1. Introduction
Let
ℝ
n
(
n
≥
2
) be the
n
-dimensional Euclidean space equipped with the Euclidean norm
|
·
|
and the Lebesgue measure
d
x
. For
1
≤
p
,
q
≤
∞
; the amalgam spaces
(
L
q
,
L
p
)
(
ℝ
n
)
of
L
p
(
ℝ
n
)
and
L
q
(
ℝ
n
)
are denoted by the set of all measurable functions
f
:
ℝ
n
→
ℂ
, which are locally in
L
q
(
ℝ
n
)
and satisfy
(1)
∥
f
∥
(
L
q
,
L
p
)
(
ℝ
n
)
∶
=
(
∫
ℝ
n
∥
f
χ
B
(
y
,
1
)
∥
q
p
d
y
)
1
/
p
<
∞
,
where
B
(
y
,
r
)
∶
=
{
x
∈
ℝ
n
:
|
x
-
y
|
<
r
}
for
r
>
0
and
y
∈
ℝ
n
. We remark that the amalgam spaces
(
L
q
,
L
p
)
(
ℝ
n
)
were introduced by Fofana in [1] in connection with the study of the continuity of the fractional maximal operator of Hardy-Littlewood and of the Fourier transformation in
ℝ
n
. In [1], Fofana also considered the subspace
(
L
q
,
L
p
)
α
(
ℝ
n
)
of
(
L
q
,
L
p
)
(
ℝ
n
)
, which consists of measurable functions
f
such that for
1
≤
α
≤
∞
,
(2)
∥
f
∥
(
L
q
,
L
p
)
α
(
ℝ
n
)
∶
=
sup
r
>
0
(
∫
ℝ
n
(
|
B
(
y
,
r
)
|
1
/
α
-
1
/
q
-
1
/
p
∥
f
χ
B
(
y
,
r
)
∥
L
q
(
ℝ
n
)
)
p
d
y
)
1
/
p
<
∞
,
1
≤
p
,
q
<
∞
and a suitable modification version for
p
=
∞
or
q
=
∞
.
By the definitions, it is clear (also see [1]) that
(
L
q
,
L
q
)
(
ℝ
n
)
=
L
q
(
ℝ
n
)
,
(
L
q
,
L
∞
)
α
(
ℝ
n
)
=
L
q
,
(
n
q
/
α
)
(
ℝ
n
)
, where
L
q
,
λ
(
ℝ
n
)
, with
1
≤
q
<
∞
and
0
<
λ
<
n
, is the classical Morrey space that consists of measurable functions
f
:
ℝ
n
→
ℂ
such that
(3)
∥
f
∥
L
q
,
λ
(
ℝ
n
)
∶
=
(
sup
y
∈
ℝ
n
,
r
>
0
|
B
(
y
,
r
)
|
λ
/
n
-
1
∫
B
(
y
,
r
)
|
f
(
x
)
|
q
d
x
)
1
/
q
<
∞
.
In this paper, we focus on the weighted version of
(
L
q
,
L
p
)
α
(
ℝ
n
)
. Precisely, letting
w
be a weight on
ℝ
n
and
1
≤
q
,
p
,
α
≤
∞
, we define the weighted amalgam spaces
(
L
w
q
,
L
p
)
α
(
ℝ
n
)
as the space of all measurable functions
f
satisfying
(4)
∥
f
∥
(
L
w
q
,
L
p
)
α
(
ℝ
n
)
∶
=
sup
r
>
0
(
∫
ℝ
n
(
w
(
B
(
y
,
r
)
)
1
/
α
-
1
/
q
-
1
/
p
s
s
s
s
s
s
h
h
i
×
∥
f
χ
B
(
y
,
r
)
∥
L
w
q
(
ℝ
n
)
)
p
d
y
)
1
/
p
<
∞
,
1
≤
p
,
q
<
∞
and a suitable modification version for
p
=
∞
or
q
=
∞
, where
L
w
q
(
ℝ
n
)
is the weighted Lebesgue space.
It is easy to check that when
κ
=
1
-
q
/
α
and
1
≤
q
<
α
<
∞
, the space
(
L
w
q
,
L
∞
)
α
(
ℝ
n
)
is nothing but the weighted Morrey space
L
w
q
,
κ
(
ℝ
n
)
, which is the set of all measurable functions
f
such that (see [2])
(5)
∥
f
∥
L
w
q
,
κ
(
ℝ
n
)
∶
=
sup
B
(
1
w
(
B
)
κ
∫
B
|
f
(
x
)
|
q
w
(
x
)
d
x
)
1
/
q
<
∞
,
si
555
ssshssss
1
≤
q
<
∞
,
0
<
κ
<
1
.
As is well known, the boundedness of the classical operators in the harmonic analysis on the weighted Morrey spaces has extensively been studied (see [2–6] and references therein). In particular, Wang and Yi [6] recently showed that the
m
-linear commutators and the iterated commutators of the
m
-linear Calderón-Zygmund operators are bounded on weighted Morrey spaces.
Based on the above, we feel that it is natural and interesting to study the boundedness of the classical operators in harmonic analysis on the amalgam spaces and the weighted versions. Indeed, a lot of attention has recently been given to this topic (e.g., see [7–10]). Here, we will continue the investigation along this line. The main purpose of this paper is to study the boundedness of the multilinear operators on the weighted amalgam spaces
(
L
v
w
→
q
,
L
p
)
α
(
ℝ
n
)
.
Let
K
(
x
,
y
1
,
…
,
y
m
)
be a locally integral function defined off the diagonal
x
=
y
1
=
⋯
=
y
m
in
(
ℝ
n
)
m
+
1
and let
T
:
𝒮
(
ℝ
n
)
×
⋯
×
𝒮
(
ℝ
n
)
→
𝒮
′
(
ℝ
n
)
be an
m
-linear operator associated with the kernel
K
(
x
,
y
1
,
…
,
y
m
)
in the following way:
(6)
〈
T
(
f
1
,
…
,
f
m
)
,
g
〉
=
∫
ℝ
n
∫
(
ℝ
n
)
m
K
(
x
,
y
1
,
…
,
y
m
)
∏
i
=
1
m
f
i
(
y
i
)
g
(
x
)
d
y
1
⋯
d
y
m
d
x
,
where
f
1
,
…
,
f
m
,
g
in
𝒮
(
ℝ
n
)
with
⋂
j
=
1
m
supp
(
f
j
)
⋂
supp
(
g
)
=
∅
.
For
b
→
=
(
b
1
,
…
,
b
m
)
∈
(
BMO
(
ℝ
n
)
)
m
, we define the
m
-linear commutator of
T
denoted by
T
Σ
b
→
as follows:
(7)
T
Σ
b
→
(
f
1
,
…
,
f
m
)
∶
=
∑
j
=
1
m
T
b
→
j
(
f
→
)
,
where each term is the commutator of
b
j
and
T
in the
j
th entry of
T
; that is
(8)
T
b
→
j
(
f
→
)
∶
=
b
j
T
(
f
1
,
…
,
f
m
)
-
T
(
f
1
,
…
,
b
j
f
j
,
…
,
f
m
)
and
f
→
=
(
f
1
,
…
,
f
m
)
, where
f
j
is a smooth function with compact support on
ℝ
n
. The iterated commutator
T
Π
b
→
is defined by
(9)
T
Π
b
→
(
f
→
)
∶
=
[
b
1
,
[
b
2
,
…
,
[
b
m
-
1
,
[
b
m
,
T
]
m
]
m
-
1
,
…
]
2
]
1
(
f
→
)
.
If
T
is associated with a distribution kernel, which coincides with the above function
K
, then we have, at a formal level,
(10)
T
(
f
→
)
(
x
)
=
∫
(
ℝ
n
)
m
K
(
x
,
y
1
,
…
,
y
m
)
f
1
(
y
1
)
⋯
f
m
(
y
m
)
d
y
1
⋯
d
y
m
;
T
b
→
j
(
f
→
)
(
x
)
=
∫
(
ℝ
n
)
m
(
b
j
(
x
)
-
b
j
(
y
j
)
)
×
K
(
x
,
y
1
,
…
,
y
m
)
f
1
(
y
1
)
⋯
f
m
(
y
m
)
d
y
1
⋯
d
y
m
,
T
Π
b
→
(
f
→
)
(
x
)
=
∫
(
ℝ
n
)
m
∏
j
=
1
m
(
b
j
(
x
)
-
b
j
(
y
j
)
)
K
(
x
,
y
1
,
…
,
y
m
)
×
f
1
(
y
1
)
⋯
f
m
(
y
m
)
d
y
1
⋯
d
y
m
.
Also, we recall the definitions of the classical Muckenhoupt classes
A
p
weights and the multilinear
A
P
→
conditions for multiple weights.
Definition 1.
A weighted
w
on
ℝ
n
, that is, a positive locally integrable function on
ℝ
n
, belongs to
A
p
(
ℝ
n
)
for
1
<
p
<
∞
if there exists a constant
C
>
0
such that
(11)
sup
Q
cube
in
ℝ
n
(
1
|
Q
|
∫
Q
w
(
x
)
d
x
)
×
(
1
|
Q
|
∫
Q
w
-
1
/
(
p
-
1
)
(
x
)
d
x
)
p
-
1
≤
C
.
The infimum of these constants
C
is called the
A
p
constant of
w
and denoted by
[
w
]
A
p
. A weight
w
belongs to the class
A
1
(
ℝ
n
)
if there exists a constant
C
>
0
such that
(12)
sup
Q
cube
in
ℝ
n
1
|
Q
|
∫
Q
w
(
x
)
d
x
(
inf
x
∈
Q
w
(
x
)
)
-
1
≤
C
,
and the infimum of these constants
C
is called the
A
1
constant of
w
and is denoted by
[
w
]
A
1
.
Definition 2.
Let
m
∈
ℕ
with
m
≥
1
and
1
≤
p
1
,
…
,
p
m
<
∞
,
1
/
m
≤
p
<
∞
, and
1
/
p
=
∑
i
=
1
m
1
/
p
i
. Let
P
→
=
(
p
1
,
…
,
p
m
)
and
w
→
=
(
w
1
,
…
,
w
m
)
. Set
(13)
v
w
→
=
∏
j
=
1
m
w
j
p
/
p
j
.
We say that
w
→
satisfies the
A
P
→
condition if
(14)
sup
Q
(
1
|
Q
|
∫
Q
v
w
→
(
x
)
d
x
)
1
/
p
×
∏
j
=
1
m
(
1
|
Q
|
∫
Q
w
j
1
-
p
j
′
(
x
)
d
x
)
1
/
p
j
′
<
∞
,
where
p
j
′
=
p
j
/
(
p
j
-
1
)
for
j
=
1
,
…
,
m
.
Obviously, for
m
=
1
,
A
P
→
is the classical Muckenhoupt classes
A
p
condition. It is not difficult to check that for
m
>
1
(see [11]),
(15)
∏
j
=
1
m
A
p
j
⊊
A
P
→
,
which implies that something more general happens for the
A
P
→
classes. Also, the authors in [11] showed that the
A
P
→
conditions are the largest classes of weights because all
m
-linear Calderón-Zygmund operators are bounded on the weighted Lebesgue spaces.
To state our main results, we still need to recall and introduce some notations. For fixed
y
∈
ℝ
n
and
r
>
0
, we set
B
=
B
(
y
,
r
)
. For any
λ
>
0
, let
λ
B
=
B
(
y
,
λ
r
)
and
χ
λ
B
be the characteristic function of the set
λ
B
. Given any positive integer
m
and
j
∈
{
1
,
…
,
m
}
, we denote by
D
j
m
the family of all finite subset
σ
=
{
σ
(
1
)
,
…
,
σ
(
j
)
}
of
{
1
,
…
,
m
}
of
j
different elements. For any
σ
∈
D
j
m
, we also denote the complementary sequence of
σ
by
τ
given by
τ
=
{
1
,
…
,
m
}
∖
σ
. We remark that
τ
=
∅
if and only if
σ
∈
D
m
m
. Letting
f
→
=
(
f
1
,
…
,
f
m
)
for a fixed
j
∈
{
0
,
…
,
m
}
and
σ
∈
D
j
m
, we set
f
σ
~
→
=
(
f
~
1
,
…
,
f
~
m
)
and
f
~
i
=
f
i
χ
(
2
B
)
c
if
i
∈
σ
and
f
~
i
=
f
i
χ
2
B
if
i
∈
τ
. Now we can formulate our main results as follows.
Theorem 3.
Let
m
∈
ℕ
with
m
≥
2
and
T
be an
m
-linear operator. Let
1
<
q
j
≤
α
j
<
p
j
<
∞
(
j
=
1
,
…
,
m
)
satisfy
1
/
q
=
∑
i
=
1
m
1
/
q
i
,
1
/
α
=
∑
i
=
1
m
1
/
α
i
,
1
/
p
=
∑
i
=
1
m
1
/
p
i
, and
p
/
p
j
=
q
/
q
j
=
α
/
α
j
(
j
=
1
,
…
,
m
)
. Assume that
w
→
=
(
w
1
,
…
,
w
m
)
∈
A
Q
→
for
Q
→
=
(
q
1
,
…
,
q
m
)
with
w
1
,
…
,
w
m
∈
A
∞
and
v
w
→
=
∏
j
=
1
m
w
j
q
/
q
j
. If
T
maps
L
w
1
q
1
(
ℝ
n
)
×
⋯
×
L
w
m
q
m
(
ℝ
n
)
to
L
v
w
→
q
(
ℝ
n
)
, then the inequality
(16)
∥
T
(
f
→
)
∥
(
L
v
w
→
q
,
L
p
)
α
(
ℝ
n
)
≤
C
∏
j
=
1
m
∥
f
j
∥
(
L
w
j
q
j
,
L
p
j
)
α
j
(
ℝ
n
)
holds provided that for any ball
B
in
ℝ
n
, any
j
∈
{
1
,
…
,
m
}
and
σ
∈
D
j
m
, there exist constants
γ
≥
0
and
β
≥
0
such that for a.e.
x
∈
B
,
(17)
|
T
(
f
σ
~
→
)
(
x
)
|
≤
C
(
∏
i
∈
τ
w
i
(
2
B
)
-
1
/
q
i
∥
f
i
χ
2
B
∥
L
w
i
q
i
)
×
∏
i
∈
σ
∑
k
=
1
∞
1
+
γ
k
2
k
β
w
i
(
2
k
+
1
B
)
-
1
/
q
i
∥
f
i
χ
2
k
+
1
B
∥
L
w
i
q
i
.
Theorem 4.
Let
m
∈
ℕ
with
m
≥
2
,
1
<
q
j
≤
α
j
<
p
j
<
∞
(
j
=
1
,
…
,
m
)
, satisfy
1
/
q
=
∑
i
=
1
m
1
/
q
i
,
1
/
α
=
∑
i
=
1
m
1
/
α
i
,
1
/
p
=
∑
i
=
1
m
1
/
p
i
, and
p
/
p
j
=
q
/
q
j
=
α
/
α
j
(
j
=
1
,
…
,
m
)
. Assume that
w
→
=
(
w
1
,
…
,
w
m
)
∈
A
Q
→
for
Q
→
=
(
q
1
,
…
,
q
m
)
with
w
1
,
…
,
w
m
∈
A
∞
,
v
w
→
=
∏
j
=
1
m
w
j
q
/
q
j
, and
b
→
=
(
b
1
,
…
,
b
m
)
∈
B
M
O
m
. If
(18)
∥
T
Σ
b
→
(
f
→
)
∥
L
v
w
→
q
(
ℝ
n
)
≤
C
(
∑
j
=
1
m
∥
b
j
∥
B
M
O
(
ℝ
n
)
)
∏
j
=
1
m
∥
f
j
∥
L
w
j
q
j
(
ℝ
n
)
,
then the inequality
(19)
∥
T
Σ
b
→
(
f
→
)
∥
(
L
v
w
→
q
,
L
p
)
α
(
ℝ
n
)
≤
C
(
∑
j
=
1
m
∥
b
j
∥
B
M
O
(
ℝ
n
)
)
∏
j
=
1
m
∥
f
j
∥
(
L
w
j
q
j
,
L
p
j
)
α
j
(
ℝ
n
)
holds provided that for any ball
B
in
ℝ
n
, any
j
∈
{
1
,
…
,
m
}
,
σ
∈
D
j
m
,
μ
∈
σ
, and
ν
∈
τ
, there exist constants
γ
≥
0
and
β
≥
0
such that for a.e.
x
∈
B
,
(20)
|
T
b
→
μ
(
f
σ
~
→
)
(
x
)
|
≤
C
(
∏
i
∈
τ
w
i
(
2
B
)
-
1
/
q
i
∥
f
i
χ
2
B
∥
L
w
i
q
i
)
×
(
∏
i
∈
σ
∖
{
μ
}
∑
k
=
1
∞
1
+
γ
k
2
k
β
w
i
(
2
k
+
1
B
)
-
1
/
q
i
×
∥
f
i
χ
2
k
+
1
B
∥
L
w
i
q
i
∏
i
∈
σ
{
μ
}
)
×
∑
k
=
1
∞
(
|
b
μ
(
x
)
-
(
b
μ
)
2
k
+
1
B
|
+
k
∥
b
μ
∥
B
M
O
)
w
μ
×
(
2
k
+
1
B
)
-
1
/
q
μ
∥
f
μ
χ
2
k
+
1
B
∥
L
w
μ
q
μ
,
(21)
|
T
b
→
ν
(
f
σ
~
→
)
(
x
)
|
≤
C
(
∏
i
∈
τ
∖
{
ν
}
w
i
(
2
B
)
-
1
/
q
i
∥
f
i
χ
2
B
∥
L
w
i
q
i
)
×
(
∏
i
∈
σ
∑
k
=
1
∞
1
+
γ
k
2
k
β
w
i
(
2
k
+
1
B
)
-
1
/
q
i
×
∥
f
i
χ
2
k
+
1
B
∥
L
w
i
q
i
∑
k
=
1
∞
1
+
γ
k
2
k
β
)
×
(
|
b
ν
(
x
)
-
(
b
ν
)
2
B
|
+
∥
b
ν
∥
B
M
O
)
w
ν
×
(
2
B
)
-
1
/
q
ν
∥
f
ν
χ
2
B
∥
L
w
ν
q
ν
.
Theorem 5.
Let
m
∈
ℕ
with
m
≥
2
,
1
<
q
j
≤
α
j
<
p
j
<
∞
(
j
=
1
,
…
,
m
)
, satisfy
1
/
q
=
∑
i
=
1
m
1
/
q
i
,
1
/
α
=
∑
i
=
1
m
1
/
α
i
,
1
/
p
=
∑
i
=
1
m
1
/
p
i
, and
p
/
p
j
=
q
/
q
j
=
α
/
α
j
(
j
=
1
,
…
,
m
)
. Suppose that
w
→
=
(
w
1
,
…
,
w
m
)
∈
A
Q
→
for
Q
→
=
(
q
1
,
…
,
q
m
)
with
w
1
,
…
,
w
m
∈
A
∞
,
v
w
→
=
∏
j
=
1
m
w
j
q
/
q
j
, and
b
→
=
(
b
1
,
…
,
b
m
)
∈
B
M
O
m
. If
(22)
∥
T
Π
b
→
(
f
→
)
∥
L
v
w
→
q
(
ℝ
n
)
≤
C
(
∏
j
=
1
m
∥
b
j
∥
B
M
O
(
ℝ
n
)
)
∏
j
=
1
m
∥
f
j
∥
L
w
j
q
j
(
ℝ
n
)
,
then the inequality
(23)
∥
T
Π
b
→
(
f
→
)
∥
(
L
v
w
→
q
,
L
p
)
α
(
ℝ
n
)
≤
C
(
∏
j
=
1
m
∥
b
j
∥
B
M
O
(
ℝ
n
)
)
∏
j
=
1
m
∥
f
j
∥
(
L
w
j
q
j
,
L
p
j
)
α
j
(
ℝ
n
)
holds provided that for any ball
B
in
ℝ
n
, any
j
∈
{
1
,
…
,
m
}
and
σ
∈
D
j
m
, there exist constants
γ
≥
0
and
β
≥
0
such that for a.e.
x
∈
B
,
(24)
|
T
Π
b
→
(
f
σ
~
→
)
(
x
)
|
≤
C
∑
η
=
0
m
∑
σ
0
∈
D
η
m
(
∏
i
∈
σ
0
|
b
i
(
x
)
-
(
b
i
)
2
B
|
)
×
(
∏
i
∈
τ
0
∥
b
i
∥
B
M
O
)
×
(
∏
i
∈
τ
w
i
(
2
B
)
-
1
/
q
i
∥
f
i
χ
2
B
∥
L
w
i
q
i
)
×
∏
i
∈
σ
∑
k
=
1
∞
1
+
γ
k
2
k
β
w
i
(
2
k
+
1
B
)
-
1
/
q
i
∥
f
i
χ
2
k
+
1
B
∥
L
w
i
q
i
,
where for any
η
∈
{
0,1
,
…
,
m
}
,
σ
0
∈
D
η
m
and
τ
0
=
{
1
,
…
,
m
}
∖
σ
0
.
Theorem 6.
Let
T
be an
m
-linear operator with kernel
K
satisfying
(25)
|
K
(
y
0
,
y
1
,
…
,
y
m
)
|
≤
A
(
∑
k
,
l
=
0
m
|
y
k
-
y
l
|
)
m
n
.
Let
1
<
q
j
≤
α
j
<
p
j
<
∞
(
j
=
1
,
…
,
m
)
satisfy
1
/
q
=
∑
i
=
1
m
1
/
q
i
,
1
/
α
=
∑
i
=
1
m
1
/
α
i
,
1
/
p
=
∑
i
=
1
m
1
/
p
i
, and
p
/
p
j
=
q
/
q
j
=
α
/
α
j
(
j
=
1
,
…
,
m
)
. Assume that
w
→
=
(
w
1
,
…
,
w
m
)
∈
∏
j
=
1
m
A
q
j
,
v
w
→
=
∏
j
=
1
m
w
j
q
/
q
j
, and
b
→
=
(
b
1
,
…
,
b
m
)
∈
B
M
O
m
. Then these inequalities (17), (20)-(21), and (24) hold.
Remark 7.
We remark that for
p
1
=
⋯
=
p
m
=
p
=
∞
, Theorems 3–6 are also true, just with the restrictive condition:
q
/
q
j
=
α
/
α
j
(
j
=
1
,
…
,
m
)
. Moreover, for
v
w
→
=
∏
j
=
1
m
w
j
q
/
q
j
=
w
∈
A
q
, that is,
w
1
=
⋯
=
w
m
=
w
∈
A
q
, we can remove the restrictive condition
p
/
p
j
=
q
/
q
j
=
α
/
α
j
(
j
=
1
,
…
,
m
)
in Theorems 3–6. See also [12, Theorem 3.5] for the unweighted case.
The rest of this paper is organized as follows. In Section 2, we will give the proofs of our main results. Some applications will be given in Section 3. Throughout this paper, the letter
C
, sometimes with additional parameters, will stand for positive constants, not necessarily the same one at each occurrence but independent of the essential variables. In what follows, we use the convention
∏
j
∈
∅
a
j
=
1
and
∑
j
∈
∅
a
j
=
0
.
2. The Proofs of Main Results
Let us begin with a lemma, which will be used in the proofs of our main results.
Lemma 8 (cf. [6, Lemma 3.1]).
Let
m
≥
2
,
p
1
,
…
,
p
m
∈
(
0
,
∞
)
and
p
∈
(
0
,
∞
)
with
1
/
p
=
∑
i
=
1
m
1
/
p
i
. Assume that
w
1
,
…
,
w
m
∈
A
∞
and
v
w
→
=
∏
i
=
1
m
w
i
p
/
p
i
, then for any ball
B
, there exists a constant
C
>
0
such that
(26)
∏
i
=
1
m
(
∫
B
w
i
(
x
)
d
x
)
p
/
p
i
≤
C
∫
B
v
w
→
(
x
)
d
x
.
Proof of Theorem 3.
For fixed
x
∈
B
, we can write
(27)
T
(
f
→
)
(
x
)
=
T
(
f
→
χ
2
B
)
(
x
)
+
∑
j
=
1
m
∑
σ
∈
D
j
m
T
(
f
σ
~
→
)
(
x
)
.
The boundedness of
T
from
L
w
1
q
1
×
⋯
×
L
w
m
q
m
to
L
v
w
→
q
, (17) and Hölder’s inequality lead to
(28)
∥
T
(
f
→
)
χ
B
∥
L
v
w
→
q
≤
C
∏
i
=
1
m
∥
f
i
χ
2
B
∥
L
w
i
q
i
+
C
∑
j
=
1
m
∑
σ
∈
D
j
m
(
∏
i
∈
τ
(
w
i
(
B
)
w
i
(
2
B
)
)
1
/
q
i
g
g
g
g
g
g
g
g
g
g
g
g
h
g
×
∥
f
i
χ
2
B
∥
L
w
i
q
i
(
w
i
(
B
)
w
i
(
2
B
)
)
)
×
∏
i
∈
σ
∑
k
=
1
∞
1
+
γ
k
2
k
β
(
w
i
(
B
)
w
i
(
2
k
+
1
B
)
)
1
/
q
i
ffffffffffffffffff
×
∥
f
i
χ
2
k
+
1
B
∥
L
w
i
q
i
.
Note that for any
k
∈
ℕ
and
w
∈
A
q
, there exists a constant
c
q
>
0
that depends only on
q
,
[
w
]
A
q
,
n
such that
(29)
w
(
B
)
w
(
2
k
+
1
B
)
≤
C
2
-
(
k
+
1
)
n
c
q
.
Hence, multiplying both sides of (28) by
v
w
→
(
B
)
1
/
α
-
1
/
q
-
1
/
p
, note that
1
/
α
-
1
/
q
-
1
/
p
<
0
and
p
/
p
i
=
q
/
q
i
=
α
/
α
i
; by Lemma 8 and (29) we obtain
(30)
v
w
→
(
B
)
1
/
α
-
1
/
q
-
1
/
p
∥
T
(
f
→
)
χ
B
∥
L
v
w
→
q
≤
C
∏
i
=
1
m
w
i
(
B
)
(
1
/
α
-
1
/
q
-
1
/
p
)
q
/
q
i
∥
T
(
f
→
)
χ
B
∥
L
v
w
→
q
≤
C
∏
i
=
1
m
w
i
(
B
)
1
/
α
i
-
1
/
q
i
-
1
/
p
i
∥
T
(
f
→
)
χ
B
∥
L
v
w
→
q
≤
C
∏
i
=
1
m
w
i
(
2
B
)
1
/
α
i
-
1
/
q
i
-
1
/
p
i
∥
f
i
χ
2
B
∥
L
w
i
q
i
+
C
∑
j
=
1
m
∑
σ
∈
D
j
m
(
∏
i
∈
τ
w
i
(
2
B
)
1
/
α
i
-
1
/
q
i
-
1
/
p
i
∥
f
i
χ
2
B
∥
L
w
i
q
i
)
×
∏
i
∈
σ
∑
k
=
1
∞
1
+
γ
k
2
k
n
c
q
i
(
1
/
α
i
-
1
/
p
i
+
β
)
w
i
(
2
k
+
1
B
)
1
/
α
i
-
1
/
q
i
-
1
/
p
i
×
∥
f
i
χ
2
k
+
1
B
∥
L
w
i
q
i
,
which combined with the fact that
α
i
<
p
i
for all
i
∈
{
1
,
…
,
m
}
leads to
(31)
∥
T
(
f
→
)
∥
(
L
v
w
→
q
,
L
p
)
α
(
ℝ
n
)
≤
C
∏
j
=
1
m
∥
f
j
∥
(
L
w
j
q
j
,
L
p
j
)
α
j
(
ℝ
n
)
.
Theorem 3 is proved.
Proof of Theorem 4.
For fixed
x
∈
B
, by linearity we can write
(32)
T
Σ
b
→
(
f
→
)
(
x
)
=
T
Σ
b
→
(
f
→
χ
2
B
)
(
x
)
+
∑
j
=
1
m
∑
σ
∈
D
j
m
(
∑
μ
∈
σ
T
b
→
μ
(
f
σ
~
→
)
(
x
)
+
∑
ν
∈
τ
T
b
→
ν
(
f
σ
~
→
)
(
x
)
)
.
Invoking (18), (20)-(21) and Hölder’s inequality, we have
(33)
∥
T
Σ
b
→
(
f
→
)
χ
B
∥
L
v
w
→
q
≤
C
(
∑
i
=
1
m
∥
b
i
∥
BMO
)
∏
i
=
1
m
∥
f
i
χ
2
B
∥
L
w
i
q
i
+
C
∑
j
=
1
m
∑
σ
∈
D
j
m
∑
μ
∈
σ
(
∏
i
∈
τ
(
w
i
(
B
)
w
i
(
2
B
)
)
1
/
q
i
×
∥
f
i
χ
2
B
∥
L
w
i
q
i
∏
i
∈
τ
)
×
(
∏
i
∈
σ
∖
{
μ
}
∑
k
=
1
∞
1
+
γ
k
2
k
β
(
w
i
(
B
)
w
i
(
2
k
+
1
B
)
)
1
/
q
i
h
h
h
h
h
h
h
h
×
∥
f
i
χ
2
k
+
1
B
∥
L
w
i
q
i
∏
i
∈
σ
{
μ
}
)
×
∑
k
=
1
∞
1
+
γ
k
2
k
β
∥
b
μ
∥
BMO
(
w
μ
(
B
)
w
μ
(
2
k
+
1
B
)
)
1
/
q
μ
×
∥
f
μ
χ
2
k
+
1
B
∥
L
w
μ
q
μ
+
C
∑
j
=
1
m
∑
σ
∈
D
j
m
∑
ν
∈
τ
(
∏
i
∈
τ
∖
{
ν
}
(
w
i
(
B
)
w
i
(
2
B
)
)
1
/
q
i
h
h
h
h
h
h
h
×
∥
f
i
χ
2
B
∥
L
w
i
q
i
∏
i
∈
τ
{
ν
}
)
×
(
∏
i
∈
σ
∑
k
=
1
∞
1
+
γ
k
2
k
β
(
w
i
(
B
)
w
i
(
2
k
+
1
B
)
)
1
/
q
i
h
h
h
h
h
h
×
∥
f
i
χ
2
k
+
1
B
∥
L
w
i
q
i
∏
i
∈
σ
)
×
∥
b
ν
∥
BMO
(
w
ν
(
B
)
w
ν
(
2
B
)
)
1
/
q
ν
∥
f
ν
χ
2
B
∥
L
w
ν
q
ν
.
By a similar argument as in getting (31), we can conclude that
(34)
∥
T
Σ
b
→
(
f
→
)
∥
(
L
v
w
→
q
,
L
p
)
α
(
ℝ
n
)
≤
C
(
∑
j
=
1
m
∥
b
j
∥
BMO
(
ℝ
n
)
)
∏
j
=
1
m
∥
f
j
∥
(
L
w
j
q
j
,
L
p
j
)
α
j
(
ℝ
n
)
,
which completes the proof of Theorem 4.
Proof of Theorem 5.
For fixed
x
∈
B
, we can write
(35)
T
Π
b
→
(
f
→
)
(
x
)
=
T
Π
b
→
(
f
→
χ
2
B
)
(
x
)
+
∑
j
=
1
m
∑
σ
∈
D
j
m
T
Π
b
→
(
f
σ
~
→
)
(
x
)
.
Applying (22), (24) and Hölder’s inequality, we get that
(36)
∥
T
Π
b
→
(
f
→
)
χ
B
∥
L
v
w
→
q
≤
C
(
∏
i
=
1
m
∥
b
i
∥
BMO
)
∏
i
=
1
m
∥
f
i
χ
2
B
∥
L
w
i
q
i
+
C
∑
j
=
1
m
∑
σ
∈
D
j
m
∑
η
=
0
m
∑
σ
0
∈
D
η
m
(
∏
i
∈
σ
0
∥
b
i
∥
BMO
)
(
∏
i
∈
τ
∥
b
i
∥
BMO
)
×
(
∏
i
∈
τ
(
w
i
(
B
)
w
i
(
2
B
)
)
1
/
q
i
∥
f
i
χ
2
B
∥
L
w
i
q
i
)
×
∏
i
∈
σ
∑
k
=
1
∞
1
+
γ
k
2
k
β
(
w
i
(
B
)
w
i
(
2
k
+
1
B
)
)
1
/
q
i
∥
f
i
χ
2
k
+
1
B
∥
L
w
i
q
i
.
By similar arguments as in getting (31) again, we can deduce that
(37)
∥
T
Π
b
→
(
f
→
)
∥
(
L
v
w
→
q
,
L
p
)
α
(
ℝ
n
)
≤
C
(
∏
j
=
1
m
∥
b
j
∥
BMO
(
ℝ
n
)
)
∏
j
=
1
m
∥
f
j
∥
(
L
w
j
q
j
,
L
p
j
)
α
j
(
ℝ
n
)
.
This completes the proof of Theorem 5.
Proof of Theorem 6.
For fixed
x
∈
B
, it is easy to check that
(38)
1
2
|
z
-
y
|
≤
|
z
-
x
|
≤
2
|
z
-
y
|
,
if
z
∈
(
2
B
)
c
.
Since
w
i
∈
A
q
i
, we have for any
i
∈
{
1
,
…
,
m
}
and
β
∈
ℕ
,
(39)
∫
2
B
|
f
i
(
z
)
|
d
z
≤
C
∥
f
i
χ
2
B
∥
L
w
i
q
i
|
2
B
|
w
i
(
2
B
)
-
1
/
q
i
,
(40)
∫
(
2
B
)
c
|
f
i
(
z
)
|
|
y
-
z
|
n
d
z
≤
∑
k
=
1
∞
|
2
k
B
|
-
1
∫
2
k
+
1
B
∖
2
k
B
|
f
i
(
z
)
|
d
z
≤
C
∑
k
=
1
∞
∥
f
i
χ
2
k
+
1
B
∥
L
w
i
q
i
w
i
(
2
k
+
1
B
)
-
1
/
q
i
,
(41)
∫
(
2
B
)
c
|
f
i
(
z
)
|
|
y
-
z
|
(
β
+
1
)
n
d
z
≤
C
∑
k
=
1
∞
∥
f
i
χ
2
k
+
1
B
∥
L
w
i
q
i
×
|
2
k
+
1
B
|
-
β
w
i
(
2
k
+
1
B
)
-
1
/
q
i
.
By Hölder’s inequality, (25) and (38), writing
σ
=
{
σ
(
1
)
,
…
,
σ
(
j
)
}
, we have
(42)
|
T
(
f
σ
~
→
)
(
x
)
|
≤
C
(
∏
i
∈
τ
∫
2
B
|
f
i
(
z
)
|
d
z
)
×
∫
(
ℝ
n
)
j
(
∑
i
∈
σ
|
x
-
y
i
|
)
-
m
n
∏
i
∈
σ
|
f
i
(
y
i
)
χ
(
2
B
)
c
(
y
i
)
|
d
y
i
≤
C
(
∏
i
∈
τ
∫
2
B
|
f
i
(
z
)
|
d
z
)
×
(
∏
i
=
1
j
-
1
∫
(
2
B
)
c
|
f
σ
(
i
)
(
z
)
|
|
y
-
z
|
n
d
z
)
∫
(
2
B
)
c
|
f
σ
(
j
)
(
z
)
|
|
y
-
z
|
(
m
-
j
+
1
)
n
d
z
.
It follows from (39)–(42) that
(43)
|
T
(
f
σ
~
→
)
(
x
)
|
≤
C
(
∏
i
∈
τ
∥
f
i
χ
2
B
∥
L
w
i
q
i
|
2
B
|
w
i
(
2
B
)
-
1
/
q
i
)
×
(
∏
i
=
1
j
-
1
∑
k
=
1
∞
w
σ
(
i
)
(
2
k
+
1
B
)
-
1
/
q
σ
(
i
)
∥
f
σ
(
i
)
χ
2
k
+
1
B
∥
L
w
σ
(
i
)
q
σ
(
i
)
)
×
∑
k
=
1
∞
|
2
k
+
1
B
|
-
m
+
j
w
σ
(
j
)
(
2
k
+
1
B
)
-
1
/
q
σ
(
j
)
×
∥
f
σ
(
j
)
χ
2
k
+
1
B
∥
L
w
σ
(
j
)
q
σ
(
j
)
≤
C
(
∏
i
∈
τ
∥
f
i
χ
2
B
∥
L
w
i
q
i
w
i
(
2
B
)
-
1
/
q
i
)
×
(
∏
i
=
1
j
-
1
∑
k
=
1
∞
w
σ
(
i
)
(
2
k
+
1
B
)
-
1
/
q
σ
(
i
)
∥
f
σ
(
i
)
χ
2
k
+
1
B
∥
L
w
σ
(
i
)
q
σ
(
i
)
)
×
∑
k
=
1
∞
2
-
k
(
m
-
j
)
w
σ
(
j
)
(
2
k
+
1
B
)
-
1
/
q
σ
(
j
)
×
∥
f
σ
(
j
)
χ
2
k
+
1
B
∥
L
w
σ
(
j
)
q
σ
(
j
)
.
This implies (17) in the case of that
β
=
0
and
γ
=
0
or
β
=
m
-
j
and
γ
=
0
.
For
T
Σ
b
→
,
σ
∈
D
j
m
and
i
∈
σ
, we have from (25) and (38) that
(44)
|
T
b
→
i
(
f
σ
~
→
)
(
x
)
|
≤
C
∫
(
ℝ
n
)
m
|
b
i
(
x
)
-
b
i
(
y
i
)
|
×
∏
k
=
1
m
|
f
~
k
(
y
k
)
|
(
∑
k
=
1
m
|
y
-
y
k
|
)
-
m
n
d
y
1
⋯
d
y
m
≤
C
(
∏
k
∈
τ
∫
2
B
|
f
k
(
z
)
|
d
z
)
∫
(
2
B
)
c
|
(
b
i
(
x
)
-
b
i
(
z
)
)
f
i
(
z
)
|
|
x
-
z
|
n
d
z
×
∫
(
ℝ
n
)
j
-
1
(
∑
k
∈
σ
∖
{
i
}
|
y
-
y
k
|
)
-
(
m
-
1
)
n
×
∏
k
∈
σ
∖
{
i
}
|
f
k
(
y
k
)
χ
(
2
B
)
c
(
y
k
)
|
d
y
k
.
Since
w
v
∈
A
q
v
, thus
w
1
-
q
v
′
∈
A
q
v
′
with
q
v
′
=
q
v
/
(
q
v
-
1
)
. By the properties of functions in
BMO
(
ℝ
n
)
and Hölder’s inequality, we have for any ball
Q
and
v
∈
{
1
,
…
,
m
}
,
(45)
∫
Q
|
b
v
(
z
)
-
(
b
v
)
Q
|
|
f
v
(
z
)
|
d
z
≤
(
∫
Q
|
f
v
(
z
)
|
q
v
w
(
z
)
d
z
)
1
/
q
×
(
∫
Q
|
b
v
(
z
)
-
(
b
v
)
Q
|
q
v
′
w
v
1
-
q
v
′
(
z
)
d
z
)
1
/
q
v
′
≤
C
∥
f
v
χ
Q
∥
L
w
v
q
v
∥
b
v
∥
BMO
|
Q
|
w
v
(
Q
)
-
1
/
q
v
.
It follows from (39) and (45) that for any
ν
∈
{
1
,
…
,
m
}
and
1
<
q
ν
<
∞
,
(46)
∫
(
2
B
)
c
|
(
b
ν
(
x
)
-
b
ν
(
z
)
)
f
ν
(
z
)
|
|
x
-
z
|
n
d
z
≤
C
∑
k
=
1
∞
∫
2
k
+
1
B
∖
2
k
B
|
(
b
ν
(
x
)
-
b
ν
(
z
)
)
f
ν
(
z
)
|
|
y
-
z
|
n
d
z
≤
C
∑
k
=
1
∞
(
|
b
ν
(
x
)
-
(
b
ν
)
2
k
+
1
B
|
+
k
∥
b
ν
∥
BMO
)
×
w
ν
(
2
k
+
1
B
)
-
1
/
q
ν
∥
f
ν
χ
2
k
+
1
B
∥
L
w
ν
q
ν
.
Let
μ
=
σ
(
i
0
)
for some
i
0
∈
{
1
,
…
,
j
}
. We now consider two cases:
Case 1 (
i
0
≠
j
). We have
(47)
|
T
b
→
μ
(
f
σ
~
→
)
(
x
)
|
≤
C
(
∏
i
∈
τ
∥
f
i
χ
2
B
∥
L
w
i
q
i
|
2
B
|
w
i
(
2
B
)
-
1
/
q
i
)
×
(
∏
i
∈
{
1
,
…
,
j
-
1
}
∖
{
i
0
}
∫
(
2
B
)
c
|
f
σ
(
i
)
(
z
)
|
|
y
-
z
|
n
d
z
)
×
∫
(
2
B
)
c
|
f
σ
(
j
)
(
z
)
|
|
y
-
z
|
(
m
-
j
+
1
)
n
d
z
×
∑
k
=
1
∞
(
|
b
μ
(
x
)
-
(
b
μ
)
2
k
+
1
B
|
+
k
∥
b
μ
∥
BMO
)
×
w
μ
(
2
k
+
1
B
)
-
1
/
q
μ
∥
f
μ
χ
2
k
+
1
B
∥
L
w
μ
q
μ
≤
C
(
∏
i
∈
τ
∥
f
i
χ
2
B
∥
L
w
i
q
i
w
(
2
B
)
-
1
/
q
i
)
×
(
∏
i
∈
{
1
,
…
,
j
-
1
}
∖
{
i
0
}
∑
k
=
1
∞
w
σ
(
i
)
(
2
k
+
1
B
)
-
1
/
q
σ
(
i
)
∥
f
σ
(
i
)
χ
2
k
+
1
B
∥
L
w
σ
(
i
)
q
σ
(
i
)
)
×
∑
k
=
1
∞
2
-
k
(
m
-
j
)
w
σ
(
j
)
(
2
k
+
1
B
)
-
1
/
q
σ
(
j
)
×
∥
f
σ
(
j
)
χ
2
k
+
1
B
∥
L
w
σ
(
j
)
q
σ
(
j
)
×
∑
k
=
1
∞
(
|
b
μ
(
x
)
-
(
b
μ
)
2
k
+
1
B
|
+
k
∥
b
μ
∥
BMO
)
×
w
μ
(
2
k
+
1
B
)
-
1
/
q
μ
∥
f
μ
χ
2
k
+
1
B
∥
L
w
μ
q
μ
,
which satisfies (20) in the case of that
β
=
0
and
γ
=
0
or
β
=
m
-
j
and
γ
=
0
.
Case 2 (
i
0
=
j
). We have
(48)
|
T
b
→
μ
(
f
σ
~
→
)
(
x
)
|
≤
C
(
∏
i
∈
τ
∥
f
i
χ
2
B
∥
L
w
i
q
i
|
2
B
|
w
i
(
2
B
)
-
1
/
q
i
)
×
(
∏
i
∈
{
1
,
…
,
j
-
2
}
∫
(
2
B
)
c
|
f
σ
(
i
)
(
z
)
|
|
y
-
z
|
n
d
z
)
×
∫
(
2
B
)
c
|
f
σ
(
j
-
1
)
(
z
)
|
|
y
-
z
|
(
m
-
j
+
1
)
n
d
z
×
∑
k
=
1
∞
(
|
b
μ
(
x
)
-
(
b
μ
)
2
k
+
1
B
|
+
k
∥
b
μ
∥
BMO
)
×
w
μ
(
2
k
+
1
B
)
-
1
/
q
μ
∥
f
μ
χ
2
k
+
1
B
∥
L
w
μ
q
μ
≤
C
(
∏
i
∈
τ
∥
f
i
χ
2
B
∥
L
w
i
q
i
w
i
(
2
B
)
-
1
/
q
i
)
×
(
∏
i
∈
{
1
,
…
,
j
-
2
}
∑
k
=
1
∞
w
σ
(
i
)
(
2
k
+
1
B
)
-
1
/
q
σ
(
i
)
∥
f
σ
(
i
)
χ
2
k
+
1
B
∥
L
w
σ
(
i
)
q
σ
(
i
)
)
×
∑
k
=
1
∞
2
-
k
(
m
-
j
)
w
σ
(
j
-
1
)
(
2
k
+
1
B
)
-
1
/
q
σ
(
j
-
1
)
×
∥
f
σ
(
j
-
1
)
χ
2
k
+
1
B
∥
L
w
σ
(
j
-
1
)
q
σ
(
j
-
1
)
×
∑
k
=
1
∞
(
|
b
μ
(
x
)
-
(
b
μ
)
2
k
+
1
B
|
+
k
∥
b
μ
∥
BMO
)
×
w
μ
(
2
k
+
1
B
)
-
1
/
q
μ
∥
f
μ
χ
2
k
+
1
B
∥
L
w
μ
q
μ
,
which satisfies (20) in the case of that
β
=
0
and
γ
=
0
or
β
=
m
-
j
.
For
ν
∈
τ
, it follows from (25) and (38) that
(49)
|
T
b
→
ν
(
f
σ
~
→
)
(
x
)
|
≤
C
∫
(
ℝ
n
)
m
|
b
ν
(
x
)
-
b
ν
(
y
ν
)
|
×
(
∑
k
=
1
m
|
y
-
y
k
|
)
-
m
n
∏
k
=
1
m
|
f
~
k
(
y
k
)
|
d
y
1
⋯
d
y
m
≤
C
(
∏
i
∈
τ
∖
{
ν
}
∫
2
B
|
f
i
(
z
)
|
d
z
)
∏
i
=
1
j
-
1
∫
(
2
B
)
c
|
f
σ
(
i
)
(
z
)
|
|
y
-
z
|
n
d
z
×
∫
(
2
B
)
c
|
f
σ
(
j
)
(
z
)
|
|
y
-
z
|
(
m
-
j
+
1
)
n
d
z
×
∫
2
B
|
(
b
ν
(
x
)
-
b
ν
(
z
)
)
f
ν
(
z
)
|
d
z
.
This together with (39)–(41), (45) and Hölder’s inequality leads to
(50)
|
T
b
→
ν
(
f
σ
~
→
)
(
x
)
|
≤
C
(
∏
i
∈
τ
∖
{
ν
}
∥
f
i
χ
2
B
∥
L
w
i
q
i
|
2
B
|
w
i
(
2
B
)
-
1
/
q
i
)
×
∏
i
=
1
j
-
1
∑
k
=
1
∞
w
σ
(
i
)
(
2
k
+
1
B
)
-
1
/
q
σ
(
i
)
∥
f
σ
(
i
)
χ
2
k
+
1
B
∥
L
w
σ
(
i
)
q
σ
(
i
)
×
∑
k
=
1
∞
|
2
k
+
1
B
|
-
m
+
j
w
σ
(
j
)
(
2
k
+
1
B
)
-
1
/
q
σ
(
j
)
∥
f
σ
(
j
)
χ
2
k
+
1
B
∥
L
w
σ
(
j
)
q
σ
(
j
)
×
(
|
b
ν
(
x
)
-
(
b
ν
)
2
B
|
+
∥
b
ν
∥
BMO
)
×
|
2
B
|
w
ν
(
2
B
)
-
1
/
q
ν
∥
f
ν
χ
2
B
∥
L
w
ν
q
ν
≤
C
(
∏
i
∈
τ
∖
{
ν
}
∥
f
i
χ
2
B
∥
L
w
i
q
i
w
i
(
2
B
)
-
1
/
q
i
)
×
∏
i
=
1
j
-
1
∑
k
=
1
∞
w
σ
(
i
)
(
2
k
+
1
B
)
-
1
/
q
σ
(
i
)
∥
f
σ
(
i
)
χ
2
k
+
1
B
∥
L
w
σ
(
i
)
q
σ
(
i
)
×
∑
k
=
1
∞
2
-
k
(
m
-
j
)
w
σ
(
j
)
(
2
k
+
1
B
)
-
1
/
q
σ
(
j
)
∥
f
σ
(
j
)
χ
2
k
+
1
B
∥
L
w
σ
(
j
)
q
σ
(
j
)
×
(
|
b
ν
(
x
)
-
(
b
ν
)
2
B
|
+
∥
b
ν
∥
BMO
)
×
w
ν
(
2
B
)
-
1
/
q
ν
∥
f
ν
χ
2
B
∥
L
w
ν
q
ν
,
which satisfies (21) in the case of that
β
=
0
and
γ
=
0
or
β
=
m
-
j
and
γ
=
0
.
For
T
Π
b
→
,
j
∈
{
1
,
…
,
m
}
,
σ
∈
D
j
m
,
σ
=
{
σ
(
1
)
,
…
,
σ
(
j
)
}
and
τ
=
{
1
,
…
,
m
}
∖
σ
, we have
(51)
T
Π
b
→
(
f
σ
~
→
)
(
x
)
=
∫
(
ℝ
n
)
m
∏
i
=
1
m
(
b
i
(
x
)
-
b
i
(
y
)
)
K
(
x
,
y
1
,
…
,
y
m
)
×
∏
i
∈
σ
f
i
(
y
i
)
χ
(
2
B
)
c
(
y
i
)
∏
i
∈
τ
f
i
(
y
i
)
χ
2
B
(
y
i
)
d
y
1
⋯
d
y
m
∶
=
∑
η
=
0
m
∑
σ
0
∈
D
η
m
T
j
,
σ
0
(
f
→
)
(
x
)
,
where
(52)
T
j
,
σ
0
(
f
→
)
(
x
)
≔
∏
i
∈
σ
0
(
b
i
(
x
)
-
(
b
i
)
2
B
)
×
∫
(
ℝ
n
)
m
∏
i
∈
τ
0
(
(
b
i
)
2
B
-
b
i
(
y
i
)
)
K
(
x
,
y
1
,
…
,
y
m
)
×
∏
i
∈
σ
f
i
(
y
i
)
χ
(
2
B
)
c
(
y
i
)
∏
i
∈
τ
f
i
(
y
i
)
χ
2
B
(
y
i
)
d
y
1
⋯
d
y
m
.
For fixed
x
∈
B
,
η
∈
{
0
,
…
,
m
}
and
σ
0
∈
D
η
m
, we set
(53)
τ
1
∶
=
τ
∩
τ
0
,
τ
2
∶
=
τ
0
∖
τ
,
τ
3
∶
=
τ
∖
τ
0
.
Then by (25) and (38), we have
(54)
|
T
j
,
σ
0
(
f
→
)
(
x
)
|
≤
C
(
∏
i
∈
σ
0
|
b
i
(
x
)
-
(
b
i
)
2
B
|
)
×
∫
(
ℝ
n
)
m
(
∏
i
∈
τ
1
|
(
b
i
(
y
i
)
-
(
b
i
)
2
B
)
f
i
(
y
i
)
χ
2
B
(
y
i
)
|
)
×
(
∏
i
∈
τ
2
|
b
i
(
y
i
)
-
(
b
i
)
2
B
|
)
(
∏
i
∈
τ
3
|
f
i
(
y
i
)
χ
2
B
(
y
i
)
|
)
×
∏
i
∈
σ
|
f
i
(
y
i
)
χ
(
2
B
)
c
(
y
i
)
|
|
K
(
x
,
y
1
,
…
,
y
m
)
|
d
y
1
⋯
d
y
m
≤
C
(
∏
i
∈
σ
0
|
b
i
(
x
)
-
(
b
i
)
2
B
|
)
×
(
∏
i
∈
τ
1
∫
2
B
|
b
i
(
z
)
-
(
b
i
)
2
B
|
|
f
i
(
z
)
|
d
z
)
×
(
∏
i
∈
τ
3
∫
2
B
|
f
i
(
z
)
|
d
z
)
×
(
∏
i
∈
τ
2
∫
(
2
B
)
c
|
b
i
(
z
)
-
(
b
i
)
2
B
|
|
f
i
(
z
)
|
|
y
-
z
|
n
d
z
)
×
∏
i
∈
σ
∖
(
τ
2
∪
{
σ
(
j
)
}
)
∫
(
2
B
)
c
|
f
i
(
z
)
|
|
y
-
z
|
n
d
z
×
∫
(
2
B
)
c
|
f
σ
(
j
)
(
z
)
|
|
y
-
z
|
(
m
-
j
+
1
)
n
d
z
.
It is easy to check that
(55)
∫
(
2
B
)
c
|
(
b
i
(
z
)
-
(
b
i
)
2
B
)
f
i
(
z
)
|
|
y
-
z
|
n
d
z
≤
C
∑
k
=
1
∞
(
k
+
1
)
∥
b
i
∥
BMO
w
i
(
2
k
+
1
B
)
-
1
/
q
i
×
∥
f
i
χ
2
k
+
1
B
∥
L
w
i
q
i
for any
i
∈
{
1
,
…
,
m
}
. This combining (39)–(41), (45) with (54) yields that
(56)
|
T
j
,
σ
0
(
f
→
)
(
x
)
|
≤
C
(
∏
i
∈
σ
0
|
b
i
(
x
)
-
(
b
i
)
2
B
|
)
(
∏
i
∈
τ
0
∥
b
i
∥
BMO
)
×
(
∏
i
∈
τ
1
w
i
(
2
B
)
-
1
/
q
i
∥
f
i
χ
2
B
∥
L
w
i
q
i
)
×
(
∏
i
∈
τ
3
w
i
(
2
B
)
-
1
/
q
i
∥
f
i
χ
2
B
∥
L
w
i
q
i
)
×
(
∏
i
∈
τ
2
∑
k
=
1
∞
(
k
+
1
)
w
i
(
2
k
+
1
B
)
-
1
/
q
i
∥
f
i
χ
2
k
+
1
B
∥
L
w
i
q
i
)
×
∏
i
∈
σ
∖
(
τ
2
∪
{
σ
(
j
)
}
)
∑
k
=
1
∞
w
(
2
k
+
1
B
)
-
1
/
q
i
∥
f
i
χ
2
k
+
1
B
∥
L
w
q
i
×
∑
k
=
1
∞
2
-
k
(
m
-
j
)
w
(
2
k
+
1
B
)
-
1
/
q
σ
(
j
)
∥
f
σ
(
j
)
χ
2
k
+
1
B
∥
L
w
σ
(
j
)
q
σ
(
j
)
.
This together with (51) implies (24) in the case of that
β
=
0
and
γ
=
1
, or
β
=
0
and
γ
=
0
, or
β
=
m
-
j
and
γ
=
0
, and completes the proof of Theorem 6.