# Weighted Estimates for Bilinear Operators

Authors: Hua Zhu; Heping Liu
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

We study the boundedness of weighted multilinear operators given by products of finite vectors of Calderón-Zygmund operators. We also investigate weighted estimates for bilinear operators related to Schrödinger operator.


---

## Body

## 1. Introduction

Bilinear (or multilinear) operators have attracted many researchers’ attention, due to their relations closely connected to the Cauchy integral along with Lipschitz curves, Calderón commutators, and compensated compactness. In [1–3] and references therein, we can see an extensive study on the Hardy space estimate of bilinear operators. In [4], we can see the bilinear operators related to a Schrödinger operator ℒ and estimates of them with respect to the Hardy type space associated with the Schrödinger operator ℒ under some general conditions. In our paper, we study weighted estimates for bilinear operators which have the same expression as the operators in [4].In this paper,N and K will denote fixed integers ≥2. Given a matrix of convolution Calderón-Zygmund kernels {
K
i
j
}
i
=
1
,
j
=
1
N
K on ℝ
d, we define {
T
i
j
}
i
,
j as the associated Calderón-Zygmund operators. We denote by L
(
f
1
,
…
,
f
K
) the K-linear operator:
(1)
L
(
f
1
,
…
,
f
K
)
=
∑
i
=
1
N
‍
(
T
i
1
f
1
)
⋯
(
T
i
K
f
K
)
,
which originally is defined for smooth compactly supported functions f
1
,
…
,
f
K. For p
≤
1, ω
∈
A
1
(
ℝ
d
), we denote by H
p
(
ω
) the usual weighted Hardy space as defined in [5, 6], that is, the set of all distributions f on ℝ
d for which the maximal function sup
⁡
t
>
0
|
ϕ
t
*
f
(
x
)
| is in L
p
(
ω
), where ϕ
t
(
x
)
=
(
1
/
t
n
)
ϕ
(
x
/
t
) and ϕ is smooth, nonzero, and compactly supported, and we also denote by H
p
,
∞
(
ω
) the weak H
p as defined in [7], that is, the set of all distributions f on ℝ
d for which the maximal function sup
⁡
t
>
0
|
ϕ
t
*
f
(
x
)
| is in weak L
p
(
ω
). If ω
(
x
)
≡
1, the weak Hardy space H
1
,
∞
(
ℝ
d
)
=
H
1
,
∞
(
ω
) was introduced by Feerman and Soria in [8]. H
p
,
∞
(
ℝ
d
) first appeared in [9] (see also [10]).Theorem 1.
Assume thatp
1
,
…
,
p
K
>
1 are given and let r
=
(
p
1
-
1
+
⋯
+
p
K
-
1
)
-
1 be their harmonic mean. We also assume that the harmonic mean of any proper subset of the p
j’s is greater than 1. If, for all (
f
1
,
…
,
f
K
)
∈
(
C
0
∞
)
K, the K-linear operator L satisfies
(2)
∫
‍
L
(
f
1
,
…
,
f
K
)
d
x
=
0
.
Then, for ω
∈
A
1
(
ℝ
d
), one has following conclusions. (1)
Ifr
>
1, L maps L
p
1
(
ω
)
×
⋯
×
L
p
K
(
ω
)
→
L
r
(
ω
).
(2)
If1
≥
r
>
d
/
(
d
+
1
), L maps L
p
1
(
ω
)
×
⋯
×
L
p
K
(
ω
)
→
H
r
(
ω
).
(3)
Ifr
=
d
/
(
d
+
1
), L maps L
p
1
(
ω
)
×
⋯
×
L
p
K
(
ω
)
→
L
r
,
∞
(
ω
).Letℒ
=
-
Δ
+
V be a Schrödinger operator on ℝ
d, d
≥
3, where V
≢
0 is a fixed nonnegative potential. We will assume that V belongs to reverse Hölder class R
H
s
(
ℝ
d
) for some s
≥
d
/
2; that is, there exists C
=
C
(
s
,
V
)
>
0 such that
(3)
(
1
|
B
|
∫
B
‍
V
(
x
)
s
d
x
)
1
/
s
≤
C
(
1
|
B
|
∫
B
‍
V
(
x
)
d
x
)
,
for every ball B
⊂
ℝ
d. In what follows, B
(
x
,
r
) denotes the ball centered at x and of the radius r. Trivially, R
H
q
(
ℝ
d
)
⊂
R
H
p
(
ℝ
d
) provided 1
<
p
≤
q
<
∞. It is well known that if V
∈
R
H
q
(
ℝ
d
) for some q
>
1, then there exists ɛ
>
0, which depends only on d and the constant C in (3), such that V
∈
R
H
q
+
ɛ
(
ℝ
d
) (see [11]). Throughout this paper, we always assume that 0
≢
V
∈
R
H
d
/
2. Thus, V
∈
R
H
q
0 for some q
0
>
d
/
2.Let{
T
t
ℒ
}
t
>
0 be the semigroup of linear operators generated by ℒ and let T
t
ℒ
(
x
,
y
) be their kernels; that is,
(4)
T
t
ℒ
f
(
x
)
=
e
-
t
ℒ
f
(
x
)
=
∫
ℝ
d
‍
T
t
ℒ
(
x
,
y
)
f
(
y
)
d
y
,
for
t
>
0
,
f
∈
L
2
(
ℝ
d
)
.
By the Trotter product formula (cf. [12]),
(5)
0
≤
T
t
ℒ
(
x
,
y
)
≤
H
t
(
x
,
y
)
=
(
4
π
t
)
-
d
/
2
exp
⁡
(
-
|
x
-
y
|
2
4
t
)
.
The maximal function with respect to the semigroup {
T
t
ℒ
}
t
>
0 is defined by
(6)
𝒯
*
f
(
x
)
=
sup
⁡
t
>
0
|
T
t
ℒ
f
(
x
)
|
.
The weighted Hardy-type space related to ℒ is naturally defined by (see [13])
(7)
H
ℒ
1
(
ω
)
=
{
f
∈
L
1
(
ω
)
:
𝒯
*
f
(
x
)
∈
L
1
(
ω
)
}
,
with
∥
f
∥
H
ℒ
1
(
ω
)
=
∥
𝒯
*
f
∥
L
1
(
ω
)
.Following [14], we define the auxiliary function ρ
(
x
,
V
)
=
ρ
(
x
) by
(8)
ρ
(
x
)
=
ρ
(
x
,
V
)
=
sup
⁡
{
r
>
0
:
1
r
d
-
2
∫
B
(
x
,
r
)
‍
V
(
y
)
d
y
≤
1
}
.
The auxiliary function ρ
(
x
) plays an important role in studying the boundedness of singular integral operators related to the Schrödinger operator ℒ as well as the atomic decomposition of H
ℒ
1 and H
ℒ
1
(
ω
) (see [13–15]).In our paper, we also consider the following bilinear operators:(9)
T
±
(
f
,
g
)
(
x
)
=
(
T
1
f
)
(
x
)
(
T
2
g
)
(
x
)
±
(
T
2
f
)
(
x
)
(
T
1
g
)
(
x
)
,
where f
∈
L
p
(
ω
), g
∈
L
q
(
ω
) with 1
<
p
,
q
<
∞ and 1
/
p
+
1
/
q
=
1, T
i
(
i
=
1,2
) are Calderón-Zygmund operators related to ℒ and satisfy the following two conditions. (i)
There exist parallel Calderón-Zygmund operatorsT
i
~ related to the Laplacian Δ and a constant δ
>
0 such that
(10)
|
T
i
(
x
,
y
)
-
T
i
~
(
x
,
y
)
|
≤
C
ρ
(
y
)
δ
|
x
-
y
|
d
-
δ
,
x
≠
y
,
where T
i
(
x
,
y
) and T
i
~
(
x
,
y
) denote the kernels of T
i and T
i
~, respectively.
(ii)
One of the parallel bilinear operators(11)
T
~
±
(
f
,
g
)
(
x
)
=
(
T
~
1
f
)
(
x
)
(
T
~
2
g
)
(
x
)
±
(
T
~
2
f
)
(
x
)
(
T
~
1
g
)
(
x
)
has the vanishing moment; that is, either T
~
+ or T
~
- satisfies
(12)
∫
ℝ
d
‍
T
~
±
(
f
,
g
)
(
x
)
d
x
=
0
∀
f
,
g
∈
C
c
∞
(
ℝ
d
)
.We will show that either T
+ or T
- is bounded from L
p
(
ω
)
×
L
q
(
ω
) to H
ℒ
1
(
ω
).Theorem 2.
Suppose that the bilinear operatorsT
± are defined as above. Let 1
<
p
,
q
<
∞ and 1
/
p
+
1
/
q
=
1. Then either T
+ or T
- (but not both), which corresponds to the parallel bilinear operator satisfying (12), maps L
p
(
ω
)
×
L
q
(
ω
) into H
ℒ
1
(
ω
) and there exists a constant C
>
0 such that
(13)
∥
T
±
(
f
,
g
)
∥
H
ℒ
1
(
ω
)
≤
C
∥
f
∥
L
p
(
ω
)
∥
g
∥
L
q
(
ω
)
.This paper is organized as follows. In Section2, we give some notation and preliminary estimates on ρ
(
x
) and the kernel T
t
(
x
,
y
) which have been proved in [14–16]. In Section 3, we prove Theorem 1, and in Section 4 we prove Theorem 2.Throughout this paper, we will useC to denote a positive constant, which is not necessarily the same at each occurrence. By A
~
B, we mean that there exists a constant C
>
1, such that C
-
1
≤
A
/
B
≤
C. For a given ball B, we denote by B
* the concentric ball with twice radius, and B
*
*
=
(
B
*
)
*.


## 2. Preliminaries

Throughout this paper, we will denoteω
(
E
)
:
=
∫
E
‍
ω
(
x
)
d
x for any set E
⊂
ℝ
d. For 1
≤
p
≤
∞, denote by p
′ the adjoint number of p; that is, 1
/
p
+
1
/
p
′
=
1.We review some needed background about Muckenhoupt weights. A weightω is a nonnegative locally integrable function on ℝ
d. We say that ω
∈
A
p
(
ℝ
d
)
,
1
<
p
<
∞, if there exists a constant C such that
(14)
(
1
|
B
|
∫
B
‍
ω
)
(
1
|
B
|
∫
B
‍
ω
1
-
p
′
)
p
-
1
≤
C
,
for every ball B
⊂
ℝ
d. The class A
1
(
ℝ
d
) is defined replacing the above inequality by
(15)
M
ω
(
x
)
≤
C
ω
(
x
)
a
.
e
.
x
∈
ℝ
d
,
where M denotes the standard Hardy-Littlewood operator.Let0
<
p
<
∞ and let ω be a locally integrable nonnegative function. We denote the weighted space L
p
(
ℝ
d
,
ω
(
x
)
d
x
) by L
p
(
ω
) and set
(16)
∥
f
∥
L
p
(
ω
)
=
(
∫
ℝ
d
‍
|
f
(
x
)
|
p
ω
(
x
)
d
x
)
1
/
p
.
We also denote the weak L
p
(
ω
) by L
p
,
∞
(
ω
) and set
(17)
∥
f
∥
L
p
,
∞
(
ω
)
=
sup
⁡
λ
>
0
λ
[
ω
(
{
x
∈
ℝ
d
:
|
f
(
x
)
|
>
λ
}
)
]
1
/
p
.We first sum up some properties of weights in the following results.Lemma 3.
One has the following properties.(i)
A
p
⊂
A
q for 1
≤
p
≤
q
<
∞.
(ii)
Letp
∈
(
1
,
∞
). Then ω
∈
A
p if and only if  ω
1
-
p
′
∈
A
p
′.
(iii)
Ifω
∈
A
p
,
1
≤
p
<
∞, then there exists ɛ
>
0 such that ω
1
+
ɛ
∈
A
p.
(iv)
Ifω
∈
A
p
,
1
≤
p
<
∞, then the measure ω
(
x
)
d
x is doubling: precisely, for all λ
>
1 and all cubes Q one has
(18)
ω
(
λ
Q
)
≤
λ
d
p
[
ω
]
A
p
ω
(
Q
)
(where [
ω
]
A
p denotes the A
p Muckenhoupt characteristic constant and λ
Q denotes the cube with the same center as Q and side length λ times the side length of Q
).
(v)
A
∞
(
ℝ
d
)
=
⋃
p
≥
1
‍
A
p
(
ℝ
d
).Proof.
Properties are standard; see, for instance, [17–19].Lemma 4.
LetB be a ball of ℝ
d and let E be any measurable subset of B. Let ω
∈
A
p
,
1
≤
p
<
∞. Then, there exist constants C
1
>
0
,
C
2
>
0, and δ
>
0 such that
(19)
C
1
(
|
E
|
|
B
|
)
p
≤
ω
(
E
)
ω
(
B
)
≤
C
2
(
|
E
|
|
B
|
)
δ
.Proof.
For the proof, we refer to [17].Next we recall some basic properties of the auxiliary functionρ
(
x
). It is known that 0
<
ρ
(
x
)
<
∞ for any x
∈
ℝ
d (cf. [14]). Therefore,
(20)
ℝ
d
=
⋃
k
=
-
∞
∞
Ω
k
,
where
(21)
Ω
k
=
{
x
∈
ℝ
d
:
2
-
k
-
1
≤
ρ
(
x
)
<
2
-
k
}
.Lemma 5 (see [15]).
There exist a constantN
=
N
(
V
) and a sequence of points {
x
(
k
,
α
)
∈
Ω
k
:
k
,
α
∈
ℤ
} such that the family of critical balls {
B
(
k
,
α
)
}, defined by B
(
k
,
α
)
=
{
x
∈
ℝ
d
:
|
x
-
x
(
k
,
α
)
|
<
ρ
(
x
(
k
,
α
)
)
}, satisfies (i)
⋃
(
k
,
α
)
‍
B
(
k
,
α
)
=
ℝ
d,
(ii)
#
{
(
k
′
,
α
′
)
:
B
(
k
′
,
α
′
)
*
*
∩
B
(
k
,
α
)
*
*
≠
∅
}
≤
N for every (
k
,
α
). Furthermore, there exists a family of C
∞ functions {
ψ
(
k
,
α
)
}
k
,
α
∈
ℤ such that (iii)
supp
⁡
ψ
(
k
,
α
)
⊆
B
(
k
,
α
)
*,
(iv)
0
≤
ψ
(
k
,
α
)
(
x
)
≤
1 and ∑
(
k
,
α
)
‍
ψ
(
k
,
α
)
(
x
)
=
1 for all x
∈
ℝ
d,
(v)
∥
∇
ψ
(
k
,
α
)
∥
L
∞
≤
C
/
ρ
(
x
(
k
,
α
)
).Lemma 6 (see [14]).
There existC
>
0 and k
0
≥
1 so that, for all x
,
y
∈
ℝ
d,
(22)
C
-
1
ρ
(
x
)
(
1
+
|
x
-
y
|
ρ
(
x
)
)
-
k
0
≤
ρ
(
y
)
≤
C
ρ
(
x
)
(
1
+
|
x
-
y
|
ρ
(
x
)
)
k
0
/
(
k
0
+
1
)
.
In particular, ρ
(
x
)
~
ρ
(
y
) when y
∈
B
(
x
,
r
) and r
≤
C
ρ
(
x
).To prove Theorem2, we need the following estimates for the kernel T
t
ℒ
(
x
,
y
).Lemma 7 (see [15, 16]).
For everyN, there is a constant C
N
>
0 such that
(23)
0
≤
T
t
ℒ
(
x
,
y
)
≤
C
N
t
-
(
d
/
2
)
(
1
+
t
ρ
(
x
)
+
t
ρ
(
y
)
)
-
N
exp
⁡
(
-
|
x
-
y
|
2
5
t
)
.
Also, there exists a constant C
>
0 such that
(24)
|
T
t
ℒ
(
x
,
y
)
-
H
t
(
x
,
y
)
|
≤
C
ρ
(
x
)
σ
|
x
-
y
|
d
-
σ
,
where σ
=
2
-
d
/
q
0
>
0.


## 3. The Proof of Theorem1

We fixp
1
,
…
,
p
K
>
1 and we let r be their harmonic mean. For r
>
1, we can get the result just by Hölder’s inequality and the L
p
(
ω
) boundedness of Calderón-Zygmund operators. Then we just need to study the case r
≤
1. Fix a smooth compactly supported function ϕ in ℝ
d, an x
0
∈
ℝ
d, and define ϕ
t
,
x
0
(
x
)
=
(
1
/
t
n
)
ϕ
(
(
x
-
x
0
)
/
t
). Without loss of generality we may assume that ϕ is supported in |
x
|
≤
1. We need to show that sup
⁡
t
>
0
|

∫
ϕ
t
,
x
0
L
(
f
1
,
…
,
f
K
)
d
x
| is in L
r
(
ω
) when r
>
n
/
(
n
+
1
) and in L
r
,
∞
(
ω
) when r
=
n
/
(
n
+
1
). We also fix a smooth cutoff η
(
x
) such that   η
≡
1 on |
x
|
<
2 and is supported in |
x
|
<
4. We call for simplicity η
0
(
x
)
=
η
(
(
x
0
-
x
)
/
t
) and η
1
(
x
)
=
1
-
η
0
(
x
). We now decompose L
(
f
1
,
…
,
f
K
)
=
L
0
+
L
1
+
⋯
+
L
K
+
1, where
(25)
L
0
=
L
(
η
0
f
1
,
η
0
f
2
,
…
,
η
0
f
K
)
,
L
1
=
∑
j
=
1
K
L
(
f
1
,
…
,
η
1
f
j
,
…
,
f
K
)
,
L
2
=
∑
1
≤
j
,
l
≤
K
j
<
l
L
(
f
1
,
…
,
η
1
f
j
,
…
,
η
1
f
l
,
…
,
f
K
)
,
⋮
L
K
+
1
=
(
-
1
)
K
L
(
η
1
f
1
,
η
1
f
2
,
…
,
η
1
f
K
)
.
In each L
u above exactly u functions among the f
j’s are multiplied by η
1 and the remaining are left intact.For any fixedi
,
k and any x such that |
x
-
x
0
|
≤
t we have
(26)
sup
⁡
t
>
0
|
T
i
k
(
η
1
f
)
(
x
)
-
T
i
k
(
η
1
f
)
(
x
0
)
|
≤
sup
⁡
t
>
0
|
∫
‍
(
K
i
k
(
x
-
y
)
-
K
i
k
(
x
0
-
y
)
)
η
1
(
y
)
f
(
y
)
d
y
|
≤
C
sup
⁡
t
>
0
∫
|
y
-
x
0
|
≥
t
‍
|
x
-
x
0
|
|
y
-
x
0
|
-
n
-
1
|
f
(
y
)
|
d
y
≤
C
M
(
|
f
|
)
(
x
0
)
,
where M
(
f
)
(
x
0
) is the Hardy-Littlewood maximal function of f at the point x
0. We denote the maximal truncated operator of T
i
j by (
T
i
j
)
*. We begin with the L
1:
(27)
L
1
=
∑
j
=
1
K
‍
L
(
f
1
,
…
,
(
η
1
f
j
)
(
x
)
-
(
η
1
f
j
)
(
x
0
)
,
…
,
f
K
)
+
∑
j
=
1
K
‍
L
(
f
1
,
…
,
(
η
1
f
j
)
(
x
0
)
,
…
,
f
K
)
.
Then we have
(28)
I
=
sup
⁡
t
>
0
|
∫
‍
ϕ
t
,
x
0
L
1
d
x
|
≤
∑
j
=
1
K
‍
∑
i
=
1
N
‍
sup
⁡
t
>
0
∫
‍
|
ϕ
t
,
x
0
|
∏
1
≤
k
≤
K
k
≠
j
|
T
i
k
f
k
|
×
(
|
T
i
j
(
η
1
f
j
)
(
x
)
-
T
i
j
(
η
1
f
j
)
(
x
0
)
|
+
|
T
i
j
(
η
1
f
j
)
(
x
0
)
|
)
d
x
≤
C
∑
j
=
1
K
‍
∑
i
=
1
N
‍
M
(
∏
1
≤
k
≤
K
k
≠
j
|
T
i
k
f
k
|
)
(
x
0
)
×
(
M
(
f
j
)
(
x
0
)
+
(
T
i
j
)
*
f
j
(
x
0
)
)
.
Define σ
j by σ
j
-
1
+
p
j
-
1
=
r
-
1. By Hölder inequality and the boundedness of M and (
T
i
j
)
*, the L
r
(
ω
) norm in x
0 of I is bounded by
(29)
C
∑
j
=
1
K
‍
∑
i
=
1
N
‍
(
∥
M
(
f
j
)
∥
L
p
j
(
ω
)
+
∥
(
T
i
j
)
*
(
f
j
)
∥
L
p
j
(
ω
)
)
×
∥
M
(
∏
1
≤
k
≤
K
k
≠
j
|
T
i
k
f
k
|
)
∥
L
σ
j
(
ω
)
≤
C
∑
j
=
1
K
‍
∑
i
=
1
N
‍
∥
f
j
∥
L
p
j
(
ω
)
∥
∏
1
≤
k
≤
K
k
≠
j
|
T
i
k
f
k
|
∥
L
σ
j
(
ω
)
≤
C
∑
j
=
1
K
‍
∑
i
=
1
N
‍
∥
f
j
∥
L
p
j
(
ω
)
∏
1
≤
k
≤
K
k
≠
j
∥
T
i
k
f
k
∥
L
p
k
(
ω
)
≤
C
∑
j
=
1
K
‍
∥
f
j
∥
L
p
j
(
ω
)
∏
1
≤
k
≤
K
k
≠
j
∥
f
k
∥
L
p
k
(
ω
)
=
C
∏
k
=
1
K
‍
∥
f
k
∥
L
p
k
(
ω
)
.
We obtain that
(30)
∥
sup
⁡
t
>
0
⁡
|
∫
ϕ
t
,
x
0
L
1
d
x
|
∥
L
r
(
ω
)
≤
C
∏
k
-
1
K
∥
f
k
∥
L
p
k
(
ω
)
,
ω
(
{
x
0
:
sup
⁡
t
>
0
|
∫
‍
ϕ
t
,
x
0
L
1
d
x
|
>
λ
}
)
≤
C
λ
-
r
∏
k
=
1
K
∥
f
k
∥
L
p
k
(
ω
)
r
.ForL
2, we write L
2
=
L
21
+
L
22
+
L
23
+
L
24 where
(31)
L
21
=
-
∑
1
≤
j
,
l
≤
K
j
<
l
L
(
f
1
,
…
,
(
η
1
f
j
)
(
x
)
-
(
η
1
f
j
)
(
x
0
)
,
…
,
(
η
1
f
l
)
(
x
)
-
(
η
1
f
l
)
(
x
0
)
,
…
,
f
K
(
η
1
f
j
)
)
,
L
22
=
-
∑
1
≤
j
,
l
≤
K
j
<
l
L
(
f
1
,
…
,
(
η
1
f
j
)
(
x
0
)
,
…
,
(
η
1
f
l
)
(
x
)
-
(
η
1
f
l
)
(
x
0
)
,
…
,
f
K
(
η
1
f
j
)
)
,
L
23
=
-
∑
1
≤
j
,
l
≤
K
j
<
l
L
(
f
1
,
…
,
(
η
1
f
j
)
(
x
)
-
(
η
1
f
j
)
(
x
0
)
,
…
,
(
η
1
f
l
)
(
x
0
)
,
…
,
f
K
)
,
L
24
=
-
∑
1
≤
j
,
l
≤
K
j
<
l
L
(
f
1
,
…
,
(
η
1
f
j
)
(
x
0
)
,
…
,
(
η
1
f
l
)
(
x
0
)
,
…
,
f
K
(
η
1
f
j
)
)
.
Using the same reason as before, we have the following estimate of L
2
u
,
u
=
1,2
,
3,4:
(32)
sup
⁡
t
>
0
|
∫
‍
ϕ
t
,
x
0
L
2
u
d
x
|
≤
C
∑
1
≤
j
,
l
≤
K
j
<
l
∑
i
=
1
K
‍
M
(
∏
1
≤
k
≤
K
k
≠
j
,
l
|
T
i
k
f
k
|
)
×
(
x
0
)
(
(
C
j
f
j
)
(
x
0
)
(
C
l
f
l
)
(
x
0
)
)
,
where each C
j
f
j is either M
(
f
j
) or (
T
i
j
)
*
f
j and therefore ∥
C
j
f
j
∥
L
p
j
(
ω
)
≤
C
∥
f
j
∥
L
p
j
(
ω
). Define σ
j
l by σ
j
l
-
1
+
p
j
-
1
+
p
l
-
1
=
r
-
1. For each u
=
1,2
,
3,4, Hölder inequality implies that
(33)
∥
sup
⁡
t
>
0
|
∫
‍
ϕ
t
,
x
0
L
2
u
d
x
|
∥
L
r
(
ω
)
≤
C
∑
1
≤
j
,
l
≤
K
j
<
l
∑
i
=
1
K
‍
∥
M
(
∏
1
≤
k
≤
K
k
≠
j
,
l
|
T
i
k
f
k
|
)
∥
L
σ
j
l
(
ω
)
×
(
∥
C
j
f
j
∥
L
p
j
(
ω
)
∥
C
l
f
l
∥
L
p
l
(
ω
)
)
≤
C
∑
1
≤
j
,
l
≤
K
j
<
l
∑
i
=
1
K
‍
∥
∏
1
≤
k
≤
K
k
≠
j
,
l
|
T
i
k
f
k
|
∥
L
σ
j
l
(
ω
)
×
(
∥
f
j
∥
L
p
j
(
ω
)
∥
f
l
∥
L
p
l
(
ω
)
)
≤
C
∑
1
≤
j
,
l
≤
K
j
<
l
∑
i
=
1
K
∏
1
≤
k
≤
K
k
≠
j
,
l
∥
T
i
k
f
k
∥
L
p
k
(
ω
)
×
(
∥
f
j
∥
L
p
j
(
ω
)
∥
f
l
∥
L
p
l
(
ω
)
)
≤
C
∑
1
≤
j
,
l
≤
K
j
<
l
∏
1
≤
k
≤
K
k
≠
j
,
l
∥
f
k
∥
L
p
k
(
ω
)
×
(
∥
f
j
∥
L
p
j
(
ω
)
∥
f
l
f
l
∥
L
p
l
(
ω
)
)
≤
C
∏
1
≤
k
≤
K
‍
∥
f
k
∥
L
p
k
(
ω
)
.
Then we get
(34)
∥
sup
⁡
t
>
0
⁡
|
∫
ϕ
t
,
x
0
L
1
d
x
|
∥
L
r
(
ω
)
≤
C
∏
k
-
1
K
∥
f
k
∥
L
p
k
(
ω
)
,
ω
(
{
x
0
:
sup
⁡
t
>
0
|
∫
‍
ϕ
t
,
x
0
L
2
d
x
|
>
λ
}
)
≤
C
λ
-
r
∏
k
=
1
K
‍
∥
f
k
∥
L
p
k
(
ω
)
r
.ForL
3
,
L
4
,
…
,
L
K
+
1, by the similar method, we get the desired conclusions. Now we turn to the term L
0, and we use two inequalities which have been proved in [3] as follows:
(35)
sup
⁡
t
>
0
|
∫
‍
ϕ
t
,
x
0
L
0
d
x
|
≤
C
∏
k
=
1
K
‍
(
M
(
|
f
k
|
s
k
)
(
x
0
)
)
1
/
s
k
when
r
>
d
(
d
+
1
)
,
(36)
sup
⁡
t
>
0
|
∫
ϕ
t
,
x
0
L
0
d
x
|
≤
C
∏
k
=
1
K
(
M
(
|
f
k
|
p
k
)
(
x
0
)
)
1
/
p
k
when
r
=
d
(
d
+
1
)
,
where 1
<
s
j
<
p
j.Then, forr
>
d
/
(
d
+
1
), we get
(37)
∥
sup
⁡
t
>
0
|
∫
‍
ϕ
t
,
x
0
L
0
d
x
|
∥
L
r
(
ω
)
≤
C
∥
∏
k
=
1
K
‍
(
M
(
|
f
k
|
s
k
)
(
x
0
)
)
1
/
s
k
∥
L
r
(
ω
)
≤
C
∏
k
=
1
K
‍
∥
(
M
(
|
f
k
|
s
k
)
(
x
0
)
)
1
/
s
k
∥
L
p
k
(
ω
)
≤
C
∏
k
=
1
K
‍
∥
f
k
∥
L
p
k
(
ω
)
,
where we use p
k
/
s
k
>
1, and we obtain conclusion (2) of Theorem 1.For conclusion (3), letε
0
=
λ
/
C, ε
K
=
1, and ε
1
,
…
,
ε
K
-
1
>
0 be arbitrary; then, by using (36) and the weak (
1,1
) result for the Hardy-Littlewood maximal function, we can get
(38)
ω
(
{
x
0
:
sup
⁡
t
>
0
|
∫
‍
ϕ
t
,
x
0
L
0
d
x
|
>
λ
}
)
≤
∑
j
=
1
K
‍
ω
(
{
x
0
:
M
(
|
f
j
|
p
j
)
(
x
0
)
>
(
ε
j
-
1
ε
j
)
p
j
}
)
≤
C
∑
j
=
1
K
‍
(
ε
j
-
1
ε
j
)
-
p
j
∫
‍
|
f
j
|
p
j
ω
(
x
)
d
x
,
and the expression minimizes when all the terms that appear in the sum are equal.Finally, we get the weak type estimate:(39)
ω
(
{
x
0
:
sup
⁡
t
>
0
|
∫
‍
ϕ
t
,
x
0
L
0
d
x
|
>
λ
}
)
≤
C
λ
-
r
∏
j
=
1
K
‍
∥
f
j
∥
L
p
j
(
ω
)
r
,and we complete the proof.


## 4. The Proof of Theorem2

We give the proof for the bilinear operatorT
- only and the proof for T
+ is similar. Let f
∈
L
p
(
ω
) and g
∈
L
q
(
ω
). Assume that the parallel Calderón-Zygmund operators T
~
i
(
i
=
1,2
) and the parallel bilinear operator T
~
- satisfy (10) and (12). We split T
-
(
f
,
g
) into three parts:
(40)
T
-
(
f
,
g
)
(
x
)
=
T
1
-
(
f
,
g
)
(
x
)
+
T
2
-
(
f
,
g
)
(
x
)
+
T
~
-
(
f
,
g
)
(
x
)
,
where
(41)
T
1
-
(
f
,
g
)
=
(
T
1
f
-
T
~
1
f
)
(
T
2
g
)
-
(
T
2
f
-
T
~
2
f
)
(
T
1
g
)
,
T
2
-
(
f
,
g
)
=
(
T
~
1
f
)
(
T
2
g
-
T
~
2
g
)
-
(
T
~
2
f
)
(
T
1
g
-
T
~
1
g
)
.
It follows from Theorem 1 that T
~
-
∈
H
1
(
ω
) and
(42)
∥
T
~
-
(
f
,
g
)
∥
H
1
(
ω
)
≤
C
∥
f
∥
L
p
(
ω
)
∥
g
∥
L
q
(
ω
)
.
Since H
1
(
ω
)
⊂
H
ℒ
1
(
ω
), we have T
~
-
(
f
,
g
) and
(43)
∥
T
~
-
(
f
,
g
)
∥
H
ℒ
1
(
ω
)
≤
C
∥
f
∥
L
p
(
ω
)
∥
g
∥
L
q
(
ω
)
.It suffices to showT
1
-
(
f
,
g
)
+
T
2
-
(
f
,
g
)
∈
H
ℒ
1
(
ω
); that is,
(44)
𝒯
*
(
T
1
-
(
f
,
g
)
+
T
2
-
(
f
,
g
)
)
(
x
)
=
sup
⁡
t
>
0
|
T
t
ℒ
(
T
1
-
(
f
,
g
)
+
T
2
-
(
f
,
g
)
)
(
x
)
|
∈
L
1
(
ω
)
.
For simplicity we write F
(
f
,
g
)
(
x
)
=
T
1
-
(
f
,
g
)
+
T
2
-
(
f
,
g
). Choose a sequence of points {
x
(
k
,
α
)
} and a family of functions {
ψ
(
k
,
α
)
} as in Lemma 5. Then
(45)
∥
T
1
-
(
f
,
g
)
+
T
2
-
(
f
,
g
)
∥
H
ℒ
1
(
ω
)
≤
∑
(
k
,
α
)
‍
(
∫
(
B
(
k
,
α
)
*
*
)
c
‍
+
∫
B
(
k
,
α
)
*
*
‍
)
×
sup
⁡
t
>
0
|
T
t
ℒ
(
ψ
(
k
,
α
)
F
(
f
,
g
)
)
(
x
)
|
ω
(
x
)
d
x
=
∑
(
k
,
α
)
‍
I
(
k
,
α
)
+
∑
(
k
,
α
)
‍
J
(
k
,
α
)
.ForI
(
k
,
α
), by using (23), we get
(46)
I
(
k
,
α
)
=
∫
(
B
(
k
,
α
)
*
*
)
c
‍
sup
⁡
t
>
0
|
∫
ℝ
d
‍
T
t
ℒ
(
x
,
y
)
ψ
(
k
,
α
)
(
y
)
F
(
f
,
g
)
(
y
)
d
y
|
×
ω
(
x
)
d
x
≤
C
N
∫
(
B
(
k
,
α
)
*
*
)
c
‍
sup
⁡
t
>
0
∫
ℝ
d
ρ
(
y
)
N
t
-
(
(
d
+
N
)
/
2
)
e
-
(
|
x
-
y
|
2
/
5
t
)
×
|
ψ
(
k
,
α
)
(
y
)
F
(
f
,
g
)
(
y
)
|
d
y
ω
(
x
)
d
x
≤
C
N
∫
(
B
(
k
,
α
)
*
*
)
c
‍
∫
ℝ
d
‍
ρ
(
x
)
N
|
x
-
y
|
d
+
N
×
|
ψ
(
k
,
α
)
(
y
)
F
(
f
,
g
)
(
y
)
|
d
y
ω
(
x
)
d
x
,
and for x
∉
B
k
,
α
*
* and y
∈
supp
⁡
ψ
(
k
,
α
)
⊂
B
(
k
,
α
)
*, Lemma 6 yields ρ
(
y
)
~
ρ
(
x
(
k
,
α
)
) and |
x
-
y
|
≥
(
1
/
2
)
|
x
-
x
(
k
,
α
)
|; then we have
(47)
I
(
k
,
α
)
≤
C
∫
B
(
k
,
α
)
*
‍
|
ψ
(
k
,
α
)
(
y
)
F
(
f
,
g
)
(
y
)
|
×
(
∫
(
B
(
k
,
α
)
*
*
)
c
‍
ρ
(
x
(
k
,
α
)
)
N
|
x
-
x
(
k
,
α
)
|
d
+
N
ω
(
x
)
d
x
)
d
y
.
For the inner integral we have
(48)
∫
(
B
(
k
,
α
)
*
*
)
c
‍
ρ
(
x
(
k
,
α
)
)
N
|
x
-
x
(
k
,
α
)
|
d
+
N
ω
(
x
)
d
x
=
∫
|
x
-
x
(
k
,
α
)
|
>
4
ρ
(
x
(
k
,
α
)
)
‍
ρ
(
x
(
k
,
α
)
)
N
|
x
-
x
(
k
,
α
)
|
d
+
N
ω
(
x
)
d
x
=
∑
j
=
2
∞
‍
∫
|
x
-
x
(
k
,
α
)
|
~
2
j
ρ
(
x
(
k
,
α
)
)
‍
ρ
(
x
(
k
,
α
)
)
N
|
x
-
x
(
k
,
α
)
|
d
+
N
ω
(
x
)
d
x
≤
∑
j
=
2
∞
‍
ρ
(
x
(
k
,
α
)
)
N
(
2
j
ρ
(
x
(
k
,
α
)
)
)
d
+
N
ω
(
B
(
x
(
k
,
α
)
,
2
j
+
1
ρ
(
x
(
k
,
α
)
)
)
)
≤
C
ω
(
B
(
k
,
α
)
*
)
|
B
(
k
,
α
)
*
|
≤
C
ω
(
y
)
,
where y
∈
B
(
k
,
α
)
*. Then we obtain
(49)
I
(
k
,
α
)
≤
C
∫
B
(
k
,
α
)
*
‍
|
ψ
(
k
,
α
)
(
y
)
F
(
f
,
g
)
(
y
)
|
ω
(
y
)
d
y
≤
C
∥
ψ
(
k
,
α
)
F
(
f
,
g
)
∥
L
1
(
ω
)
,
and hence
(50)
∑
(
k
,
α
)
‍
I
(
k
,
α
)
≤
C
∑
(
k
,
α
)
‍
∥
ψ
(
k
,
α
)
F
(
f
,
g
)
∥
L
1
(
ω
)
≤
C
∥
F
(
f
,
g
)
∥
L
1
(
ω
)
≤
C
∥
f
∥
L
p
(
ω
)
∥
g
∥
L
q
(
ω
)
.
To estimate ∑
(
k
,
α
)
‍
J
(
k
,
α
), we write
(51)
∑
(
k
,
α
)
‍
J
(
k
,
α
)
≤
∑
(
k
,
α
)
‍
∫
B
(
k
,
α
)
*
*
‍
sup
⁡
t
≥
ρ
(
x
(
k
,
α
)
)
2
|
T
t
ℒ
(
ψ
(
k
,
α
)
F
(
f
,
g
)
)
(
x
)
|
×
ω
(
x
)
d
x
+
∑
(
k
,
α
)
‍
∫
B
(
k
,
α
)
*
*
‍
sup
⁡
0
<
t
<
ρ
(
x
(
k
,
α
)
)
2
|
T
t
ℒ
(
ψ
(
k
,
α
)
F
(
f
,
g
)
)
(
x
)
|
×
ω
(
x
)
d
x
=
J
1
+
J
2
.ForJ
1, we apply (23) and Lemma 6 to obtain
(52)
∫
B
(
k
,
α
)
*
*
‍
sup
⁡
t
≥
ρ
(
x
(
k
,
α
)
)
2
|
T
t
ℒ
(
ψ
(
k
,
α
)
F
(
f
,
g
)
)
(
x
)
|
ω
(
x
)
d
x
≤
C
N
∫
B
(
k
,
α
)
*
*
‍
sup
⁡
t
≥
ρ
(
x
(
k
,
α
)
)
2
∫
ℝ
d
‍
ρ
(
y
)
N
t
-
(
(
d
+
N
)
/
2
)
e
-
(
|
x
-
y
|
2
/
5
t
)
×
|
ψ
(
k
,
α
)
(
y
)
F
(
f
,
g
)
(
y
)
|
d
y
ω
(
x
)
d
x
≤
C
N
∫
B
(
k
,
α
)
*
*
‍
sup
⁡
t
≥
ρ
(
x
(
k
,
α
)
)
2
∫
ℝ
d
‍
ρ
(
x
(
k
,
α
)
)
N
t
-
(
(
d
+
N
)
/
2
)
×
|
ψ
(
k
,
α
)
(
y
)
F
(
f
,
g
)
(
y
)
|
d
y
ω
(
x
)
d
x
≤
C
N
∫
B
(
k
,
α
)
*
‍
|
ψ
(
k
,
α
)
(
y
)
F
(
f
,
g
)
(
y
)
|
×
(
1
ρ
(
x
(
k
,
α
)
)
d
∫
B
(
k
,
α
)
*
*
‍
ω
(
x
)
d
x
)
d
y
≤
C
∫
B
(
k
,
α
)
*
‍
|
ψ
(
k
,
α
)
(
y
)
F
(
f
,
g
)
(
y
)
|
ω
(
y
)
d
y
≤
C
∥
ψ
(
k
,
α
)
F
(
f
,
g
)
∥
L
1
(
ω
)
,
and hence
(53)
J
1
≤
C
∑
(
k
,
α
)
‍
∥
ψ
(
k
,
α
)
F
(
f
,
g
)
∥
L
1
(
ω
)
≤
C
∥
f
∥
L
p
(
ω
)
∥
g
∥
L
q
(
ω
)
.
To estimate J
2, we denote
(54)
J
2
≤
∑
(
k
,
α
)
‍
∫
B
(
k
,
α
)
*
*
‍
sup
⁡
0
<
t
<
ρ
(
x
(
k
,
α
)
)
2
|
(
T
t
ℒ
-
H
t
)
(
ψ
(
k
,
α
)
F
(
f
,
g
)
)
(
x
)
|
×
ω
(
x
)
d
x
+
∑
(
k
,
α
)
‍
∫
B
(
k
,
α
)
*
*
‍
sup
⁡
0
<
t
<
ρ
(
x
(
k
,
α
)
)
2
|
H
t
(
ψ
(
k
,
α
)
(
T
2
-
(
f
,
g
)
T
1
-
(
f
,
g
)
+
T
2
-
(
f
,
g
)
)
(
x
)
)
|
×
ω
(
x
)
d
x
=
J
21
+
J
22
.
For x
∈
B
k
,
α
*
* and y
∈
supp
⁡
ψ
(
k
,
α
)
⊂
B
(
k
,
α
)
*, by Lemma 6 we have |
x
-
y
|
<
6
ρ
(
x
(
k
,
α
)
)
~
ρ
(
x
); then the estimate (24) gives
(55)
∫
B
(
k
,
α
)
*
*
‍
sup
⁡
0
<
t
<
ρ
(
x
(
k
,
α
)
)
2
|
(
T
t
ℒ
-
H
t
)
(
ψ
(
k
,
α
)
F
(
f
,
g
)
)
(
x
)
|
ω
(
x
)
d
x
≤
C
∫
B
(
k
,
α
)
*
*
‍
∫
B
(
k
,
α
)
*
‍
1
ρ
(
x
)
σ
|
x
-
y
|
d
-
σ
×
|
ψ
(
k
,
α
)
(
y
)
F
(
f
,
g
)
(
y
)
|
d
y
ω
(
x
)
d
x
≤
C
∫
B
(
k
,
α
)
*
‍
|
ψ
(
k
,
α
)
(
y
)
F
(
f
,
g
)
(
y
)
|
×
(
∫
B
(
k
,
α
)
*
*
‍
ω
(
x
)
d
x
ρ
(
x
(
k
,
α
)
)
σ
|
x
-
y
|
d
-
σ
)
d
y
.
For the inner integral, by properties of weight, we have
(56)
∫
B
(
k
,
α
)
*
*
‍
ω
(
x
)
ρ
(
x
(
k
,
α
)
)
σ
|
x
-
y
|
d
-
σ
d
x
≤
∫
|
x
-
y
|
<
8
ρ
(
x
(
k
,
α
)
)
‍
ω
(
x
)
ρ
(
x
(
k
,
α
)
)
σ
|
x
-
y
|
d
-
σ
d
x
≤
∑
j
=
-
2
∞
‍
∫
|
x
-
y
|
~
2
-
j
ρ
(
x
(
k
,
α
)
)
‍
ω
(
x
)
ρ
(
x
(
k
,
α
)
)
σ
|
x
-
y
|
d
-
σ
d
x
≤
C
∑
j
=
-
2
∞
‍
1
(
2
σ
)
j
ω
(
B
(
y
,
2
-
j
ρ
(
x
(
k
,
α
)
)
)
)
(
2
-
j
ρ
(
x
(
k
,
α
)
)
)
d
≤
C
ω
(
y
)
,
which yield
(57)
J
21
≤
C
∑
(
k
,
α
)
‍
∥
ψ
(
k
,
α
)
F
(
f
,
g
)
∥
L
1
(
ω
)
≤
C
∥
f
∥
L
p
(
ω
)
∥
g
∥
L
q
(
ω
)
.Now let us estimateJ
22. For t
>
0, let Γ
0
=
{
y
∈
ℝ
d
:
|
x
-
y
|
<
t
} and Γ
n
=
{
y
∈
ℝ
d
:
2
n
-
1
t
≤
|
x
-
y
|
<
2
n
t
}
,
n
∈
ℕ. Then
(58)
H
t
(
ψ
(
k
,
α
)
T
m
-
(
f
,
g
)
)
(
x
)
=
∑
n
=
0
∞
‍
H
t
(
ψ
(
k
,
α
)
T
m
-
(
f
,
g
)
χ
Γ
n
)
(
x
)
,
m
=
1,2
,
where χ
Γ
n denotes the characteristic function of the set Γ
n. Let η
(
y
)
∈
C
0
∞
(
ℝ
d
) with 0
≤
η
≤
1 satisfy η
(
y
)
=
1 for |
y
|
<
2 and η
(
y
)
=
0 for |
y
|
>
4. Set η
0
n
(
y
)
=
η
(
(
x
-
y
)
/
2
n
t
) and η
1
n
(
y
)
=
1
-
η
0
n
(
y
)
,
n
∈
ℕ
∪
{
0
}. We split the operator T
m
-
(
f
,
g
) into four parts:
(59)
T
m
-
(
f
,
g
)
=
T
m
-
(
f
,
η
1
n
g
)
+
T
m
-
(
η
1
n
f
,
g
)
-
T
m
-
(
η
1
n
f
,
η
1
n
g
)
+
T
m
-
(
η
0
n
f
,
η
0
n
g
)
,
m
=
1,2
.
We first estimate
(60)
∫
B
(
k
,
α
)
*
*
‍
sup
⁡
0
<
t
<
ρ
(
x
(
k
,
α
)
)
2
|
H
t
(
ψ
(
k
,
α
)
T
1
-
(
f
,
η
1
n
g
)
χ
Γ
n
)
(
x
)
|
ω
(
x
)
d
x
.
Write
(61)
T
1
-
(
f
,
η
1
n
g
)
=
(
U
1
1
f
)
U
1
2
(
η
1
n
g
)
-
(
U
2
1
f
)
U
2
2
(
η
1
n
g
)
,
where
(62)
U
1
1
=
T
1
-
T
~
1
,
U
2
1
=
T
2
-
T
~
2
,
U
1
2
=
T
2
,
U
2
2
=
T
1
.
Then we have
(63)
|
H
t
(
ψ
(
k
,
α
)
T
1
-
(
f
,
η
1
n
g
)
χ
Γ
n
)
(
x
)
|
≤
∑
l
=
1
2
‍
∫
Γ
n
‍
H
t
(
x
-
y
)
|
U
l
1
f
(
y
)
|
×
|
U
l
2
(
η
1
n
g
)
(
y
)
-
U
l
2
(
η
1
n
g
)
(
x
)
|
d
y
+
∑
l
=
1
2
‍
∫
Γ
n
‍
H
t
(
x
-
y
)
|
U
l
1
f
(
y
)
|
|
U
l
2
(
η
1
n
g
)
(
x
)
|
d
y
≤
C
2
n
d
e
-
2
2
n
-
4
∑
l
=
1
2
‍
1
(
2
n
t
)
d
×
∫
|
x
-
y
|
<
2
n
t
‍
|
U
l
1
f
(
y
)
|
×
|
U
l
2
(
η
1
n
g
)
(
y
)
-
U
l
2
(
η
1
n
g
)
(
x
)
|
d
y
+
C
2
n
d
e
-
2
2
n
-
4
∑
l
=
1
2
‍
1
(
2
n
t
)
d
×
∫
|
x
-
y
|
<
2
n
t
‍
|
U
l
1
f
(
y
)
|
|
U
l
2
(
η
1
n
g
)
(
x
)
|
d
y
,
where we have used (5) in the last inequality. Since U
l
j
(
j
,
l
=
1,2
) are Calderón-Zygmund operators, their kernels satisfy the standard kernel estimate for some δ
>
0. For |
x
-
y
|
≤
2
n
t,
(64)
|
U
l
2
(
η
1
n
g
)
(
y
)
-
U
l
2
(
η
1
n
g
)
(
x
)
|
≤
∫
|
x
-
z
|
≥
2
n
+
1
t
‍
|
x
-
y
|
δ
|
x
-
z
|
d
+
δ
|
(
η
1
n
g
)
(
z
)
|
d
z
≤
C
M
(
η
1
n
g
)
(
x
)
,
where M is the Hardy-Littlewood maximal operator. Thus,
(65)
|
H
t
(
ψ
(
k
,
α
)
T
1
-
(
f
,
η
1
n
g
)
χ
Γ
n
)
(
x
)
|
≤
C
2
n
d
e
-
2
2
n
-
4
∑
l
=
1
2
‍
(
M
(
U
l
1
f
)
(
x
)
M
(
η
1
n
g
)
(
x
)
+
M
(
U
l
1
f
)
(
x
)
|
U
l
2
(
η
1
n
g
)
(
x
)
|
)
,

and hence
(66)
∑
(
k
,
α
)
‍
∫
B
(
k
,
α
)
*
*
‍
sup
⁡
0
<
t
<
ρ
(
x
(
k
,
α
)
)
2
|
H
t
(
ψ
(
k
,
α
)
T
1
-
(
f
,
η
1
n
g
)
χ
Γ
n
)
(
x
)
|
×
ω
(
x
)
d
x
≤
C
2
n
d
e
-
2
2
n
-
4
∫
ℝ
d
‍
∑
l
=
1
2
‍
M
(
U
l
1
f
)
(
x
)
×
(
M
(
η
1
n
g
)
(
x
)
+
|
U
l
2
(
η
1
n
g
)
(
x
)
|
)
×
ω
(
x
)
d
x
≤
C
2
n
d
e
-
2
2
n
-
4
∑
l
=
1
2
∥
M
(
U
l
1
f
)
(
x
)
∥
L
p
(
ω
)
×
(
∥
M
(
g
)
∥
L
q
(
ω
)
+
∥
U
l
2
g
∥
L
q
(
ω
)
)
≤
C
2
n
d
e
-
2
2
n
-
4
∥
f
∥
L
p
(
ω
)
∥
g
∥
L
q
(
ω
)
.
A similar argument shows that
(67)
∑
(
k
,
α
)
‍
∫
B
(
k
,
α
)
*
*
‍
sup
⁡
0
<
t
<
ρ
(
x
(
k
,
α
)
)
2
|
H
t
(
ψ
(
k
,
α
)
T
1
-
(
η
1
n
f
,
g
)
χ
Γ
n
)
(
x
)
|
ω
(
x
)
d
x
≤
C
2
n
d
e
-
2
2
n
-
4
∥
f
∥
L
p
(
ω
)
∥
g
∥
L
q
(
ω
)
,
∑
(
k
,
α
)
‍
∫
B
(
k
,
α
)
*
*
‍
sup
⁡
0
<
t
<
ρ
(
x
(
k
,
α
)
)
2
|
H
t
(
ψ
(
k
,
α
)
T
1
-
(
η
1
n
f
,
η
1
n
g
)
χ
Γ
n
)
(
x
)
|
×
ω
(
x
)
d
x
≤
C
2
n
d
e
-
2
2
n
-
4
∥
f
∥
L
p
(
ω
)
∥
g
∥
L
q
(
ω
)
.
Finally we consider the term T
1
-
(
η
0
n
f
,
η
0
n
g
). By using (5) and (10) we have
(68)
sup
⁡
0
<
t
<
ρ
(
x
(
k
,
α
)
)
2
|
H
t
(
ψ
(
k
,
α
)
T
1
-
(
η
0
n
f
,
η
0
n
g
)
χ
Γ
n
)
(
x
)
|
≤
∑
l
=
1
2
‍
sup
⁡
0
<
t
<
ρ
(
x
(
k
,
α
)
)
2
∫
Γ
n
‍
H
t
(
x
-
y
)
|
U
l
1
(
η
0
n
f
)
(
y
)
|
×
|
ψ
(
k
,
α
)
(
y
)
|
|
U
l
2
(
η
0
n
g
)
(
y
)
|
d
y
≤
sup
⁡
0
<
t
<
ρ
(
x
(
k
,
α
)
)
2
∫
Γ
n
‍
H
t
(
x
-
y
)
(
∫
ℝ
d
‍
|
T
1
(
y
,
z
)
-
T
~
1
(
y
,
z
)
|
×
|
(
η
0
n
f
)
(
z
)
|
d
z
∫
ℝ
d
‍
|
T
1
(
y
,
z
)
-
T
~
1
(
y
,
z
)
|
)
×
|
T
2
(
η
0
n
g
)
(
y
)
|
d
y
+
sup
⁡
0
<
t
<
ρ
(
x
(
k
,
α
)
)
2
∫
Γ
n
‍
H
t
(
x
-
y
)
×
(
∫
ℝ
d
‍
|
T
2
(
y
,
z
)
-
T
~
2
(
y
,
z
)
|
×
|
(
η
0
n
f
)
(
z
)
|
d
z
∫
ℝ
d
‍
|
T
2
(
y
,
z
)
-
T
~
2
(
y
,
z
)
|
)
|
T
1
(
η
0
n
g
)
(
y
)
|
d
y
≤
C
e
-
2
2
n
-
4
sup
⁡
0
<
t
<
ρ
(
x
(
k
,
α
)
)
2
t
-
d
/
2
∥
η
0
n
g
∥
L
p
1
(
ℝ
d
)
×
{
∫
Γ
n
‍
(
∫
ℝ
d
‍
|
(
η
0
n
f
)
(
z
)
|
ρ
(
z
)
δ
|
y
-
z
|
d
-
δ
d
z
)
p
1
′
d
y
}
1
/
p
1
′
,
where 1
/
p
1
′
+
1
/
p
1
=
1. For z
∈
supp
⁡
η
0
n
=
{
y
∈
ℝ
d
:
|
x
-
y
|
≤
2
n
+
2
t
}, x
∈
B
(
k
,
α
)
*
*, and s
<
ρ
(
x
(
k
,
α
)
)
2, we have |
z
-
x
(
k
,
α
)
|
≤
2
n
+
3
ρ
(
x
(
k
,
α
)
). It follows from Lemma 6 that ρ
(
z
)
-
δ
≤
C
2
n
δ
k
0
ρ
(
x
(
k
,
α
)
)
-
δ for a fixed constant k
0
>
0. Hence, a well-known result for fractional integrals gives
(69)
sup
⁡
0
<
t
<
ρ
(
x
(
k
,
α
)
)
2
|
H
t
(
ψ
(
k
,
α
)
T
1
-
(
η
0
n
f
,
η
0
n
g
)
χ
Γ
n
)
(
x
)
|
≤
C
2
n
δ
k
0
e
-
2
2
n
-
4
sup
⁡
0
<
t
<
ρ
(
x
(
k
,
α
)
)
2
t
-
d
/
2
ρ
(
x
(
k
,
α
)
)
-
δ
×
∥
|
η
0
n
f
|
*
1
|
·
|
d
-
δ
∥
L
p
1
′
(
ℝ
d
)
∥
η
0
n
g
∥
L
p
1
(
ℝ
d
)
≤
C
2
n
δ
k
0
e
-
2
2
n
-
4
sup
⁡
0
<
t
<
ρ
(
x
(
k
,
α
)
)
2
(
2
n
t
)
d
+
δ
t
d
ρ
(
x
(
k
,
α
)
)
δ
1
(
2
n
t
)
d
+
δ
×
∥
η
0
n
f
∥
L
q
1
(
ℝ
d
)
∥
η
0
n
g
∥
L
p
1
(
ℝ
d
)
≤
C
2
n
(
d
+
δ
+
δ
k
0
)
e
-
2
2
n
-
4
M
(
|
f
|
q
1
)
1
/
q
1
(
x
)
M
(
|
g
|
p
1
)
1
/
p
1
(
x
)
,
where 1
/
p
1
′
=
1
/
q
1
-
δ
/
d. Since 1
/
p
1
+
1
/
q
1
=
1
+
δ
/
d
>
1
/
p
+
1
/
q, we are always able to choose p
1 and q
1 such that 1
<
p
1
<
q and 1
<
q
1
<
p. Then we get
(70)
∑
(
k
,
α
)
‍
∫
B
(
k
,
α
)
*
*
‍
sup
⁡
0
<
t
<
ρ
(
x
(
k
,
α
)
)
2
|
H
t
(
ψ
(
k
,
α
)
T
1
-
(
η
0
n
f
,
η
0
n
g
)
χ
Γ
n
)
(
x
)
|
×
ω
(
x
)
d
x
≤
C
2
n
(
d
+
δ
+
δ
k
0
)
e
-
2
2
n
-
4
×
∫
ℝ
d
‍
M
(
|
f
|
q
1
)
1
/
q
1
(
x
)
M
(
|
g
|
p
1
)
1
/
p
1
(
x
)
ω
(
x
)
d
x
≤
C
2
n
(
d
+
δ
+
δ
k
0
)
e
-
2
2
n
-
4
×
(
∫
ℝ
d
‍
M
(
|
f
|
q
1
)
p
/
q
1
(
x
)
ω
(
x
)
d
x
)
1
/
p
×
(
∫
ℝ
d
‍
M
(
|
g
|
p
1
)
q
/
p
1
(
x
)
ω
(
x
)
d
x
)
1
/
q
≤
C
2
n
(
d
+
δ
+
δ
k
0
)
e
-
2
2
n
-
4
∥
f
∥
L
p
(
ω
)
∥
g
∥
L
q
(
ω
)
.
Therefore, we have
(71)
∑
(
k
,
α
)
‍
∫
B
(
k
,
α
)
*
*
‍
sup
⁡
0
<
t
<
ρ
(
x
(
k
,
α
)
)
2
|
H
t
(
ψ
(
k
,
α
)
T
1
-
(
f
,
g
)
)
(
x
)
|
ω
(
x
)
d
x
≤
∑
(
k
,
α
)
‍
∫
B
(
k
,
α
)
*
*
‍
sup
⁡
0
<
t
<
ρ
(
x
(
k
,
α
)
)
2
∑
n
=
0
∞
‍
|
H
t
(
ψ
(
k
,
α
)
T
1
-
(
f
,
g
)
χ
Γ
n
)
(
x
)
|
×
ω
(
x
)
d
x
≤
C
∥
f
∥
L
p
(
ω
)
∥
g
∥
L
q
(
ω
)
∑
n
=
0
∞
‍
2
n
(
d
+
δ
+
δ
k
0
)
e
-
2
2
n
-
4
≤
C
∥
f
∥
L
p
(
ω
)
∥
g
∥
L
q
(
ω
)
.
Similarly, we get
(72)
∑
(
k
,
α
)
‍
∫
B
(
k
,
α
)
*
*
‍
sup
⁡
0
<
t
<
ρ
(
x
(
k
,
α
)
)
2
|
H
t
(
ψ
(
k
,
α
)
T
2
-
(
f
,
g
)
)
(
x
)
|
ω
(
x
)
d
x
≤
C
∥
f
∥
L
p
(
ω
)
∥
g
∥
L
q
(
ω
)
.
Then we obtain
(73)
J
22
≤
C
∥
f
∥
L
p
(
ω
)
∥
g
∥
L
q
(
ω
)
,
and we complete the proof of Theorem 2.



---

*Source: 797956-2014-02-06.xml*