# Compactness of the Commutator of Multilinear Fourier Multiplier Operator on Weighted Lebesgue Space

Authors: Jiang Zhou; Peng Li
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

LetT
σ be the multilinear Fourier multiplier operator associated with multiplier σ satisfying the Sobolev regularity that s
u
p
l
∈
Z
σ
l
W
s
1
,
…
,
s
m
(
R
m
n
)
<
∞ for some s
k
∈
(
n
/
2
,
n
]
(
k
=
1
,
…
,
m
). The authors prove that if b
1
,
…
,
b
m
∈
B
M
O
(
R
n
) and w
⃗
∈
∏
k
=
1
m
‍
A
p
k
/
t
k
(
t
k
=
n
/
s
k
), then the commutator T
σ
,
Σ
b is bounded from L
p
1
(
R
n
,
w
1
)
×
⋯
×
L
p
m
(
R
n
,
w
m
) to L
p
(
R
n
,
ν
w
⃗
). Moreover, the authors also prove that if b
1
,
…
,
b
m
∈
V
M
O
(
R
n
) and w
⃗
∈
∏
k
=
1
m
‍
A
p
k
/
t
k
(
t
k
=
n
/
s
k
), then the commutator T
σ
,
Σ
b is compact operator from L
p
1
(
R
n
,
w
1
)
×
⋯
×
L
p
m
(
R
n
,
w
m
) to L
p
(
R
n
,
ν
w
⃗
).


---

## Body

## 1. Introduction

The study of the multilinear Fourier multiplier operator was originated by Coifman and Meyer in their celebrated work [1, 2]. Let σ
∈
L
∞
(
R
m
n
); the multilinear Fourier multiplier operator T
σ is defined by
(1)
T
σ
(
f
→
)
(
x
)
≔
∫
R
m
n
‍
exp
⁡
(
2
π
i
x
(
ξ
l
+
⋯
+
ξ
m
)
)
×
σ
(
ξ
l
,
…
,
ξ
m
)
f
^
1
(
ξ
l
)
⋯
f
^
m
(
ξ
m
)
d
ξ
→
for all f
1
,
…
,
f
m
∈
S
(
R
n
), where d
ξ
→
=
d
ξ
1
⋯
d
ξ
m and f
^ is the Fourier transform of f. Coifman and Meyer [2] proved that if σ
∈
C
s
(
R
m
n
∖
{
0
}
) satisfies
(2)
|
∂
ξ
1
α
1
⋯
ξ
k
α
k
σ
(
ξ
1
,
…
,
ξ
m
)
|
≤
C
α
1
,
…
,
α
m
(
|
ξ
1
|
+
⋯
+
|
ξ
m
|
)
-
(
|
α
1
|
+
⋯
|
α
k
|
)
for all |
α
1
|
+
⋯
|
α
m
|
≤
s with s
≥
2
m
n
+
1, then T
σ is bounded from L
p
1
(
R
n
)
×
⋯
×
L
p
m
(
R
n
) to L
p
(
R
n
) for all 1
<
p, p
1
,
…
,
p
m
<
∞ with 1
/
p
=
∑
k
=
1
m
1
/
p
k. For the case of s
≥
m
n
+
1, Kenig and Stein [3] and Grafakos and Torres [4] improved Coifman and Meyer’s multiplier theorem to the indices 1
/
m
≤
p
≤
1 by the multilinear Calderón-Zygmund operator theory. In the last several years, considerable attention has been paid to the behavior on function spaces for T
σ when the multiplier satisfies certain Sobolev regularity condition. Let Φ
∈
S
(
R
m
n
) satisfy
(3)
supp
⁡
Φ
⊂
{
(
ξ
1
,
…
,
ξ
m
)
:
1
2
≤
∑
‍
k
=
1
m
|
ξ
k
|
≤
2
}
,
∑
‍
l
∈
Z
Φ
(
2
l
ξ
1
,
…
,
2
l
ξ
m
)
=
1
∀
(
ξ
1
,
…
,
ξ
m
)
∈
R
m
n
∖
{
0
}
.
For l
∈
Z, set
(4)
σ
l
(
ξ
1
,
…
,
ξ
m
)
:
=
σ
(
2
-
l
ξ
1
,
…
,
2
-
l
ξ
m
)
Φ
(
ξ
1
,
…
,
ξ
m
)
,
∥
σ
l
∥
W
s
(
R
m
n
)
≔
(
∫
R
2
n
‍
(
1
+
|
ξ
1
|
2
+
⋯
+
|
ξ
m
|
2
)
s
∫
R
2
n
‍
×
|
σ
^
(
ξ
1
,
…
,
ξ
m
)
|
2
d
ξ
→
)
1
/
2
.
Tomita [5] proved that if
(5)
sup
⁡
l
∈
Z
∥
σ
l
∥
W
s
(
R
m
n
)
<
∞
for some s
∈
(
m
n
/
2
,
∞
), then T
σ is bounded from L
p
1
(
R
n
)
×
⋯
×
L
p
m
(
R
n
) to L
p
(
R
n
) provided that 1
<
p
,
p
1
,
…
,
p
m
<
∞ and 1
/
p
=
∑
k
=
1
m
1
/
p
k. Grafakos and Si [6] considered the mapping properties from L
p
1
(
R
n
)
×
⋯
×
L
p
m
(
R
n
) to L
p
(
R
n
) for T
σ when p
≤
1. Let σ satisfy the Sobolev regularity that
(6)
∥
σ
l
∥
W
s
1
,
…
,
s
m
(
R
m
n
)
≔
(
∫
R
m
n
‍
〈
ξ
1
〉
2
s
1
⋯
〈
ξ
m
〉
2
s
2
∫
R
m
n
‍
×
|
σ
^
l
(
ξ
1
,
…
,
ξ
m
)
|
2
d
ξ
→
)
1
/
2
,
where 〈
ξ
k
〉
:
=
(
1
+
|
ξ
k
|
2
)
1
/
2. Miyachi and Tomita [7] proved that if
(7)
sup
⁡
l
∈
Z
∥
σ
l
∥
W
s
1
,
…
,
s
m
(
R
m
n
)
<
∞
for some s
k
∈
(
n
/
2
,
n
]
(
k
=
1
,
…
,
m
), then T
σ is bounded from L
p
1
(
R
n
)
×
⋯
×
L
p
m
(
R
n
) to L
p
(
R
n
) provided that 1
<
p
1
,
…
,
p
m
<
∞ with 1
/
p
=
∑
k
=
1
m
‍
1
/
p
k.As well known, whenσ satisfies (3) for some s
≥
m
n
+
1, then T
σ is a standard multilinear Calderón-Zygmund operator, and then by the weighted estimates with multiple weights for multilinear Calderón-Zygmund operators, which were estimated by Lerner et al. [8], we know that, for any p
1
,
…
,
p
m
∈
[
1
,
∞
) and p
∈
(
0
,
∞
) with 1
/
p
=
∑
k
=
1
m
1
/
p
k and weights w
1
,
…
,
w
m such that w
→
=
(
w
1
,
…
,
w
m
)
∈
A
p
→
(
R
m
n
),
(8)
∥
T
σ
(
f
1
,
…
,
f
m
)
∥
L
p
,
∞
(
R
n
,
ν
w
→
)
≲
∏
k
=
1
m
∥
f
k
∥
L
p
k
(
R
n
,
w
k
)
.
By a suitable kernel estimate and the theory of multilinear singular integral operator, Bui and Duong [9] established the weighted estimates with multiple weights for T
σ when σ satisfies (3) for m
=
2 and s
∈
(
n
,
2
n
]. Hu and Yi [10] considered the behavior on L
p
1
(
R
n
)
×
⋯
×
L
p
m
(
R
n
) for T
σ
,
Σ
b when σ satisfies (5) for s
1
,
…
,
s
m
∈
(
n
/
2
,
n
] and showed that T
σ
,
Σ
b enjoys the same L
p
1
(
R
n
)
×
⋯
×
L
p
m
(
R
n
)
→
L
p
(
R
n
) mapping properties as that of the operator T
σ.Now, considerable attention has been paid to the behavior on the compactness of multilinear Fourier multipliers operator with Sobolev regularity. LetVMO
(
R
n
) be the closure of C
c
∞ in the BMO
(
R
n
) topology, which coincides with the space of functions of vanishing mean oscillation (see [11, 12]). Bényi and Torres [13] proved that if b
1
,
…
,
b
m
∈
VMO
(
R
n
) and T is multilinear Calderón-Zygmund operator, then, for p
1
,
…
,
p
m
∈
(
1
,
∞
)
,
p
∈
[
1
,
∞
) with 1
/
p
=
∑
k
=
1
m
‍
1
/
p
k, the commutator T
σ
,
Σ
b is compact operator from L
p
1
(
R
n
)
×
⋯
×
L
p
m
(
R
n
) to L
p
(
R
n
). Hu [14] proved that if σ is a multilinear multiplier which satisfies (5) for some s
∈
(
m
n
/
2
,
m
n
], t
1
,
…
,
t
m
∈
[
1,2
)
t
k
=
n
/
s
k, b
1
,
…
,
b
m
∈
VMO
(
R
n
), and p
k
∈
(
t
k
,
∞
) for k
=
1
,
…
,
m and p
∈
(
1
,
∞
) with 1
/
p
=
∑
k
=
1
m
‍
1
/
p
k, then T
σ
,
Σ
b is compact operators from L
p
1
(
R
n
)
×
⋯
×
L
p
m
(
R
n
) to L
p
(
R
n
). Bényi et al. [15] proved that if b
→
∈
VMO
(
R
n
)
×
⋯
×
VMO
(
R
n
) and T is multilinear Calderón-Zygmund operator, w
→
∈
A
p
×
⋯
×
A
p, then, for p
1
,
…
,
p
m
∈
(
1
,
∞
)
,
p
∈
(
1
,
∞
) with 1
/
p
=
∑
k
=
1
m
‍
1
/
p
k, the commutator T
σ
,
Σ
b is compact operator from L
p
1
(
R
n
,
w
1
)
×
⋯
×
L
p
m
(
R
n
,
w
m
) to L
p
(
R
n
,
ν
w
→
).Inspired by the above, we consider the weighted compactness of the commutatorT
σ
,
Σ
b of the multilinear Fourier multiplier operator on L
p
(
R
n
).Given a multilinear Fourier multiplier operatorT
σ, b
1
,
…
,
b
m
∈
BMO
(
R
n
) and b
→
=
(
b
1
,
…
,
b
m
), the commutator T
σ
,
Σ
b
(
f
→
)
(
x
) is defined by
(9)
T
σ
,
Σ
b
(
f
→
)
(
x
)
≔
∑
‍
k
=
1
m
[
b
k
,
T
σ
]
k
(
f
1
,
…
,
f
m
)
(
x
)
,
with
(10)
[
b
k
,
T
σ
]
k
(
f
1
,
…
,
f
k
,
…
,
f
m
)
(
x
)
≔
b
k
(
x
)
T
σ
(
f
1
,
…
,
f
k
,
…
,
f
m
)
(
x
)
-
T
σ
(
f
1
,
…
,
b
k
f
k
,
…
,
f
m
)
(
x
)
.Our main results are stated as follows.Theorem 1.
Suppose thatσ be a multilinear multiplier which satisfies (7) for some s
k
∈
(
n
/
2
,
n
]
(
k
=
1
,
…
,
m
) and t
1
,
…
,
t
m
∈
[
1,2
). Let t
k
=
n
/
s
k, p
k
∈
(
t
k
,
∞
) for k
=
1
,
…
,
m and 1
/
p
=
∑
k
=
1
m
‍
1
/
p
k with p
>
1. If the weights w
1
,
…
,
w
m satisfy w
→
∈
∏
k
=
1
m
‍
A
p
k
/
t
k, then, for any b
1
,
…
,
b
m
∈
B
M
O
(
R
n
), T
σ
,
Σ
b is bounded from L
p
1
(
R
n
,
w
1
)
×
⋯
×
L
p
m
(
R
n
,
w
m
) to L
p
(
R
n
,
ν
w
→
).Theorem 2.
Suppose thatσ be a multilinear multiplier which satisfies (7) for some s
k
∈
(
n
/
2
,
n
]
(
k
=
1
,
…
,
m
) and t
1
,
…
,
t
m
∈
[
1,2
). Let t
k
=
n
/
s
k, p
k
∈
(
t
k
,
∞
) for k
=
1
,
…
,
m and 1
/
p
=
∑
k
=
1
m
‍
1
/
p
k with p
>
1. If the weights w
1
,
…
,
w
m satisfy w
→
∈
∏
k
=
1
m
‍
A
p
k
/
t
k, then, for any b
1
,
…
,
b
m
∈
V
M
O
(
R
n
), T
σ
,
Σ
b is a compact operator from L
p
1
(
R
n
,
w
1
)
×
⋯
×
L
p
m
(
R
n
,
w
m
) to L
p
(
R
n
,
ν
w
→
).Because the regularity condition∥
σ
l
∥
W
s
(
R
m
n
)
<
∞ is stronger than ∥
σ
l
∥
W
s
1
,
…
,
s
m
(
R
m
n
)
<
∞, we have the following corollaries.Corollary 3.
Suppose thatσ be a multilinear multiplier which satisfies (5) for some s
∈
(
m
n
/
2
,
m
n
]. Let r
=
m
n
/
s, p
k
∈
(
m
n
/
s
,
∞
) for k
=
1
,
…
,
m and 1
/
p
=
∑
k
=
1
m
‍
1
/
p
k with p
>
1. If the weights w
1
,
…
,
w
m satisfy w
→
∈
∏
k
=
1
m
‍
A
p
k
/
r, then, for any b
1
,
…
,
b
m
∈
B
M
O
(
R
n
), T
σ
,
Σ
b is bounded from L
p
1
(
R
n
,
w
1
)
×
⋯
×
L
p
m
(
R
n
,
w
m
) to L
p
(
R
n
,
ν
w
→
).Corollary 4.
Suppose thatσ be a multilinear multiplier which satisfies (5) for some s
∈
(
m
n
/
2
,
m
n
]. Let r
=
m
n
/
s, p
k
∈
(
m
n
/
s
,
∞
) for k
=
1
,
…
,
m and 1
/
p
=
∑
k
=
1
m
‍
1
/
p
k with p
>
1. If the weights w
1
,
…
,
w
m satisfy w
→
∈
∏
k
=
1
m
‍
A
p
k
/
r, then, for any b
1
,
…
,
b
m
∈
V
M
O
(
R
n
), T
σ
,
Σ
b is a compact operator from L
p
1
(
R
n
,
w
1
)
×
⋯
×
L
p
m
(
R
n
,
w
m
) to L
p
(
R
n
,
ν
w
→
).The paper is organized as follows. In Section2, we give some necessary notion and lemmas. In Section 3, we prove our main results, Theorems 1 and 2. Throughout the paper, C always denotes a positive constant that may vary from line to line but remains independent of the main variables. We use the symbol A
≲
B to indicate that there exists a positive constant C such that A
≤
C
B. We use B
(
x
,
R
) to denote a ball centered at x with radius R. For a ball B
⊂
R
n and λ
>
0, we use λ
B to denote the ball concentric with B whose radius is λ times of B
′
s. As usual, |
E
| denotes the Lebesgue measure of a measurable set E in R
n and χ
E denotes thecharacteristic function of E. For p
≥
1, we denote by p
′
=
p
/
(
p
-
1
) the dual exponent of p.


## 2. Some Notations and Lemmas

Let us first introduce some definitions below.Definition 5.
Letm
≥
1 be an integer, and let w
1
,
…
,
w
m be weights, p
1
,
…
,
p
m, p
∈
(
0
,
∞
), with 1
/
p
=
∑
k
=
1
m
‍
1
/
p
k, r
k
∈
(
0
,
p
k
]
(
1
≥
k
≥
m
). Set w
→
=
(
w
1
,
…
,
w
m
) and ν
w
→
:
=
∏
k
=
1
m
‍
w
k
p
/
p
k. Then
(11)
w
→
∈
A
p
1
/
r
1
×
⋯
×
A
p
m
/
r
m
⟹
{
ϑ
k
≔
w
k
1
-
(
p
k
/
r
k
)
′
∈
A
m
(
p
k
/
r
k
)
′
,
k
=
1
,
…
,
m
,
ν
w
→
∈
A
m
p
/
r
.Definition 6.
For anyf
→
:
=
(
f
1
,
…
,
f
m
), r
→
:
=
(
r
1
,
…
,
r
m
) and p
≥
1, M is defined by
(12)
M
r
→
(
f
→
)
(
x
)
≔
sup
⁡
x
∈
Q
∏
‍
k
=
1
m
(
1
|
Q
|
∫
Q
‍
|
f
k
(
y
k
)
|
r
k
d
y
k
)
1
/
r
k
.
For δ
>
0, M
δ is the maximal function
(13)
M
δ
(
f
)
(
x
)
≔
M
(
|
f
|
δ
)
1
/
δ
(
x
)
=
(
sup
⁡
x
∈
Q
1
|
Q
|
∫
Q
‍
|
f
(
y
)
|
δ
d
y
)
1
/
δ
.
The sharp maximal function M
# of Fefferman-Stein is defined by
(14)
M
#
(
f
)
(
x
)
≔
sup
⁡
x
∈
Q
inf
⁡
c
1
|
Q
|
∫
Q
‍
|
f
(
y
)
-
c
|
d
y
=
sup
⁡
x
∈
Q
1
|
Q
|
∫
Q
‍
|
f
(
y
)
-
f
Q
|
d
y
,
where f
Q
:
=
(
1
/
|
Q
|
)
∫
Q
‍
f
(
y
)
d
y.Next, we give some symbols.Letσ
∈
L
∞
(
R
m
n
) and Φ
∈
S
(
R
m
n
) satisfy (3). For l
∈
Z, define
(15)
σ
~
l
(
ξ
1
,
…
,
ξ
m
)
≔
Φ
(
2
l
ξ
1
,
…
,
2
l
ξ
m
)
σ
(
ξ
1
,
…
,
ξ
m
)
.
Then
(16)
σ
~
l
(
ξ
1
,
…
,
ξ
m
)
=
Φ
(
2
l
ξ
1
,
…
,
2
l
ξ
m
)
,
σ
~
ˇ
l
=
2
-
l
m
n
σ
ˇ
l
(
2
-
l
ξ
1
,
…
,
2
-
l
ξ
m
)
,
where f
ˇ denotes the inverse Fourier transform of f.ForN
∈
N, let
(17)
σ
N
(
ξ
1
,
…
,
ξ
m
)
≔
∑
‍
|
l
|
≤
N
σ
~
l
(
ξ
1
,
…
,
ξ
m
)
and denote by T
σ
N the multiplier operator associated with σ
N. It is obvious that T
σ
N is an m-linear singular operator with kernel
(18)
K
N
(
x
;
y
1
,
…
,
y
m
)
:
=
σ
ˇ
N
(
x
-
y
1
,
…
,
x
-
y
m
)
.
For an integer k with 1
≤
k
≤
2 and x
,
x
′
,
y
1
,
…
,
y
m
∈
R
n, let
(19)
W
N
(
x
,
x
′
;
y
1
,
…
,
y
m
)
≔
K
N
(
x
;
y
1
,
…
,
y
m
)
-
K
N
(
x
′
;
y
1
,
…
,
y
m
)
.Assume thatT is a multilinear operator initially defined on the m-fold product of Schwartz spaces, and, taking values in the space of tempered distributions,
(20)
T
:
S
(
R
n
)
×
⋯
×
S
(
R
n
)
⟶
S
′
(
R
n
)
.
By the associated kernel K
(
x
,
y
1
,
…
,
y
m
), we mean that K is a function defined off the diagonal x
=
y
1
=
⋯
=
y
m in R
(
m
+
1
)
n, satisfying
(21)
T
(
f
1
,
…
,
f
m
)
(
x
)
≔
∫
R
m
n
‍
K
(
x
;
y
1
,
…
,
y
m
)
f
1
(
y
1
)
⋯
f
m
×
(
y
m
)
d
y
→
,
for all functions f
k
∈
S
(
R
n
) and all x
∉
⋂
k
=
1
m
supp
⁡
f
k. It is easy to see that the associated kernel K
(
x
,
y
1
,
…
,
y
m
) to Fourier multiplier operator T
σ is given by
(22)
K
(
x
,
y
1
,
…
,
y
m
)
≔
σ
ˇ
(
x
-
y
1
,
…
,
x
-
y
m
)
.To prove main results, we need the following lemmas.By the reverse Hölder inequality, we have the first lemma.Lemma 7.
Assume thatw
→
∈
∏
k
=
1
m
‍
A
p
k
/
t
k, with t
1
,
…
,
t
m
∈
[
1,2
), p
k
∈
(
t
k
,
∞
)
(
k
=
1
,
…
,
m
), and 1
/
p
=
∑
k
=
1
m
1
/
p
k with p
>
1. Let s
k
∈
(
n
/
2
,
n
]; then there exists a constant ϵ
k
∈
(
1
,
min
⁡
{
p
k
/
t
k
,
s
k
/
(
s
k
-
1
)
,
2
s
k
/
n
}
) such that w
→
k
∈
A
p
k
/
(
t
j
ϵ
k
).Forp
1
,
…
,
p
m
∈
(
0
,
∞
) and s
1
,
…
,
s
m
∈
R, the weighted Lebesgue space of mixed type L
(
p
1
,
…
,
p
m
)
(
ω
(
s
1
,
…
,
s
m
)
) is defined by the norm
(23)
∥
F
∥
L
(
p
1
,
…
,
p
m
)
(
ω
(
s
1
,
…
,
s
m
)
)
=
(
(
∫
R
n
‍
|
F
(
x
)
|
p
1
〈
x
1
〉
s
1
d
x
1
)
p
2
/
q
1
〈
x
2
〉
s
2
d
x
2
)
p
3
/
q
2
∫
R
n
‍
⋯
(
∫
R
n
‍
(
∫
R
n
‍
|
F
(
x
)
|
p
1
〈
x
1
〉
s
1
d
x
1
)
p
2
/
q
1
00000
(
∫
R
n
‍
|
F
(
x
)
|
p
1
〈
x
1
〉
s
1
d
x
1
)
p
2
/
q
1
00000
×
〈
x
2
〉
s
2
d
x
2
)
p
3
/
q
2
⋯
〈
x
m
〉
s
m
d
x
m
)
1
/
p
m
,
where x
:
=
(
x
1
,
…
,
x
m
)
∈
R
n
×
⋯
×
R
n.Lemma 8 (see [16]).
Letr
>
0, 2
≤
p
j
<
∞, and s
j
≥
0 for 1
≤
j
≤
m. Then there exists a constant C
>
0 such that
(24)
∥
F
^
∥
L
(
p
1
,
…
,
p
m
)
(
ω
(
s
1
,
…
,
s
m
)
)
≤
C
∥
F
∥
W
s
1
/
p
1
,
…
,
s
m
/
p
m
for all F
∈
W
s
1
/
p
1
,
…
,
s
m
/
p
m
(
R
m
n
) with supp
⁡
F
⊂
{
|
x
1
|
2
⋯
+
|
x
m
|
2
≤
r
}.The following lemma is the key to our main lemma.Lemma 9.
Suppose thatσ be a multilinear multiplier which satisfies (7) for some s
k
∈
(
n
/
2
,
n
]
(
k
=
1
,
…
,
m
). Let 0
<
δ
<
r, 1
/
r
=
1
/
r
1
+
⋯
+
1
/
r
m, r
j
=
ϵ
j
t
j, and ϵ
j is the same as that appears in Lemma 7. Then for all f
→
∈
L
p
1
(
R
n
)
×
⋯
×
L
p
m
(
R
n
) with r
j
≤
p
j
<
∞ for 1
≤
j
≤
m,
(25)
M
δ
#
(
T
σ
(
f
→
)
)
(
x
)
≤
C
M
r
→
(
f
→
)
(
x
)
,
where r
→
=
(
r
1
,
…
,
r
m
).Proof.
By Lemma8, 1
<
t
j
ϵ
j
≤
2; then r
j
/
m
≤
1. Fix a point x and a cube Q such that x
∈
Q. It suffices to prove
(26)
(
1
|
Q
|
∫
Q
‍
|
T
σ
(
f
→
)
(
z
)
-
c
Q
|
δ
d
z
)
1
/
δ
≤
C
M
r
→
(
f
→
)
(
x
)
,
for some constant c
Q. We decompose f
j
=
f
j
0
+
f
j
∞ with f
j
0
=
f
j
χ
Q
⋆ for all j
=
1
,
…
,
m and Q
⋆
=
4
n
Q. Then
(27)
∏
j
=
1
m
‍
f
j
(
y
j
)
=
∏
j
=
1
m
‍
(
f
j
0
(
y
j
)
+
f
j
∞
(
y
j
)
)
=
∑
α
1
,
…
,
α
m
∈
{
0
,
∞
}
‍
f
1
α
1
(
y
1
)
⋯
f
m
α
m
(
y
m
)
=
∏
j
=
1
m
‍
f
j
0
(
y
j
)
+
∑
α
1
,
…
,
α
m
∈
I
‍
f
1
α
1
(
y
1
)
⋯
f
m
α
m
(
y
m
)
,
where I
=
{
α
1
,
…
,
α
m
:
there
is
at
least
one
α
j
≠
0
}. Then we can write
(28)
T
σ
(
f
→
)
(
z
)
=
T
σ
(
f
→
0
)
(
z
)
+
∑
α
1
,
…
,
α
m
∈
I
T
σ
(
f
1
α
1
⋯
f
m
α
m
)
(
z
)
≔
I
+
I
I
.
Applying Kolmogorov’s inequality to I, we have
(29)
(
1
|
Q
|
∫
Q
‍
|
T
σ
(
f
→
0
)
(
z
)
|
δ
d
z
)
1
/
δ
≤
C
∥
T
σ
(
f
→
0
)
∥
L
r
,
∞
(
Q
,
d
x
/
|
Q
|
)
≤
C
∏
j
=
1
m
‍
(
1
|
Q
⋆
|
∫
Q
⋆
‍
|
f
j
(
y
j
)
|
r
l
d
y
j
)
r
j
≤
C
M
r
→
(
f
→
)
(
x
)
,
since T
σ is bounded from L
r
1
×
⋯
×
L
r
m to L
r.
Take(30)
c
Q
=
∑
α
1
,
…
,
α
m
∈
I
T
σ
(
f
1
α
1
⋯
f
m
α
m
)
(
x
)
.
We claim that, for any z
∈
Q,
(31)
∑
α
1
,
…
,
α
m
∈
I
|
T
σ
(
f
1
α
1
⋯
f
m
α
m
)
(
z
)
-
T
σ
(
f
1
α
1
⋯
f
m
α
m
)
(
x
)
|
≤
C
M
r
→
(
f
→
)
(
x
)
.
Let
(32)
W
l
(
x
,
z
;
y
1
,
…
,
y
m
)
=
σ
ˇ
l
(
x
-
y
1
,
…
,
x
-
y
m
)
-
σ
ˇ
l
(
z
-
y
1
,
…
,
z
-
y
m
)
.
At first, we consider the case α
1
=
⋯
=
α
m. We get
(33)
|
T
σ
(
f
1
∞
⋯
f
m
∞
)
(
z
)
-
T
σ
(
f
1
∞
⋯
f
m
∞
)
(
x
)
|
≤
∑
l
∈
Z
|
T
σ
l
(
f
1
∞
⋯
f
m
∞
)
(
z
)
-
T
σ
l
(
f
1
∞
⋯
f
m
∞
)
(
x
)
|
≤
∑
l
∈
Z
∫
R
m
n
∖
(
Q
⋆
)
m
‍
|
W
l
(
x
,
z
;
y
1
,
…
,
y
m
)
|
∏
j
=
1
m
‍
f
j
(
y
j
)
d
y
→
≤
∑
l
∈
Z
∑
k
=
0
∞
∬
(
2
k
+
1
Q
⋆
)
∖
(
2
k
Q
⋆
)
|
W
l
(
x
,
z
;
y
1
,
…
,
y
m
)
|
∏
j
=
1
m
f
j
(
y
j
)
d
y
→
≤
∑
k
=
0
∞
∑
l
∈
Z
∏
j
=
1
m
(
∫
(
2
k
+
1
Q
⋆
)
m
‍
|
f
j
(
y
j
)
|
r
j
d
y
j
)
1
/
r
j
×
(
∫
(
2
k
+
1
Q
⋆
)
∖
(
2
k
Q
⋆
)
‍
W
l
×
(
x
,
z
;
y
1
,
…
,
y
m
)
|
r
1
′
d
y
1
)
r
2
′
/
r
1
′
⋯
)
r
m
′
/
r
m
-
1
′
∫
(
2
k
+
1
Q
⋆
)
∖
(
2
k
Q
⋆
)
(
∫
(
2
k
+
1
Q
⋆
)
∖
(
2
k
Q
⋆
)
‍
W
l
×
(
x
,
z
;
y
1
,
…
,
y
m
)
|
r
1
′
d
y
1
)
r
2
′
/
r
1
′
∫
(
2
k
+
1
Q
⋆
)
∖
(
2
k
Q
⋆
)
⋯
(
∫
(
2
k
+
1
Q
⋆
)
∖
(
2
k
Q
⋆
)
‍
|
W
l
(
x
,
z
;
y
1
,
…
,
y
m
)
|
r
1
′
∫
(
2
k
+
1
Q
⋆
)
∖
(
2
k
Q
⋆
)
‍
d
y
1
)
r
2
′
/
r
1
′
⋯
)
r
m
′
/
r
m
-
1
′
d
y
m
)
1
/
r
m
′
≔
∑
k
=
0
∞
∑
l
∈
Z
‍
∏
j
=
1
m
‍
(
∫
(
2
k
+
1
Q
⋆
)
m
‍
|
f
j
(
y
j
)
|
r
j
d
y
j
)
1
/
r
j
I
I
k
,
l
∞
,
…
,
∞
.
Denote h
=
z
-
x, Q
~
=
x
-
Q
⋆, and l
(
Q
) the side length of a cube Q; it follows from Lemma 7 that
(34)
I
I
k
,
l
∞
,
…
,
∞
=
(
∫
(
2
k
+
1
Q
~
)
∖
(
2
k
Q
~
)
⋯
(
∫
(
2
k
+
1
Q
~
)
∖
(
2
k
Q
~
)
⋯
)
r
m
′
/
r
m
-
1
′
∫
(
2
k
+
1
Q
~
)
∖
(
2
k
Q
~
)
(
∫
(
2
k
+
1
Q
~
)
∖
(
2
k
Q
~
)
⋯
(
∫
(
2
k
+
1
Q
~
)
∖
(
2
k
Q
~
)
∫
(
2
k
+
1
Q
~
)
∖
(
2
k
Q
~
)
|
σ
ˇ
l
(
h
+
y
1
,
…
,
h
+
y
m
)
-
σ
ˇ
l
(
y
1
,
…
,
y
m
)
|
r
1
′
d
y
1
)
r
2
′
/
r
1
′
∫
(
2
k
+
1
Q
~
)
∖
(
2
k
Q
~
)
⋯
(
∫
(
2
k
+
1
Q
~
)
∖
(
2
k
Q
~
)
⋯
)
r
m
′
/
r
m
-
1
′
d
y
m
)
1
/
r
m
′
≤
2
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
⋯
)
r
m
′
/
r
m
-
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
|
<
c
2
2
k
+
1
l
(
Q
)
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
⋯
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
|
σ
ˇ
l
(
y
1
,
…
,
y
m
)
|
r
1
′
|
σ
ˇ
l
(
y
1
,
…
,
y
m
)
|
r
1
′
d
y
1
)
r
2
′
/
r
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
⋯
)
r
m
′
/
r
m
-
1
′
d
y
m
)
1
/
r
m
′
≤
C
(
2
k
l
(
Q
)
)
-
(
s
1
+
⋯
+
s
m
)
×
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
⋯
⋯
)
r
m
′
/
r
m
-
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
|
<
c
2
2
k
+
1
l
(
Q
)
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
⋯
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
|
σ
ˇ
l
(
y
1
,
…
,
y
m
)
|
r
1
′
〈
y
1
〉
s
1
r
1
′
d
y
1
)
r
2
′
/
r
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
⋯
⋯
)
r
m
′
/
r
m
-
1
′
〈
y
m
〉
s
m
r
m
′
d
y
m
)
1
/
r
m
′
≤
C
(
2
k
l
(
Q
)
)
-
(
s
1
+
⋯
+
s
m
)
2
l
(
s
1
+
⋯
+
s
m
)
×
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
‍
⋯
)
r
m
′
/
r
m
-
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
|
<
c
2
2
k
+
1
l
(
Q
)
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
‍
⋯
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
|
2
-
l
m
n
σ
ˇ
l
(
2
-
l
y
1
,
…
,
2
-
l
y
m
)
|
r
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
×
〈
2
-
l
y
1
〉
s
1
r
1
′
d
y
1
)
r
2
′
/
r
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
‍
⋯
)
r
m
′
/
r
m
-
1
′
〈
2
-
l
y
m
〉
s
m
r
m
′
d
y
m
)
1
/
r
m
′
≤
C
(
2
k
l
(
Q
)
)
-
(
s
1
+
⋯
+
s
m
)
2
l
(
s
1
+
⋯
+
s
m
)
2
-
l
m
n
2
-
l
(
n
/
r
1
′
+
⋯
+
n
/
r
m
′
)
×
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
‍
|
σ
ˇ
l
(
z
1
,
…
,
z
m
)
|
r
1
′
×
〈
z
1
〉
s
1
r
1
′
d
z
1
)
r
2
′
/
r
1
′
⋯
)
r
m
′
/
r
m
-
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
|
<
c
2
2
k
+
1
l
(
Q
)
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
‍
|
σ
ˇ
l
(
z
1
,
…
,
z
m
)
|
r
1
′
×
〈
z
1
〉
s
1
r
1
′
d
z
1
)
r
2
′
/
r
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
⋯
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
‍
|
σ
ˇ
l
(
z
1
,
…
,
z
m
)
|
r
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
‍
|
σ
ˇ
l
(
z
1
,
…
,
z
m
)
|
r
1
′
×
〈
z
1
〉
s
1
r
1
′
d
z
1
)
r
2
′
/
r
1
′
⋯
)
r
m
′
/
r
m
-
1
′
〈
z
m
〉
s
m
r
m
′
d
z
m
)
1
/
r
m
′
≤
C
(
2
k
l
(
Q
)
)
-
(
s
1
+
⋯
+
s
m
)
2
-
l
(
n
/
r
1
+
⋯
+
n
/
r
m
-
s
1
-
⋯
-
s
m
)
∥
σ
l
∥
W
s
1
,
…
,
s
m
.
Given that 2
l
0
≤
l
(
Q
)
≤
2
l
0
+
1, we have that
(35)
∑
l
<
l
0
I
I
k
,
l
∞
,
…
,
∞
≤
C
sup
⁡
l
∥
σ
l
∥
W
s
1
,
…
,
s
m
×
∑
l
<
l
0
‍
(
2
k
l
(
Q
)
)
-
(
s
1
+
⋯
+
s
m
)
2
-
l
(
n
/
r
1
+
⋯
+
n
/
r
1
-
s
1
-
⋯
-
s
m
)
≤
C
sup
⁡
l
∥
σ
l
∥
W
s
1
,
…
,
s
m
2
-
k
(
s
1
+
⋯
+
s
m
)
l
(
Q
)
-
(
n
/
r
1
+
⋯
+
n
/
r
m
)
.
On the other hand, a similar process follows that in [17]; we get that
(36)
I
I
k
,
l
∞
,
…
,
∞
=
(
∫
(
2
k
+
1
Q
~
)
∖
(
2
k
Q
~
)
⋯
(
∫
(
2
k
+
1
Q
~
)
∖
(
2
k
Q
~
)
‍
⋯
)
r
m
′
/
r
m
-
1
′
∫
(
2
k
+
1
Q
~
)
∖
(
2
k
Q
~
)
(
∫
(
2
k
+
1
Q
~
)
∖
(
2
k
Q
~
)
⋯
(
∫
(
2
k
+
1
Q
~
)
∖
(
2
k
Q
~
)
‍
∫
(
2
k
+
1
Q
~
)
∖
(
2
k
Q
~
)
‍
|
σ
ˇ
l
(
h
+
y
1
,
…
,
h
+
y
m
)
-
σ
ˇ
l
(
y
1
,
…
,
y
m
)
|
r
1
′
d
y
1
)
r
2
′
/
r
1
′
∫
(
2
k
+
1
Q
~
)
∖
(
2
k
Q
~
)
⋯
(
∫
(
2
k
+
1
Q
~
)
∖
(
2
k
Q
~
)
‍
⋯
)
r
m
′
/
r
m
-
1
′
d
y
m
)
1
/
r
m
′
≤
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
d
y
1
)
r
2
′
/
r
1
′
⋯
)
r
m
′
/
r
m
-
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
|
<
c
2
2
k
+
1
l
(
Q
)
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
d
y
1
)
r
2
′
/
r
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
⋯
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
(
∫
0
1
|
h
→
·
∇
σ
ˇ
l
(
y
1
+
θ
h
,
…
,
y
m
∫
0
1
|
h
→
·
∇
σ
ˇ
l
h
→
+
θ
h
)
|
d
θ
)
r
1
′
d
y
1
)
r
2
′
/
r
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
d
y
1
)
r
2
′
/
r
1
′
⋯
)
r
m
′
/
r
m
-
1
′
d
y
m
)
1
/
r
m
′
≤
∫
0
1
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
|
h
→
·
∇
σ
ˇ
l
(
y
1
+
θ
h
,
…
,
y
m
+
θ
h
)
|
r
1
′
d
y
1
)
(
r
_
2
^
′
)
/
(
r
_
1
^
′
)
⋯
)
(
r
_
m
^
′
)
/
(
r
_
(
m
-
1
)
^
′
)
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
|
<
c
2
2
k
+
1
l
(
Q
)
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
|
h
→
·
∇
σ
ˇ
l
(
y
1
+
θ
h
,
…
,
y
m
+
θ
h
)
|
r
1
′
d
y
1
)
(
r
_
2
^
′
)
/
(
r
_
1
^
′
)
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
|
h
→
·
∇
σ
ˇ
l
(
y
1
+
θ
h
,
…
,
y
m
+
θ
h
)
|
r
1
′
|
h
→
·
∇
σ
ˇ
l
(
y
1
+
θ
h
,
…
,
y
m
+
θ
h
)
|
r
1
′
d
y
1
)
r
2
′
/
r
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
|
h
→
·
∇
σ
ˇ
l
(
y
1
+
θ
h
,
…
,
y
m
+
θ
h
)
|
r
1
′
d
y
1
)
(
r
_
2
^
′
)
/
(
r
_
1
^
′
)
⋯
)
r
m
′
/
r
m
-
1
′
d
y
m
)
1
/
r
m
′
d
θ
≤
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
⋯
)
r
m
′
/
r
m
-
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
|
<
c
2
2
k
+
1
l
(
Q
)
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
⋯
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
|
h
→
·
∇
σ
ˇ
l
(
y
1
,
…
,
y
m
)
|
r
1
′
d
y
1
)
r
2
′
/
r
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
⋯
)
r
m
′
/
r
m
-
1
′
d
y
m
)
1
/
r
m
′
,
where h
→
=
(
h
,
…
,
h
)
∈
(
R
n
)
m. Since
(37)
h
→
·
∇
σ
ˇ
l
(
y
1
,
…
,
y
m
)
=
∑
j
=
1
m
‍
h
j
∂
j
∇
σ
ˇ
l
(
y
1
,
…
,
y
m
)
,
we have
(38)
I
I
k
,
l
∞
,
…
,
∞
≤
∑
j
=
1
m
‍
l
(
Q
)
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
⋯
⋯
)
r
m
′
/
r
m
-
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
|
<
c
2
2
k
+
1
l
(
Q
)
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
⋯
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
|
∂
j
·
σ
ˇ
l
(
y
1
,
…
,
y
m
)
|
r
1
′
d
y
1
)
r
2
′
/
r
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
⋯
⋯
)
r
m
′
/
r
m
-
1
′
d
y
m
)
1
/
r
m
′
≤
∑
j
=
1
m
‍
l
(
Q
)
(
2
k
l
(
Q
)
)
-
(
s
1
+
⋯
+
s
m
)
×
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
⋯
⋯
)
r
m
′
/
r
m
-
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
|
<
c
2
2
k
+
1
l
(
Q
)
‍
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
⋯
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
‍
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
‍
|
∂
j
σ
ˇ
l
(
y
1
,
…
,
y
m
)
|
r
1
′
〈
y
1
〉
s
1
r
1
′
d
y
1
)
r
2
′
/
r
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
⋯
⋯
)
r
m
′
/
r
m
-
1
′
〈
y
m
〉
s
m
r
m
′
d
y
m
)
1
/
r
m
′
≤
C
(
2
k
l
(
Q
)
)
-
(
s
1
+
⋯
+
s
m
)
2
l
(
s
1
+
⋯
+
s
m
)
×
(
×
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
‍
〈
2
-
l
y
1
〉
s
1
r
1
′
d
y
1
)
r
2
′
/
r
1
′
⋯
)
r
m
′
/
r
m
-
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
|
<
c
2
2
k
+
1
l
(
Q
)
‍
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
‍
×
〈
2
-
l
y
1
〉
s
1
r
1
′
d
y
1
)
r
2
′
/
r
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
⋯
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
|
2
-
l
m
n
·
∂
j
σ
ˇ
l
(
2
-
l
y
1
,
…
,
2
-
l
y
m
)
|
r
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
‍
×
〈
2
-
l
y
1
〉
s
1
r
1
′
d
y
1
)
r
2
′
/
r
1
′
⋯
)
r
m
′
/
r
m
-
1
′
×
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
|
<
c
2
2
k
+
1
l
(
Q
)
‍
〈
2
-
l
y
m
〉
s
m
r
m
′
d
y
m
×
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
‍
〈
2
-
l
y
1
〉
s
1
r
1
′
d
y
1
)
r
2
′
/
r
1
′
⋯
)
r
m
′
/
r
m
-
1
′
)
1
/
r
m
′
≤
C
(
2
k
l
(
Q
)
)
-
(
s
1
+
⋯
+
s
m
)
2
l
(
s
1
+
⋯
+
s
m
)
2
-
l
m
n
2
-
l
(
n
/
r
1
′
+
⋯
+
n
/
r
m
′
)
×
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
d
z
1
)
r
2
′
/
r
1
′
⋯
)
r
m
′
/
r
m
-
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
|
<
c
2
2
k
+
1
l
(
Q
)
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
d
z
1
)
r
2
′
/
r
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
m
-
1
|
<
c
2
2
k
+
1
l
(
Q
)
‍
⋯
(
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
|
∂
j
σ
ˇ
l
(
z
1
,
…
,
z
m
)
|
r
1
′
〈
z
1
〉
s
1
r
1
′
d
z
1
)
r
2
′
/
r
1
′
∫
c
1
2
k
l
(
Q
)
≤
|
y
1
|
<
c
2
2
k
+
1
l
(
Q
)
d
z
1
)
r
2
′
/
r
1
′
⋯
)
r
m
′
/
r
m
-
1
′
〈
z
m
〉
s
m
r
m
′
d
z
m
)
1
/
r
m
′
≤
C
(
2
k
l
(
Q
)
)
-
(
s
1
+
⋯
+
s
m
)
2
-
l
(
n
/
r
1
+
⋯
+
n
/
r
m
+
1
-
s
1
-
⋯
-
s
m
)
×
∥
σ
l
∥
W
s
1
,
…
,
s
m
.
From Lemma 8, n
/
r
1
+
⋯
+
n
/
r
1
>
s
1
+
⋯
+
s
m
-
1. It is deduced that
(39)
∑
l
≥
l
0
‍
I
I
k
,
l
∞
,
…
,
∞
≤
C
sup
⁡
l
∥
σ
l
∥
W
s
1
,
…
,
s
m
2
-
l
(
s
1
+
⋯
+
s
m
)
l
(
Q
)
-
(
n
/
r
1
+
⋯
+
n
/
r
m
)
.
So
(40)
∑
α
1
,
…
,
α
m
∈
I
‍
|
T
σ
(
f
1
α
1
⋯
f
m
α
m
)
(
z
)
-
T
σ
(
f
1
α
1
⋯
f
m
α
m
)
(
x
)
|
≤
C
∑
k
=
0
∞
2
-
k
(
s
1
+
⋯
+
s
m
-
n
/
r
1
-
⋯
-
n
/
r
m
)
M
r
→
(
f
→
)
(
x
)
≤
C
M
r
→
(
f
→
)
(
x
)
.
It remains to consider the case that there exists a proper subset {
j
1
,
…
,
j
γ
} of {
1
,
…
,
m
}, 1
≤
γ
<
m, such that α
j
1
=
⋯
=
α
j
γ
=
0. We have
(41)
|
T
σ
(
f
1
α
1
,
…
,
f
m
α
m
)
(
z
)
-
T
σ
(
f
1
α
1
,
…
,
f
m
α
m
)
(
x
)
|
≤
∑
k
=
0
∞
∑
l
∈
Z
∏
j
=
1
m
(
∫
2
k
+
1
Q
⋆
‍
|
f
j
(
y
j
)
|
r
j
d
y
→
)
1
/
r
j
×
(
∫
2
k
+
1
Q
⋆
∖
2
k
Q
⋆
⋯
(
∫
2
k
+
1
Q
⋆
∖
2
k
Q
⋆
‍
(
(
∫
Q
⋆
‍
|
W
0
,
l
(
x
,
z
;
y
1
,
…
,
y
m
)
|
r
1
′
d
y
1
)
r
2
′
/
r
1
′
∫
Q
⋆
‍
⋯
(
∫
Q
⋆
|
W
0
,
l
(
x
,
z
;
y
1
,
…
,
y
m
)
|
r
1
′
d
y
1
)
r
2
′
/
r
1
′
∫
Q
⋆
‍
⋯
⋯
d
y
γ
(
∫
Q
⋆
‍
|
W
0
,
l
(
x
,
z
;
y
1
,
…
,
y
m
)
|
r
1
′
d
y
1
)
r
2
′
/
r
1
′
)
r
γ
+
1
′
/
r
γ
′
d
y
γ
+
1
)
⋯
d
y
m
)
1
/
r
m
′
.
By the same argument as that of the case α
1
=
⋯
=
α
m, we have that
(42)
|
T
σ
(
f
1
α
1
,
…
,
f
m
α
m
)
(
z
)
-
T
σ
(
f
1
α
1
,
…
,
f
m
α
m
)
(
x
)
|
≤
C
M
r
→
(
f
→
)
(
x
)
.
This completes the proof.Lemma 10.
Suppose thatσ be a multilinear multiplier which satisfies (7) for some s
k
∈
(
n
/
2
,
n
]
(
k
=
1
,
…
,
m
). Let 0
<
δ
<
r, 1
/
r
=
1
/
r
1
+
⋯
+
1
/
r
m, and r
j
=
ϵ
j
t
j, and ϵ
j is the same as that appears in Lemma 7. Then, for b
→
∈
(
B
M
O
(
R
n
)
)
m and any γ
→
>
r
→, that is, γ
j
>
r
j
,
j
=
1
,
…
,
m, there exists some constant C
>
0 such that
(43)
M
δ
#
(
T
σ
,
Σ
b
(
f
→
)
)
(
x
)
≤
C
∥
b
→
∥
B
M
O
m
(
M
ε
(
T
σ
(
f
→
)
)
(
x
)
+
M
γ
→
(
f
→
)
(
x
)
)
,
for all m-tuples f
→
=
(
f
1
,
…
,
f
m
) of bounded measurable functions with compact support.The proof of the above lemma is standard. A statement similar to Lemma 2.7 in [17] with minor modifications deduces the estimates. We omit the details here.Lemma 11.
Letσ be a multilinear multiplier which satisfies (7) for some s
k
∈
(
n
/
2
,
n
]
(
k
=
1
,
…
,
m
), r
1
,
…
,
r
m
∈
[
1,2
) such that r
j
s
j
>
n
(
j
=
2
,
…
,
m
). Then, for every ι
∈
(
0
,
s
1
], R
>
0 and x
∈
R
n
∖
2
R,
(44)
∫
R
n
‍
⋯
∫
R
n
‍
∫
|
y
1
|
<
R
‍
|
σ
~
˘
l
(
x
-
y
1
,
…
,
x
-
y
m
)
×
f
1
(
y
1
)
⋯
f
m
(
y
m
)
|
d
y
1
⋯
d
y
m
≲
1
|
x
|
ι
2
-
l
(
ι
-
n
/
r
1
)
∥
f
1
∥
L
p
1
(
R
n
,
w
1
)
∏
‍
i
=
2
m
M
r
k
f
k
1
|
x
|
ι
2
-
l
(
ι
-
n
/
r
1
)
∥
f
1
∥
L
p
1
(
R
n
,
w
1
)
×
(
x
)
ϑ
(
R
)
(
1
/
r
1
)
-
(
1
/
p
1
)
.By a similar way in the proof of the Lemma 2.4 in [14], with slight changes, we can get the conclusion of Lemma 11 and we omit the details.About the proof of compactness, as in [18] we will rely on the Fréchet-Kolmogorov theorem characterizing the precompactness of a set in L
p. More precisely, see Yosida’s book [19]. For more about compactness, we refer to [20, 21].Lemma 12.
A setH is precompact in L
p, ≤
p
<
∞ if and only if(i)
sup
⁡
h
∈
H
⁡
∥
h
∥
L
p
<
∞,
(ii)
lim
⁡
A
→
∞
⁡
∥
h
∥
L
p
(
|
x
|
>
A
)
=
0 uniformly in h
∈
H,
(iii)
lim
⁡
t
→
0
∥
h
(
·
+
t
)
-
h
(
·
)
∥
L
p
=
0 uniformly in h
∈
H.


## 3. Proof of Theorems1 and 2

Proof of Theorem1.
We only present the case thatm
=
2. We have by Lemma 10 and Theorem 3.2 in [8]
(45)
∥
T
Σ
b
(
f
1
,
…
,
f
m
)
∥
L
p
(
R
n
,
ν
w
→
)
≲
∥
M
δ
(
T
Σ
b
(
f
→
)
)
∥
L
p
(
R
n
,
ν
w
→
)
≲
∥
M
δ
#
(
T
Σ
b
(
f
→
)
)
∥
L
p
(
R
n
,
ν
w
→
)
≲
∥
b
→
∥
BM
O
2
(
∥
M
γ
→
(
f
→
)
(
x
)
∥
L
p
(
R
n
,
ν
w
→
)
+
∥
M
γ
→
(
f
→
)
(
x
)
∥
L
p
(
R
n
,
ν
w
→
)
∥
M
ϵ
(
T
)
(
f
→
)
(
x
)
∥
L
p
(
R
n
,
ν
w
→
)
)
≲
∥
b
→
∥
BM
O
2
(
∥
M
γ
→
(
f
→
)
(
x
)
∥
L
p
(
R
n
,
ν
w
→
)
+
∥
M
γ
→
(
f
→
)
(
x
)
∥
L
p
(
R
n
,
ν
w
→
)
∥
M
ϵ
#
(
T
)
(
f
→
)
(
x
)
∥
L
p
(
R
n
,
ν
w
→
)
)
≲
∥
b
→
∥
BM
O
2
(
∥
M
γ
→
(
f
→
)
(
x
)
∥
L
p
(
R
n
,
ν
w
→
)
+
∥
M
γ
→
(
f
→
)
(
x
)
∥
L
p
(
R
n
,
ν
w
→
)
∥
M
(
f
→
)
(
x
)
∥
L
p
(
R
n
,
ν
w
→
)
)
≲
∥
b
→
∥
BM
O
2
∏
‍
k
=
1
m
∥
f
k
∥
L
p
k
(
R
n
,
w
k
)
,
which completes the proof of Theorem 1.Proof of Theorem2.
We will employ some ideas of Bényi and Torres [13]. Without loss of the generality, we only prove the case m
=
2. Let p
k
∈
(
r
k
,
∞
)
(
k
=
1,2
), p
∈
(
1
,
∞
), with 1
/
p
=
∑
‍
1
/
p
k, and b
1
,
b
2
∈
C
c
∞
(
R
n
). Note that, for any f
1
,
f
2
∈
S
(
R
n
) and almost every x
∈
R
n,
(46)
lim
⁡
N
→
∞
T
σ
N
,
Σ
b
(
f
1
,
f
2
)
(
x
)
=
T
σ
,
Σ
b
(
f
1
,
f
2
)
(
x
)
.
It is enough to prove that the following conditions hold:(a)
T
σ
N
,
Σ
b is bounded from L
p
1
(
R
n
,
w
1
)
×
L
p
2
(
R
n
,
w
2
) to L
p
(
R
n
,
ν
w
→
);
(b)
for each fixedϵ
>
0, there exists a constant A
=
A
(
ϵ
) which is independent of N, f
1, and f
2 such that
(47)
(
∫
|
x
|
>
A
‍
|
T
σ
N
,
Σ
b
(
f
1
,
f
2
)
|
p
ν
w
→
(
x
)
d
x
)
1
/
p
≲
ϵ
∏
‍
k
=
1
2
∥
f
k
∥
L
p
k
(
R
n
,
w
k
)
;
(c)
for each fixedϵ
>
0, there exists a constant ρ
=
ρ
(
ϵ
) which is independent of N, f
1, and f
2 such that, for all t with 0
<
|
t
|
<
ρ,
(48)
∥
T
σ
N
,
Σ
b
(
f
1
,
f
2
)
(
·
)
-
T
σ
N
,
Σ
b
(
f
1
,
f
2
)
(
·
+
t
)
∥
L
p
(
R
n
,
ν
w
→
)
≲
ϵ
∏
‍
k
=
1
2
∥
f
k
∥
L
p
k
(
R
n
,
w
k
)
.Then by the Fatou Lemma, the conditions (a), (b), and (c) still hold true if T
σ
N
,
Σ
b
(
f
1
,
f
2
) is replaced by T
σ
,
Σ
b
(
f
1
,
f
2
).
It is clear that the first condition (a) holds according to Theorem1.
Then, we prove the conclusion (b). LetD
>
0 be large enough such that supp
⁡
b
1
⊂
B
D
(
0
)
≔
B
(
0
,
D
) and let A
≥
max
⁡
{
2
D
,
1
}. Then for every x with |
x
|
>
2
A, we have by Lemma 11 that
(49)
∫
R
n
∫
supp
⁡
b
1
|
K
N
(
x
;
y
1
,
y
2
)
f
1
(
y
1
)
f
1
(
y
1
)
|
d
y
1
d
y
2
=
∫
R
n
∫
supp
⁡
b
1
|
∑
‍
|
l
|
≤
N
σ
N
~
˘
(
x
-
y
1
,
x
-
y
2
)
∑
‍
|
l
|
≤
N
×
f
1
(
y
1
)
f
1
(
y
1
)
|
d
y
1
d
y
2
≲
∑
‍
0
≤
l
≤
N
1
|
x
|
s
1
2
-
l
(
s
1
-
n
/
r
1
)
∥
f
1
∥
L
p
1
(
R
n
,
w
1
)
×
M
r
2
f
2
(
x
)
ϑ
(
B
D
(
0
)
)
(
1
/
r
1
)
-
(
1
/
p
1
)
+
∑
‍
-
N
≤
l
<
0
1
|
x
|
θ
2
-
l
(
θ
-
n
/
r
1
)
∥
f
1
∥
L
p
1
(
R
n
,
w
1
)
×
M
r
2
f
2
(
x
)
ϑ
(
B
D
(
0
)
)
(
1
/
r
1
)
-
(
1
/
p
1
)
≲
(
1
|
x
|
s
1
+
1
|
x
|
θ
)
∥
f
1
∥
L
p
1
(
R
n
,
w
1
)
×
M
r
2
f
2
(
x
)
ϑ
(
B
D
(
0
)
)
(
1
/
r
1
)
-
(
1
/
p
1
)
,
if we choose ι
=
s
1 and ι
=
θ
∈
(
n
/
(
p
1
ϵ
1
)
,
n
/
r
1
) in Lemma 11 respectively. Therefore,
(50)
(
∫
|
x
|
>
A
|
T
σ
N
,
b
1
(
f
1
,
f
2
)
|
p
ν
w
→
(
x
)
d
x
)
1
/
p
≲
∥
b
1
∥
L
∞
(
R
n
)
∏
‍
k
=
1
2
∥
f
k
∥
L
p
k
(
R
n
,
w
k
)
ϑ
(
B
D
(
0
)
)
(
1
/
r
1
)
-
(
1
/
p
1
)
×
(
(
∫
|
x
|
>
A
w
1
(
x
)
|
x
|
s
1
p
1
d
x
)
1
/
p
1
+
(
∫
|
x
|
>
A
‍
w
1
(
x
)
|
x
|
θ
p
1
d
x
)
1
/
p
1
)
≲
∥
b
1
∥
L
∞
(
R
n
)
∏
‍
k
=
1
2
∥
f
k
∥
L
p
k
(
R
n
,
w
k
)
ϑ
(
B
D
(
0
)
)
(
1
/
r
1
)
-
(
1
/
p
1
)
×
(
(
∫
|
x
|
>
A
‍
w
1
(
x
)
|
x
|
n
(
p
1
/
t
1
)
d
x
)
1
/
p
1
+
(
∫
|
x
|
>
A
w
1
(
x
)
|
x
|
n
(
p
1
/
(
t
1
ϵ
1
)
)
d
x
)
1
/
p
1
)
≲
ϵ
∥
b
1
∥
L
∞
(
R
n
)
∏
‍
k
=
1
2
∥
f
k
∥
L
p
k
(
R
n
,
w
k
)
,
where the last inequality holds by the fact (see [22, 23]) that for υ
∈
A
p, p
>
1
(51)
∫
R
n
υ
(
x
)
(
1
+
|
x
|
)
n
p
d
x
<
∞
.
This in turn leads to conclusion (b) directly.
We turn our attention to conclusion (c). We write(52)
T
σ
N
,
b
1
(
f
1
,
f
2
)
(
x
)
-
T
σ
N
,
b
1
(
f
1
,
f
2
)
(
x
+
t
)
=
∑
j
=
1
4
E
j
(
x
,
t
)
,
with
(53)
E
1
(
x
,
t
)
=
∫
max
⁡
1
≤
k
≤
2
|
x
-
y
k
|
≥
δ
t
K
N
(
x
;
y
1
,
y
2
)
×
(
b
1
(
x
+
t
)
-
b
1
(
x
)
)
f
1
(
y
1
)
f
2
(
y
2
)
d
y
1
d
y
2
,
E
2
(
x
,
t
)
=
∫
max
⁡
1
≤
k
≤
2
|
x
-
y
k
|
≥
δ
t
(
K
N
(
x
;
y
1
,
y
2
)
-
K
N
(
x
+
t
;
y
1
,
y
2
)
)
×
(
b
1
(
y
1
)
-
b
1
(
x
+
t
)
)
f
1
(
y
1
)
f
2
(
y
2
)
d
y
1
d
y
2
,
E
3
(
x
,
t
)
=
∫
max
⁡
1
≤
k
≤
2
|
x
-
y
k
|
<
δ
t
K
N
(
x
;
y
1
,
y
2
)
×
(
b
1
(
y
1
)
-
b
1
(
x
)
)
f
1
(
y
1
)
f
2
(
y
2
)
d
y
1
d
y
2
,
E
4
(
x
,
t
)
=
∫
max
⁡
1
≤
k
≤
2
|
x
-
y
k
|
<
δ
t
K
N
(
x
+
t
;
y
1
,
y
2
)
×
(
b
1
(
x
+
t
)
-
b
1
(
y
1
)
)
f
1
(
y
1
)
f
2
(
y
2
)
d
y
1
d
y
2
,
with δ
t
>
4
|
t
| a convenient choice to be determined later.
In a completely same way in the proof of Theorem  1.1 in [14], we can obtain the estimate of |
E
j
(
x
,
t
)
|
(
j
=
1
,
…
,
4
). We only list the results and omit the details. 							(54)
|
E
1
(
x
,
t
)
|
≲
|
t
|
∥
∇
b
1
∥
L
∞
(
R
n
)
|
E
1
(
x
,
t
)
|
≲
×
(
∏
‍
k
=
1
2
M
r
k
f
k
(
x
)
+
M
δ
(
T
(
f
1
,
f
2
)
)
(
x
)
)
,
|
E
2
(
x
,
t
)
|
≲
(
|
t
|
δ
t
-
1
)
ϱ
∥
b
1
∥
L
∞
(
R
n
)
|
E
1
(
x
,
t
)
|
≲
×
∏
‍
k
=
1
2
(
M
r
k
f
k
(
x
)
+
M
r
k
f
k
(
x
+
t
)
)
,
|
E
3
(
x
,
t
)
|
≲
δ
t
∥
∇
b
1
∥
L
∞
(
R
n
)
∏
‍
k
=
1
2
M
r
k
f
k
(
x
)
,
|
E
4
(
x
,
t
)
|
≲
δ
t
∥
∇
b
1
∥
L
∞
(
R
n
)
∏
‍
k
=
1
2
M
r
k
f
k
(
x
+
t
)
.
Fix eachϵ
>
0, set
(55)
ρ
=
A
ϵ
2
(
1
+
∥
∇
b
1
∥
L
∞
(
R
n
)
)
,
with
(56)
A
=
min
⁡
{
1
,
(
ϵ
2
(
1
+
∥
b
1
∥
L
∞
(
R
n
)
)
)
1
/
ϱ
}
and δ
t
=
|
t
|
A
-
1 for each t
∈
R
n, where constant ϱ
>
0. Our estimates for those terms of E
j
(
j
=
1
,
…
,
4
) then lead to that when 0
<
|
t
|
<
ρ,
(57)
∥
T
σ
N
,
b
1
(
f
1
,
f
2
)
(
·
)
-
T
σ
N
,
∏
‍
b
(
f
1
,
f
2
)
(
·
+
t
)
∥
L
p
(
R
n
,
ν
w
→
)
≲
(
(
|
t
|
+
δ
t
)
∥
∇
b
1
∥
L
∞
(
R
n
)
+
(
|
t
|
δ
t
-
1
)
ϱ
∥
b
1
∥
L
∞
(
R
n
)
)
×
∏
‍
k
=
1
2
∥
f
k
∥
L
p
k
(
R
n
,
w
k
)
≲
ϵ
∏
‍
k
=
1
2
∥
f
k
∥
L
p
k
(
R
n
,
w
k
)
;
this establishes conclusion (c) and we conclude that T
σ
N
,
b
1 is compact.
In a completely analogous way, ifb
2
∈
C
c
∞, then T
σ
N
,
b
2 is compact. Moreover, then T
σ
N
,
Σ
b is compact, thus we complete the proof of Theorem 2.



---

*Source: 606504-2014-06-01.xml*