C
sup
⁡
Q
⊃
Q
0
∏
j
=
1
m
‍
⨏
Q
|
f
j
(
y
j
)
|
d
y
j
.
By Hölder’s inequality, we have
(67)
sup
⁡
Q
⊃
Q
0
∏
j
=
1
m
‍
⨏
Q
|
f
j
(
y
j
)
|
d
y
j
≤
sup
⁡
Q
⊃
Q
0
∏
j
=
1
m
‍
(
⨏
Q
|
f
j
(
y
j
)
|
p
j
/
a
w
j
(
y
j
)
p
j
/
a
d
y
j
)
a
/
p
j
·
(
⨏
Q
w
j
(
y
j
)
-
(
p
j
/
a
)
′
d
y
j
)
1
/
(
p
j
/
a
)
′
≤
sup
⁡
Q
⊃
Q
0
∏
j
=
1
m
‍
(
⨏
Q
|
f
j
(
y
j
)
|
p
j
w
j
(
y
j
)
p
j
d
y
j
)
1
/
p
j
·
(
⨏
Q
w
j
(
y
j
)
-
(
p
j
/
a
)
′
d
y
j
)
1
/
(
p
j
/
a
)
′
.
Taking the multiple weighted Morrey quantity, we obtain
(68)
sup
⁡
Q
⊃
Q
0
∏
j
=
1
m
‍
(
⨏
Q
|
f
j
(
y
j
)
|
p
j
w
j
(
y
j
)
p
j
d
y
j
)
1
/
p
j
·
(
⨏
Q
w
j
(
y
j
)
-
(
p
j
/
a
)
′
d
y
j
)
1
/
(
p
j
/
a
)
′
≤
C
∥
f
→
∥
ℳ
P
→
p
0
(
W
,
w
→
P
→
)
·
sup
⁡
Q
⊃
Q
0
1
|
Q
|
1
/
p
W
(
Q
)
1
/
p
-
1
/
p
0
×
∏
j
=
1
m
‍
(
⨏
Q
w
j
(
y
j
)
-
(
p
j
/
a
)
′
d
y
j
)
1
/
(
p
j
/
a
)
′
.
This implies that
(69)
W
(
Q
0
)
1
/
q
0
(
1
W
(
Q
0
)
∫
Q
0
‍
ℳ
(
f
→
l
→
)
(
x
)
p
v
(
x
)
p
d
x
)
1
/
p
≤
C
∥
f
→
∥
ℳ
P
→
p
0
(
W
,
w
→
P
→
)
sup
⁡
Q
⊃
Q
0
W
(
Q
0
)
1
/
q
0
W
(
Q
)
1
/
p
0
(
W
(
Q
)
W
(
Q
0
)
)
1
/
p
×
(
|
Q
0
|
|
Q
|
)
1
/
p
(
⨏
Q
0
v
(
x
)
p
d
x
)
1
/
p
×
∏
j
=
1
m
‍
(
⨏
Q
w
j
(
y
j
)
-
(
p
j
/
a
)
′
d
y
j
)
1
/
(
p
j
/
a
)
′
.
By condition (28), we obtain
(70)
W
(
Q
0
)
1
/
p
0
(
1
W
(
Q
0
)
∫
Q
0
‍
ℳ
(
f
→
l
→
)
(
x
)
p
v
(
x
)
p
d
x
)
1
/
p
≤
C
[
v
,
W
,
w
→
]
p
0
,
p
,
P
→
/
a
∥
f
→
∥
ℳ
P
→
p
0
(
W
,
w
→
P
→
)
.
Therefore we obtain the desired result.


## 4.1. Onelinear Version

We prove Theorem11. We omit the proof of Theorems 6, 8, and 10.Proof of Theorem11.
For every cubeQ
0
⊂
ℝ
n, by sublinearity, M
f
(
x
)
≤
M
(
f
χ
3
Q
0
)
(
x
)
+
M
(
f
χ
(
3
Q
0
)
c
)
(
x
). Let f
0
=
f
χ
3
Q
0 and f
∞
=
f
χ
(
3
Q
0
)
c. By Lemma 14,
(36)
∥
(
M
f
0
)
v
∥
L
p
(
Q
0
)
≤
C
∥
M
~
p
Q
0
(
f
,
v
)
∥
L
p
(
Q
0
)
.
By Hölder’s inequality, we have
(37)
M
~
p
Q
0
(
f
,
v
)
(
x
)
≤
sup
⁡
x
∈
Q
⊂
Q
0
(
⨏
3
Q
|
f
(
y
)
w
2
(
y
)
|
p
/
a
d
y
)
a
/
p
×
(
⨏
3
Q
w
2
(
y
)
-
(
p
/
a
)
′
d
y
)
1
/
(
p
/
a
)
′
(
⨏
Q
v
(
y
)
p
d
y
)
1
/
p
=
sup
⁡
x
∈
Q
⊂
Q
0
w
1
(
Q
)
1
/
q
0
w
1
(
3
Q
)
1
/
p
0
(
w
1
(
3
Q
)
w
1
(
Q
)
)
1
/
p
(
|
Q
|
|
3
Q
|
)
1
/
p
×
(
⨏
Q
v
(
y
)
p
d
y
)
1
/
p
(
⨏
3
Q
w
2
(
y
)
-
(
p
/
a
)
′
d
y
)
1
/
(
p
/
a
)
′
×
w
1
(
Q
)
-
1
/
q
0
w
1
(
3
Q
)
-
1
/
p
0
(
w
1
(
3
Q
)
w
1
(
Q
)
)
-
1
/
p
(
|
3
Q
|
|
Q
|
)
1
/
p
×
(
⨏
3
Q
|
f
(
y
)
w
2
(
y
)
|
p
/
a
d
y
)
a
/
p
.
By condition (26), we have
(38)
M
~
p
Q
0
(
f
,
v
)
(
x
)
≤
C
[
w
1
,
w
2
,
v
]
p
0
,
q
0
,
p
,
p
/
a
sup
⁡
x
∈
Q
⊂
Q
0
w
1
(
Q
)
-
1
/
q
0
w
1
(
3
Q
)
-
1
/
p
0
×
(
w
1
(
3
Q
)
w
1
(
Q
)
)
-
1
/
p
(
⨏
3
Q
|
f
(
y
)
w
2
(
y
)
|
p
/
a
d
y
)
a
/
p
.
Since w
1
(
Q
)
≤
w
1
(
3
Q
), we have
(39)
sup
⁡
x
∈
Q
⊂
Q
0
w
1
(
Q
)
-
1
/
q
0
w
1
(
3
Q
)
-
1
/
p
0
(
w
1
(
3
Q
)
w
1
(
Q
)
)
-
1
/
p
×
(
⨏
3
Q
|
f
(
y
)
w
2
(
y
)
|
p
/
a
d
y
)
a
/
p
=
sup
⁡
x
∈
Q
⊂
Q
0
w
1
(
3
Q
)
1
/
p
0
w
1
(
Q
)
1
/
q
0
(
w
1
(
Q
)
w
1
(
3
Q
)
)
1
/
p
×
(
⨏
3
Q
|
f
(
y
)
w
2
(
y
)
|
p
/
a
d
y
)
a
/
p
≤
sup
⁡
x
∈
Q
⊂
Q
0
w
1
(
3
Q
)
1
/
p
0
w
1
(
Q
)
1
/
q
0
(
⨏
3
Q
|
f
(
y
)
w
2
(
y
)
|
p
/
a
d
y
)
a
/
p
.
Since w
1 satisfies the doubling condition, we obtain the following inequality; for every cube Q
⊂
ℝ
n,
(40)
w
1
(
3
Q
)
1
/
p
0
w
1
(
Q
)
1
/
q
0
≤
C
1
/
q
0
w
1
(
3
Q
)
1
/
p
0
w
1
(
3
Q
)
1
/
q
0
=
C
1
/
q
0
w
1
(
3
Q
)
1
/
p
0
-
1
/
q
0
.
Since p
0
≤
q
0, for pair of cubes Q
⊂
Q
0, we have
(41)
w
1
(
3
Q
)
(
1
/
p
0
)
-
(
1
/
q
0
)
≤
w
1
(
3
Q
0
)
(
1
/
p
0
)
-
(
1
/
q
0
)
.
Hence we have
(42)
sup
⁡
x
∈
Q
⊂
Q
0
w
1
(
Q
)
-
1
/
q
0
w
1
(
3
Q
)
-
1
/
p
0
(
w
1
(
3
Q
)
w
1
(
Q
)
)
-
1
/
p
×
(
⨏
3
Q
|
f
(
y
)
w
2
(
y
)
|
p
/
a
d
y
)
a
/
p
≤
C
1
/
q
0
w
1
(
3
Q
0
)
1
/
p
0
-
1
/
q
0
M
[
|
f
w
2
|
p
/
a
χ
3
Q
0
]
(
x
)
a
/
p
.
This implies that
(43)
(
∫
Q
0
‍
M
f
0
(
x
)
p
v
(
x
)
p
d
x
)
1
/
p
≤
C
[
w
1
,
w
2
,
v
]
p
0
,
q
0
,
p
,
p
/
a
·
w
1
(
3
Q
0
)
1
/
p
0
-
1
/
q
0
×
(
∫
Q
0
‍
M
[
|
f
w
2
|
p
/
a
·
χ
3
Q
0
]
(
x
)
(
a
/
p
)
·
p
d
x
)
1
/
p
=
C
[
w
1
,
w
2
,
v
]
p
0
,
q
0
,
p
,
p
/
a
·
w
1
(
3
Q
0
)
1
/
p
0
-
1
/
q
0
×
(
∫
Q
0
‍
M
[
|
f
w
2
|
p
/
a
χ
3
Q
0
]
(
x
)
a
d
x
)
1
/
p
.
By the boundedness of M
:
L
a
(
ℝ
n
)
→
L
a
(
ℝ
n
), we obtain
(44)
(
∫
Q
0
‍
M
[
|
f
w
2
|
p
/
a
χ
3
Q
0
]
(
x
)
a
d
x
)
1
/
p
≤
C
(
∫
3
Q
0
‍
|
f
(
x
)
w
2
(
x
)
|
p
d
x
)
1
/
p
.
Since p
≤
q
0, by the doubling condition of w
1, we have
(45)
w
1
(
Q
0
)
1
/
q
0
-
1
/
p
(
∫
Q
0
‍
M
f
0
(
x
)
p
v
(
x
)
p
d
x
)
1
/
p
≤
C
[
w
1
,
w
2
,
v
]
p
0
,
q
0
,
p
,
p
/
a
·
w
1
(
Q
0
)
1
/
q
0
-
1
/
p
w
1
(
3
Q
0
)
1
/
p
0
-
1
/
q
0
×
(
∫
3
Q
0
‍
|
f
(
x
)
w
2
(
x
)
|
p
d
x
)
1
/
p
≤
C
[
w
1
,
w
2
,
v
]
p
0
,
q
0
,
p
,
p
/
a
·
w
1
(
3
Q
0
)
1
/
p
0
-
1
/
p
(
∫
3
Q
0
‍
|
f
(
x
)
w
2
(
x
)
|
p
d
x
)
1
/
p
≤
C
[
w
1
,
w
2
,
v
]
p
0
,
q
0
,
p
,
p
/
a
·
w
1
(
3
Q
0
)
1
/
p
0
-
1
/
p
(
∫
3
Q
0
‍
|
f
(
x
)
w
2
(
x
)
|
p
d
x
)
1
/
p
.
Taking the weighted Morrey norm, we obtain the following inequality:
(46)
∥
M
f
0
∥
ℳ
p
q
0
(
w
1
,
v
p
)
≤
C
∥
f
∥
ℳ
p
p
0
(
w
1
,
w
2
p
)
.
Next, we estimateM
f
∞
(
x
). By routine geometric observation, for x
∈
Q
0, we have
(47)
M
f
∞
(
x
)
≤
C
sup
⁡
Q
⊃
Q
0
⨏
Q
|
f
(
y
)
|
d
y
.
For pair of cubes Q
⊃
Q
0, by Hölder’s inequality, we obtain
(48)
M
f
∞
(
x
)
≤
C
sup
⁡
Q
⊃
Q
0
(
⨏
Q
|
f
(
y
)
|
p
/
a
w
2
(
y
)
p
/
a
d
y
)
a
/
p
×
(
⨏
Q
w
2
(
y
)
-
(
p
/
a
)
′
d
y
)
1
/
(
p
/
a
)
′
≤
C
sup
⁡
Q
⊃
Q
0
w
1
(
Q
)
1
/
p
0
-
1
/
p
×
(
∫
Q
‍
|
f
(
y
)
w
2
(
y
)
|
p
d
y
)
1
/
p
1
|
Q
|
1
/
p
×
(
⨏
Q
w
2
(
y
)
-
(
p
/
a
)
′
d
y
)
1
/
(
p
/
a
)
′
w
1
(
Q
)
1
/
p
-
1
/
p
0
≤
C
∥
f
∥
ℳ
p
p
0
(
w
1
,
w
2
p
)
·
sup
⁡
Q
⊃
Q
0
1
|
Q
|
1
/
p
×
(
⨏
Q
w
2
(
y
)
-
(
p
/
a
)
′
d
y
)
1
/
(
p
/
a
)
′
w
1
(
Q
)
1
/
p
-
1
/
p
0
.
Therefore we have
(49)
w
1
(
Q
0
)
1
/
q
0
-
1
/
p
(
∫
Q
0
‍
M
f
∞
(
x
)
p
v
(
x
)
p
d
x
)
1
/
p
≤
C
∥
f
∥
ℳ
p
p
0
(
w
1
,
w
2
p
)
sup
⁡
Q
⊃
Q
0
w
1
(
Q
0
)
1
/
q
0
w
1
(
Q
)
1
/
p
0
(
w
1
(
Q
)
w
1
(
Q
0
)
)
1
/
p
×
(
|
Q
0
|
|
Q
|
)
1
/
p
(
⨏
Q
0
v
(
x
)
p
d
x
)
1
/
p
×
(
⨏
Q
w
2
(
y
)
-
(
p
/
a
)
′
d
y
)
1
/
(
p
/
a
)
′
.
By condition (26), we have
(50)
sup
⁡
Q
⊃
Q
0
Q
,
Q
0
:
cubes
w
1
(
Q
0
)
1
/
q
0
w
1
(
Q
)
1
/
p
0
(
w
1
(
Q
)
w
1
(
Q
0
)
)
1
/
p
(
|
Q
0
|
|
Q
|
)
1
/
p
×
(
⨏
Q
0
v
(
x
)
p
d
x
)
1
/
p
(
⨏
Q
w
2
(
y
)
-
(
p
/
a
)
′
d
y
)
1
/
(
p
/
a
)
′
≤
[
v
,
w
1
,
w
2
]
p
0
,
q
0
,
p
,
p
/
a
.
This implies that
(51)
∥
M
f
∞
∥
ℳ
p
q
0
(
w
1
,
v
p
)
≤
C
[
v
,
w
1
,
w
2
]
p
0
,
q
0
,
p
,
p
/
a
∥
f
∥
ℳ
p
p
0
(
w
1
,
w
2
p
)
.
Therefore we obtain the desired result.


## 4.2. Multilinear Version

We prove Theorem13.Proof of Theorem13.
For every cubeQ
⊂
ℝ
n, we have f
j
(
y
j
)
=
f
j
(
y
j
)
χ
3
Q
0
(
y
j
)
+
f
j
(
y
j
)
χ
(
3
Q
0
)
c
(
y
j
)
=
f
j
0
(
y
j
)
+
f
j
∞
(
y
j
). Let f
→
0
=
(
f
1
0
,
…
,
f
m
0
), f
→
l
→
=
(
f
1
l
1
,
…
,
f
m
l
m
), and l
→
=
(
l
1
,
…
,
l
m
):
(52)
ℳ
(
f
→
)
(
x
)
≤
ℳ
(
f
→
0
)
(
x
)
+
∑
l
→
≠
0
→
‍
ℳ
(
f
→
l
→
)
(
x
)
.
Firstly, we estimate ℳ
(
f
→
0
)
(
x
). By Lemma 15, we have
(53)
∥
ℳ
(
f
→
0
)
v
∥
L
p
(
Q
0
)
≤
C
∥
M
~
p
Q
0
(
f
→
,
v
)
∥
L
p
(
Q
0
)
,
where
(54)
M
~
p
Q
0
(
f
→
,
v
)
(
x
)
:
=
sup
⁡
Q
0
⊃
Q
∋
x
∏
j
=
1
m
‍
(
⨏
3
Q
|
f
j
(
y
j
)
|
d
y
j
)
(
⨏
Q
v
(
y
)
p
d
y
)
1
/
p
.
By Hölder’s inequality, we obtain
(55)
M
~
p
Q
0
(
f
→
,
v
)
(
x
)
≤
sup
⁡
Q
0
⊃
Q
∋
x
∏
j
=
1
m
‍
(
⨏
3
Q
|
f
j
(
y
j
)
|
p
j
/
a
w
j
(
y
j
)
p
j
/
a
d
y
j
)
a
/
p
j
×
(
⨏
3
Q
w
j
(
y
j
)
-
(
p
j
/
a
)
′
d
y
j
)
1
/
(
p
j
/
a
)
′
(
⨏
Q
v
(
y
)
p
d
y
)
1
/
p
.
By condition (30), we obtain
(56)
M
~
p
Q
0
(
f
→
,
v
)
(
x
)
≤
C
[
v
,
W
,
w
→
]
p
0
,
q
0
,
p
,
P
→
/
a
×
sup
⁡
x
∈
Q
⊂
Q
0
∏
j
=
1
m
‍
(
⨏
3
Q
|
f
j
(
y
j
)
|
p
j
/
a
w
j
(
y
j
)
p
j
/
a
d
y
j
)
a
/
p
j
×
W
(
3
Q
)
1
/
p
0
W
(
Q
)
1
/
q
0
(
W
(
Q
)
W
(
3
Q
)
)
1
/
p
.
Since W
(
Q
)
≤
W
(
3
Q
), we have
(57)
(
W
(
Q
)
W
(
3
Q
)
)
1
/
p
≤
1
.
Since W satisfies the doubling condition, we have
(58)
W
(
3
Q
)
1
/
p
0
W
(
Q
)
1
/
q
0
≤
C
W
(
3
Q
)
1
/
p
0
-
1
/
q
0
.
Since p
0
≤
q
0, for pair of cubes Q
⊂
Q
0, we have
(59)
W
(
3
Q
)
1
/
p
0
-
1
/
q
0
≤
W
(
3
Q
0
)
1
/
p
0
-
1
/
q
0
.
Therefore we obtain
(60)
M
~
p
Q
0
(
f
→
,
v
)
(
x
)
≤
C
[
v
,
W
,
w
→
]
p
0
,
q
0
p
,
P
→
/
a
·
W
(
3
Q
0
)
1
/
p
0
-
1
/
q
0
·
sup
⁡
x
∈
Q
⊂
Q
0
∏
j
=
1
m
‍
(
⨏
3
Q
|
f
j
(
y
j
)
|
p
j
/
a
w
j
(
y
j
)
p
j
/
a
d
y
j
)
a
/
p
j
≤
C
[
v
,
W
,
w
→
]
p
0
,
q
0
,
p
,
P
→
/
a
·
W
(
3
Q
0
)
1
/
p
0
-
1
/
q
0
·
∏
j
=
1
m
‍
M
[
|
f
j
w
j
|
p
j
/
a
χ
3
Q
0
]
(
x
)
a
/
p
j
.
This implies that
(61)
(
∫
Q
0
‍
M
~
p
Q
0
(
f
→
,
v
)
(
x
)
p
d
x
)
1
/
p
≤
C
[
v
,
W
,
w
→
]
p
0
,
q
0
,
p
,
P
→
/
a
·
W
(
3
Q
0
)
1
/
p
0
-
1
/
q
0
×
(
∫
Q
0
‍
∏
j
=
1
m
‍
M
[
|
f
j
w
j
|
p
j
/
a
χ
3
Q
0
]
(
x
)
(
a
/
p
j
)
p
d
x
)
1
/
p
.
Since 1
/
p
=
1
/
p
1
+
⋯
+
1
/
p
m, by Hölder’s inequality, we have
(62)
(
∫
Q
0
‍
M
~
p
Q
0
(
f
→
,
v
)
(
x
)
p
d
x
)
1
/
p
≤
C
[
v
,
W
,
w
→
]
p
0
,
q
0
,
p
,
P
→
/
a
·
W
(
3
Q
0
)
1
/
p
0
-
1
/
q
0
×
∏
j
=
1
m
‍
(
∫
Q
0
‍
M
[
|
f
j
w
j
|
p
j
/
a
χ
3
Q
0
]
(
x
)
a
d
x
)
1
/
p
j
.
By the boundedness of M on L
a
(
ℝ
n
)
→
L
a
(
ℝ
n
), we have
(63)
(
∫
Q
0
‍
M
~
p
Q
0
(
f
→
,
v
)
(
x
)
p
d
x
)
1
/
p
≤
C
[
v
,
W
,
w
→
]
p
0
,
q
0
,
p
,
P
→
/
a
·
W
(
3
Q
0
)
1
/
p
0
-
1
/
q
0
×
∏
j
=
1
m
‍
(
∫
3
Q
0
‍
|
f
j
(
x
)
|
p
j
w
j
(
x
)
p
j
d
x
)
1
/
p
j
.
Since p
≤
q
0, by the doubling condition of W, we have
(64)
W
(
Q
0
)
1
/
q
0
(
1
W
(
Q
0
)
∫
Q
0
‍
ℳ
(
f
→
0
)
(
x
)
p
v
(
x
)
p
d
x
)
1
/
p
≤
C
[
v
,
W
,
w
→
]
p
0
,
q
0
,
p
,
P
→
/
a
·
W
(
3
Q
0
)
1
/
p
0
-
1
/
p
×
∏
j
=
1
m
‍
(
∫
3
Q
0
‍
|
f
j
(
x
)
|
p
j
w
j
(
x
)
p
j
d
x
)
1
/
p
j
.
Taking the multiple weighted Morrey quantity, we have
(65)
W
(
Q
0
)
1
/
p
0
(
1
W
(
Q
0
)
∫
Q
0
‍
ℳ
(
f
→
0
)
(
x
)
p
v
(
x
)
p
d
x
)
1
/
p
≤
C
[
v
,
W
,
w
→
]
p
0
,
q
0
,
p
,
P
→
/
a
·
∥
f
→
∥
ℳ
P
→
p
0
(
W
,
w
→
P
→
)
.
Next, we estimateℳ
(
f
→
l
→
)
(
x
). By routine geometric observation, for x
∈
Q
0, we have
(66)
ℳ
(
f
→
l
→
)
(
x
)
≤
C
sup
⁡
Q
⊃
Q
0
∏
j
=
1
m
‍
⨏
Q
|
f
j
(
y
j
)
|
d
y
j
.
By Hölder’s inequality, we have
(67)
sup
⁡
Q
⊃
Q
0
∏
j
=
1
m
‍
⨏
Q
|
f
j
(
y
j
)
|
d
y
j
≤
sup
⁡
Q
⊃
Q
0
∏
j
=
1
m
‍
(
⨏
Q
|
f
j
(
y
j
)
|
p
j
/
a
w
j
(
y
j
)
p
j
/
a
d
y
j
)
a
/
p
j
·
(
⨏
Q
w
j
(
y
j
)
-
(
p
j
/
a
)
′
d
y
j
)
1
/
(
p
j
/
a
)
′
≤
sup
⁡
Q
⊃
Q
0
∏
j
=
1
m
‍
(
⨏
Q
|
f
j
(
y
j
)
|
p
j
w
j
(
y
j
)
p
j
d
y
j
)
1
/
p
j
·
(
⨏
Q
w
j
(
y
j
)
-
(
p
j
/
a
)
′
d
y
j
)
1
/
(
p
j
/
a
)
′
.
Taking the multiple weighted Morrey quantity, we obtain
(68)
sup
⁡
Q
⊃
Q
0
∏
j
=
1
m
‍
(
⨏
Q
|
f
j
(
y
j
)
|
p
j
w
j
(
y
j
)
p
j
d
y
j
)
1
/
p
j
·
(
⨏
Q
w
j
(
y
j
)
-
(
p
j
/
a
)
′
d
y
j
)
1
/
(
p
j
/
a
)
′
≤
C
∥
f
→
∥
ℳ
P
→
p
0
(
W
,
w
→
P
→
)
·
sup
⁡
Q
⊃
Q
0
1
|
Q
|
1
/
p
W
(
Q
)
1
/
p
-
1
/
p
0
×
∏
j
=
1
m
‍
(
⨏
Q
w
j
(
y
j
)
-
(
p
j
/
a
)
′
d
y
j
)
1
/
(
p
j
/
a
)
′
.
This implies that
(69)
W
(
Q
0
)
1
/
q
0
(
1
W
(
Q
0
)
∫
Q
0
‍
ℳ
(
f
→
l
→
)
(
x
)
p
v
(
x
)
p
d
x
)
1
/
p
≤
C
∥
f
→
∥
ℳ
P
→
p
0
(
W
,
w
→
P
→
)
sup
⁡
Q
⊃
Q
0
W
(
Q
0
)
1
/
q
0
W
(
Q
)
1
/
p
0
(
W
(
Q
)
W
(
Q
0
)
)
1
/
p
×
(
|
Q
0
|
|
Q
|
)
1
/
p
(
⨏
Q
0
v
(
x
)
p
d
x
)
1
/
p
×
∏
j
=
1
m
‍
(
⨏
Q
w
j
(
y
j
)
-
(
p
j
/
a
)
′
d
y
j
)
1
/
(
p
j
/
a
)
′
.
By condition (28), we obtain
(70)
W
(
Q
0
)
1
/
p
0
(
1
W
(
Q
0
)
∫
Q
0
‍
ℳ
(
f
→
l
→
)
(
x
)
p
v
(
x
)
p
d
x
)
1
/
p
≤
C
[
v
,
W
,
w
→
]
p
0
,
p
,
P
→
/
a
∥
f
→
∥
ℳ
P
→
p
0
(
W
,
w
→
P
→
)
.
Therefore we obtain the desired result.



---

*Source: 648251-2014-03-16.xml*