Proof of Theorem 13.
For every cube
Q
⊂
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 estimate
ℳ
(
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.