# Commutators of Multilinear Singular Integrals on Weighted Herz Spaces

Authors: Baozhen Wang; Huimin Wang
Journal: Journal of Function Spaces
 (2015)

---

## Abstract

Supposeb
→
=
(
b
1
,
…
,
b
m
)
∈
(
B
M
O
)
m, T
Π
b is the iterated commutator of b
→ and the m-linear Calderón-Zygmund operator T. The purpose of this paper is to discuss the boundedness properties of T
Π
b on weighted Herz spaces with general Muckenhoupt weights.


---

## Body

## 1. Introduction and Results

In recent years, the study of multilinear integrals has received more attention, which is motivated by not only the generalization of the linear theory but also the natural appearance of singular integral theory. The initial work on the class of multilinear Calderón-Zygmund operators was done by Coifman and Meyer in [1] and was later systematically studied by Grafakos and Torres in [2–4]. More results on the multilinear Calderón-Zygmund operators and their commutators can be found in [5–10].LetR
n be the n-dimensional Euclidean space, and let (
R
n
)
m
=
R
n
×
⋯
×
R
n be the m-fold product space (
m
∈
N
). We denote by S
(
R
n
) the space of all Schwartz functions on R
n and by S
′
(
R
n
) its dual space, the set of all tempered distributions on R
n.We say that a locally integrable functionK
(
y
0
,
y
1
,
…
,
y
m
) defined away from the diagonal y
0
=
y
1
=
⋯
=
y
m in (
R
n
)
m
+
1 is a kernel in the class m-C
Z
K
(
A
,
ϵ
) if it satisfies the size estimate(1)
K
y
0
,
y
1
,
…
,
y
m
≤
A
∑
k
,
l
=
0
m
y
l
-
y
k
m
nfor some A
>
0 and all (
y
0
,
y
1
,
…
,
y
m
)
∈
(
R
n
)
m
+
1 with y
l
≠
y
k for some l
,
k. Moreover, assume that for some ϵ
>
0 and all 0
≤
j
≤
m it satisfies the smoothness estimates(2)
K
y
0
,
…
,
y
j
,
…
,
y
m
-
K
y
0
,
…
,
y
j
′
,
…
,
y
m
≤
A
y
j
-
y
j
′
ϵ
∑
k
,
l
=
0
m
y
l
-
y
k
m
n
+
ϵwhenever(3)
y
j
-
y
j
′
≤
1
2
max
0
≤
k
≤
m
⁡
y
j
-
y
k
.LetT be a multilinear operator initially defined on the m-fold product of Schwartz spaces and taking values into the space of tempered distributions,(4)
T
:
S
R
n
×
⋯
×
S
R
n
⟶
S
′
R
n
.We say that T is a m-linear Calderón-Zygmund operator if, for some q
1
,
…
,
q
m
∈
[
1
,
∞
) and q
∈
(
0
,
∞
) with 1
/
q
=
∑
i
=
1
m
1
/
q
i, it extends to a bounded multilinear operator from L
q
1
R
n
×
⋯
×
L
q
m
(
R
n
) into L
q
(
R
n
) and if there exists a kernel function K
(
x
,
y
1
,
…
,
y
m
) in the class m-C
Z
K
(
A
,
ϵ
), defined away from the diagonal x
=
y
1
=
⋯
=
y
m in (
R
n
)
m
+
1, satisfying(5)
T
f
→
x
=
T
f
1
,
…
,
f
m
x
=
∫
R
n
m
K
x
,
y
1
,
…
,
y
m
·
f
1
y
1
⋯
f
m
y
m
d
y
1
⋯
d
y
mfor all x
∉
⋂
j
=
1
m
s
u
u
p
f
j and f
1
,
…
,
f
m are C
∞ functions with compact support.Given a locally integrable vector functionb
→
=
(
b
1
,
…
,
b
m
)
∈
(
B
M
O
)
m, the iterated commutator of b
→ and the m-linear Calderón-Zygmund operator T, denoted by T
Π
b, was introduced in [6] and is defined by (6)
T
Π
b
f
→
=
b
1
,
…
,
b
m
-
1
,
b
m
,
T
m
m
-
1
⋯
1
f
→
.To clarify the notation, ifT is associated in the usual way with a Calderón-Zygmund kernel K, then at a formal level(7)
T
Π
b
f
→
x
=
∫
R
n
m
∏
i
=
1
m
b
i
x
-
b
i
y
i
·
K
x
,
y
1
,
…
,
y
m
f
1
y
1
⋯
f
m
y
m
d
y
1
⋯
d
y
m
.SupposeT is an m-linear Calderón-Zygmund operator. Let ω
→
∈
A
q
→, let ν
ω
→
=
∏
i
=
1
m
ω
i
q
/
q
i with 1
/
q
=
1
/
q
1
+
⋯
+
1
/
q
m, and let b
→
∈
(
B
M
O
)
m
. In 2014, Pérez et al. [6] proved that the commutator T
Π
b is bounded from L
q
1
(
ω
1
)
×
⋯
×
L
q
m
(
ω
m
) to L
q
(
ν
ω
→
).LetB
k
=
{
x
∈
R
n
:
x
≤
2
k
} and C
k
=
B
k
∖
B
k
-
1 for any k
∈
Z
. Denote χ
k
=
χ
C
k for k
∈
Z, where χ
C
k is the characteristic function of the set C
k. The following weighted Herz space is introduced by Lu and Yang in [11].Letα
∈
R
,
0
<
p
,
q
<
∞, and ω
1
,
ω
2 be two weight functions on R
n. The homogeneous weighted Herz space K
˙
q
α
,
p
(
ω
1
,
ω
2
)
(
R
n
) is defined by (8)
K
˙
q
α
,
p
ω
1
,
ω
2
R
n
=
f
∈
L
l
o
c
q
R
n
∖
0
,
ω
2
:
f
K
˙
q
α
,
p
ω
1
,
ω
2
<
∞
,where(9)
f
K
˙
q
α
,
p
ω
1
,
ω
2
=
∑
k
=
-
∞
∞
ω
1
B
k
α
p
/
n
f
χ
k
L
ω
2
q
R
n
p
1
/
p
.Obviously, ifα
=
0, then K
˙
q
0
,
q
(
ω
1
,
ω
2
)
(
R
n
)
=
L
ω
2
q
(
R
n
) for any 0
<
q
<
∞
. Thus, the weighted Herz spaces are generalizations of the weighted Lebesgue spaces.In 2000, Lu et al. in [12] proved boundedness results for sublinear operators on weighted Herz spaces with general Muckenhoupt weights. Recently, many authors considered the boundedness of operators and their commutators on weighted Herz spaces. Wang in [13] proved that the intrinsic square functions are bounded on weighted Herz spaces and in [14] he also obtained the boundedness of the intrinsic square functions on weighted Herz type Hardy spaces. In [15], Kuang considered the boundedness of generalized Hausdorff operators on weighted Herz spaces; Hu et al. in [16] established the weighted boundedness for the commutator of fractional integral operators on Herz spaces. The authors also have studied the boundedness of the multilinear integrals on weighted (unweighted) Herz spaces in [17–19]. In this paper, we focus on the boundedness of the iterated commutator of B
M
O and the m-linear Calderón-Zygmund operator on weighted Herz spaces. Our main results in this paper are formulated as follows.Theorem 1.
Supposem
≥
2
,
(
b
1
,
…
,
b
m
)
∈
(
B
M
O
)
m, T is an m-linear Calderón-Zygmund operator, ν
ω
→
=
∏
i
=
1
m
ω
i
q
/
q
i, and ν
∈
A
r
(
1
≤
r
<
∞
)
. For i
=
1
,
…
,
m, let 0
<
p
i
≤
∞, let 1
<
q
i
<
∞, and let ω
i
∈
A
r
i
(
1
≤
r
i
<
q
i
)
. If p
,
q
,
α satisfy (10)
1
p
=
∑
i
=
1
m
1
p
i
,
1
q
=
∑
i
=
1
m
1
q
1
,
α
=
∑
i
=
1
m
α
i
,then T
Π
b is bounded from K
˙
q
1
α
1
,
p
1
ν
,
ω
1
R
n
×
⋯
×
K
˙
q
m
α
m
,
p
m
(
ν
,
ω
m
)
(
R
n
) into K
˙
q
α
,
p
(
ν
,
ν
ω
→
)
(
R
n
) provided that (11)
0
<
α
i
r
<
n
1
-
r
i
q
ifor i
=
1
,
…
,
m
.


## 2. Some Preliminaries

A weightω is a nonnegative, locally integrable function on R
n. The symbol B
=
B
(
x
0
,
r
B
) denotes the ball with center x
0, radius r
B and for any λ
>
0, λ
B will stand for the ball B
(
x
0
,
λ
r
B
). For a given weight function ω and a measurable set E, we also denote the Lebesgue measure of E by E and set weighted measure ω
(
E
)
=
∫
E
ω
(
x
)
d
x.A weightω is said to belong to A
q for 1
<
q
<
∞, if there exists a constant C such that, for every ball B
⊂
R
n,(12)
1
B
∫
B
ω
x
d
x
1
B
∫
B
ω
x
1
-
q
′
d
x
q
-
1
≤
C
,where q
′ is the dual of q such that 1
/
q
+
1
/
q
′
=
1
. The class A
1 is defined by replacing the above inequality with(13)
1
B
∫
B
w
y
d
y
≤
C
·
ess
inf
x
∈
B
⁡
w
x
f
o
r
e
v
e
r
y
b
a
l
l
B
⊂
R
n
.A weightω is said to belong to A
∞ if there are positive numbers C and δ so that(14)
ω
E
ω
B
≤
C
E
B
δfor all balls B and all measurable E
⊂
B. It is well known that(15)
A
∞
=
⋃
1
≤
q
<
∞
A
q
.By (12), we have(16)
∫
B
ω
x
d
x
1
/
q
∫
B
ω
x
1
-
q
′
d
x
1
/
q
′
≤
C
Bfor 1
<
q
<
∞
.The classicalA
q weight theory was first introduced by Muckenhoupt in the study of weighted L
q boundedness of Hardy-Littlewood maximal function in [20].Lemma 2 (see [20, 21]).
Supposeω
∈
A
q
(
R
n
) and the following statements hold.(i)
For any1
≤
q
<
∞, there are positive numbers C and δ such that(17)
ω
B
k
ω
B
j
≤
C
2
n
q
k
-
j
f
o
r
k
>
j
,
(18)
ω
B
k
ω
B
j
≤
C
2
δ
k
-
j
f
o
r
k
<
j
.
(ii)
A
q
1
(
R
n
)
⊂
A
q
2
(
R
n
) for any 1
≤
q
1
<
q
2
≤
∞.
(iii)
For any1
<
q
<
∞, one has ω
1
-
q
′
∈
A
q
′
(
R
n
).A locally integrable functionb is said to be in B
M
O
(
R
n
) if (19)
s
u
p
B
⊂
R
n
1
B
∫
B
b
x
-
b
B
d
x
=
b
∗
<
∞
,where b
B
=
|
B
|
-
1
∫
B
b
(
y
)
d
y.Lemma 3 (see [22]).
Supposeω
∈
A
∞ and b
∈
B
M
O. Then, for any 1
≤
q
<
∞ and r
1
,
r
2
>
0, we have(20)
1
ω
B
x
0
,
r
1
∫
B
x
0
,
r
1
b
x
-
b
B
x
0
,
r
2
q
ω
x
d
x
1
/
q
≤
C
b
∗
1
+
ln
⁡
r
1
r
2
.Now let us recall the definition of multiple weights. Form exponents q
1
,
…
,
q
m, we write q
→
=
(
q
1
,
…
,
q
m
)
. Let q
1
,
…
,
q
m
∈
[
1
,
∞
), and let q
∈
(
0
,
∞
) with 1
/
q
=
∑
k
=
1
m
1
/
q
k
. Given ω
→
=
(
ω
1
,
…
,
ω
m
), set ν
ω
→
=
∏
i
=
1
m
ω
i
q
/
q
i
. We say that ω
→ satisfies the A
q
→ condition if it satisfies(21)
sup
B
⁡
1
B
∫
B
ν
ω
→
x
d
x
1
/
q
·
∏
i
=
1
m
1
B
∫
B
ω
i
x
1
-
q
i
′
x
d
x
1
/
q
i
′
<
∞
.When q
i
=
1, (
1
/
|
B
|
)
∫
B
ω
i
(
x
)
1
-
q
i
′
(
x
)
d
x is understood as (
i
n
f
x
∈
B
ω
i
(
x
)
)
-
1
.Letq
1
,
…
,
q
m
∈
[
1
,
∞
), and let 1
/
q
=
∑
k
=
1
m
1
/
q
k
. By Hölder’s inequality,(22)
ν
ω
→
B
1
/
q
≤
∏
i
=
1
m
ω
i
B
1
/
q
i
.Supposing ω
i
∈
A
q
i
,
i
=
1
,
…
,
m, then by (22) and the definition of multiple weights we have ω
→
∈
A
q
→
.Lemma 4 (see [6]).
Letq
1
,
…
,
q
m
∈
[
1
,
∞
), and let 1
/
q
=
∑
k
=
1
m
1
/
q
k. Then ω
→
=
(
ω
1
,
…
,
ω
m
)
∈
A
q
→ if and only if(23)
ν
ω
→
∈
A
q
m
,
ω
i
1
-
q
i
′
∈
A
m
q
i
′
f
o
r
i
=
1
,
…
,
m
,where ν
ω
→
=
∏
i
=
1
m
ω
i
q
/
q
i and the condition ω
i
1
-
q
i
′
∈
A
m
q
i
′ in the case q
i
=
1 is understood as ω
i
1
/
m
∈
A
1.


## 3. Proof of Theorem1

Without loss of generality, we only prove the theorem for the casem
=
2. The method can be extended for any m-linear case without any essential difficulty.Letν
∈
A
r, and let ω
l
∈
A
r
l
,
l
=
1,2. If k
>
i
,
k
>
j, then from (17) we have(24)
ν
B
k
ν
B
i
≤
C
2
k
-
i
n
r
,
ν
B
k
ν
B
j
≤
C
2
k
-
j
n
r
,
ω
1
B
k
ω
1
B
i
≤
C
2
k
-
i
n
r
1
,
ω
2
B
k
ω
2
B
j
≤
C
2
k
-
j
n
r
2
.If k
<
i
,
k
<
j, then from (18) we have(25)
ν
B
k
ν
B
i
≤
C
2
k
-
i
δ
,
ν
B
k
ν
B
j
≤
C
2
k
-
j
δ
,
ω
1
B
k
ω
1
B
i
≤
C
2
k
-
i
δ
1
,
ω
2
B
k
ω
2
B
j
≤
C
2
k
-
j
δ
2
.We have the following conclusions.Theorem 5.
Suppose(
b
1
,
b
2
)
∈
(
B
M
O
)
2, T is an 2-linear Calderón-Zygmund operator, and ν
∈
A
r
(
1
≤
r
<
∞
). For i
=
1,2, let 1
<
q
i
<
∞, and let ω
i
∈
A
r
i
(
1
≤
r
i
<
q
i
). If q
,
α satisfy 1
/
q
=
1
/
q
1
+
1
/
q
2 and α
=
α
1
+
α
2, then(26)
ν
B
k
α
/
n
T
Π
b
f
χ
i
,
g
χ
j
χ
k
L
q
ν
ω
→
≤
C
b
1
∗
D
1
k
,
i
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
×
b
2
∗
D
2
k
,
j
ν
B
j
α
2
/
n
g
χ
j
L
q
2
ω
2
,where ν
ω
→
=
∏
i
=
1
2
ω
i
q
/
q
i, and(27)
D
l
k
,
i
=
k
-
i
2
k
-
i
α
1
r
+
n
r
1
/
q
1
-
n
i
f
i
≤
k
-
2
;
1
i
f
k
-
1
≤
i
≤
k
-
1
;
i
-
k
2
k
-
i
δ
α
1
/
n
+
δ
1
/
q
1
i
f
i
≥
k
+
2for l
=
1,2.Proof.
Whenk
-
1
≤
i
,
j
≤
k
+
1, then, by the boundedness of T
Π
b from L
q
1
(
ω
1
)
×
L
q
2
(
ω
2
) to L
q
(
ν
ω
→
) and ν
(
B
k
)
~
ν
(
B
i
)
~
ν
(
B
j
), we obtain(28)
ν
B
k
α
/
n
T
Π
b
f
χ
i
,
g
χ
j
χ
k
L
q
ν
ω
→
≤
C
b
1
∗
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
×
b
2
∗
ν
B
j
α
2
/
n
g
χ
j
L
q
2
ω
2
.This means D
1
(
k
,
i
)
=
D
2
(
k
,
j
)
=
1 if k
-
1
≤
i
,
j
≤
k
+
1.
In the other case, we see that(29)
x
-
y
1
+
x
-
y
2
~
2
max
⁡
i
,
j
,
kfor x
∈
C
k
,
y
1
∈
C
i
,
y
2
∈
C
j.
Taking(30)
λ
1
i
=
b
1
B
i
=
1
B
i
∫
B
i
b
1
x
d
x
,
λ
2
j
=
b
2
B
j
=
1
B
j
∫
B
j
b
2
x
d
x
,then(31)
T
Π
b
f
χ
i
,
g
χ
j
x
=
b
1
x
-
λ
1
i
b
2
x
-
λ
2
j
T
f
χ
i
,
g
χ
j
-
b
1
x
-
λ
1
i
T
f
χ
i
,
b
2
-
λ
2
j
g
χ
j
-
b
2
x
-
λ
2
j
T
b
1
-
λ
1
i
f
χ
i
,
g
χ
j
x
+
T
b
1
-
λ
1
i
f
χ
i
,
b
2
-
λ
2
j
g
χ
j
x
=
L
1
x
+
L
2
x
+
L
3
x
+
L
4
x
.Thus(32)
ν
B
k
α
/
n
T
Π
b
f
χ
i
,
g
χ
j
χ
k
L
q
ν
ω
→
≤
∑
m
=
1
4
ν
B
k
α
/
n
L
m
χ
k
L
q
ν
ω
→
=
∑
m
=
1
4
J
m
.
Now, we will estimate eachJ
m
(
m
=
1,2
,
3,4
), separately. Applying (29), (33)
L
1
x
=
b
1
x
-
λ
1
i
b
2
x
-
λ
2
j
T
f
χ
i
,
g
χ
j
x
≤
C
2
-
2
max
⁡
i
,
j
,
k
n
b
1
x
-
λ
1
i
b
2
x
-
λ
2
j
·
f
χ
i
L
1
g
χ
j
L
1
.By Hölder’s inequality and (16),we have(34)
L
1
x
≤
C
2
-
2
max
⁡
i
,
j
,
k
n
+
i
n
+
j
n
b
1
x
-
λ
1
i
·
b
2
x
-
λ
2
j
×
f
χ
i
L
q
1
ω
1
g
χ
j
L
q
2
ω
2
·
∫
B
i
ω
1
x
1
-
q
1
′
d
x
1
/
q
1
′
∫
B
j
ω
2
x
1
-
q
2
′
d
x
1
/
q
2
′
≤
C
2
-
2
max
⁡
i
,
j
,
k
n
+
i
n
+
j
n
b
1
x
-
λ
1
i
b
2
x
-
λ
2
j
×
f
χ
i
L
q
1
ω
1
g
χ
j
L
q
2
ω
2
ω
1
B
i
-
1
/
q
1
ω
2
B
j
-
1
/
q
2
.Then(35)
ν
B
k
α
/
n
L
1
χ
k
L
q
ν
ω
→
≤
C
2
-
2
m
a
x
i
,
j
,
k
n
+
i
n
+
j
n
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
·
ν
B
j
α
2
/
n
g
χ
j
L
q
2
ω
2
×
ν
B
k
ν
B
i
α
1
/
n
·
ν
B
k
ν
B
j
α
2
/
n
ω
1
B
i
-
1
/
q
1
ω
2
B
j
-
1
/
q
2
×
∫
B
k
b
1
x
-
λ
1
i
b
2
x
-
λ
2
j
q
·
ν
ω
→
x
d
x
1
/
q
.By Lemma 4 we know ν
ω
→
∈
A
∞. If q
≥
1
/
2, then by Hölder’s inequality, Lemma 3, and (22) we have(36)
∫
B
k
b
1
x
-
λ
1
i
b
2
x
-
λ
2
j
q
ν
ω
→
x
d
x
1
/
q
≤
∫
B
k
b
1
x
-
λ
1
i
2
q
ν
ω
→
x
d
x
1
/
2
q
·
∫
B
k
b
2
x
-
λ
1
j
2
q
ν
ω
→
x
d
x
1
/
2
q
≤
C
b
1
∗
·
b
2
∗
k
-
i
k
-
j
ν
ω
→
B
k
1
/
q
≤
C
b
1
∗
b
2
∗
·
k
-
i
k
-
j
ω
1
B
k
1
/
q
1
ω
2
B
k
1
/
q
2
.If 0
<
q
<
1
/
2, then by Hölder’s and Lemma 3 we have(37)
∫
B
k
b
1
x
-
λ
1
i
b
2
x
-
λ
2
j
q
ν
ω
→
x
d
x
1
/
q
≤
∫
B
k
b
1
x
-
λ
1
i
ν
ω
→
x
d
x
·
∫
B
k
b
2
x
-
λ
1
j
q
/
1
-
q
ν
ω
→
x
d
x
1
/
q
-
1
≤
C
b
1
∗
k
-
i
ν
ω
→
B
k
·
∫
B
k
b
2
x
-
λ
2
j
ν
ω
→
x
d
x
ν
ω
→
B
k
1
/
q
-
2
≤
C
b
1
∗
b
2
∗
k
-
i
k
-
j
ω
1
B
k
1
/
q
1
ω
2
B
k
1
/
q
2
.
Thus(38)
ν
B
k
α
/
n
L
1
χ
k
L
q
ν
ω
→
≤
C
b
1
∗
k
-
i
·
2
-
max
⁡
i
,
k
n
+
i
n
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
ν
B
k
ν
B
i
α
1
/
n
·
ω
1
B
k
ω
1
B
i
1
/
q
1
×
b
2
∗
k
-
j
·
2
-
m
a
x
j
,
k
n
+
j
n
ν
B
j
α
2
/
n
g
χ
j
L
q
2
ω
2
ν
B
k
ν
B
j
α
2
/
n
·
ω
2
B
k
ω
2
B
j
1
/
q
2
.
Wheni
≤
k
-
2
,
j
≤
k
-
2. Then by (38) and (24), we get(39)
ν
B
k
α
/
n
L
1
χ
k
L
q
ν
ω
→
≤
C
b
1
∗
k
-
i
·
2
k
-
i
α
1
r
+
n
r
1
/
q
1
-
n
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
×
b
2
∗
·
k
-
j
2
k
-
j
α
2
r
+
n
r
1
/
q
2
-
n
ν
B
i
α
2
/
n
f
χ
i
L
q
2
ω
2
.
Wheni
≥
k
+
2
,
j
≥
k
+
2, by (38) and (25), we get(40)
ν
B
k
α
/
n
T
Π
b
f
χ
i
,
g
χ
j
χ
k
L
q
ν
ω
→
≤
C
b
1
∗
i
-
k
·
2
k
-
i
α
1
δ
/
n
+
δ
1
/
q
1
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
×
b
2
∗
·
j
-
k
2
k
-
j
α
2
δ
/
n
+
δ
2
/
q
2
ν
B
j
α
2
/
n
g
χ
i
L
q
2
ω
2
.
Wheni
≤
k
-
2
,
j
≥
k
+
2
. Then by (38) and (24)-(25) we get (41)
ν
B
k
α
/
n
T
Π
b
f
χ
i
,
g
χ
j
χ
k
L
q
ν
ω
→
≤
C
b
1
∗
k
-
i
·
2
k
-
i
α
1
r
+
n
r
1
/
q
1
-
n
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
×
b
2
∗
·
j
-
k
2
k
-
j
α
2
δ
/
n
+
δ
2
/
q
2
ν
B
j
α
2
/
n
g
χ
j
L
q
2
ω
2
.By symmetry, when j
≤
k
-
2
,
i
≥
k
+
2, we have(42)
ν
B
k
α
/
n
T
Π
b
f
χ
i
,
g
χ
j
χ
k
L
q
ν
ω
→
≤
C
b
1
∗
i
-
k
·
2
k
-
i
α
1
δ
/
n
+
δ
1
/
q
1
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
×
b
2
∗
·
k
-
j
2
k
-
j
α
2
r
+
n
r
1
/
q
2
-
n
ν
B
i
α
2
/
n
f
χ
i
L
q
2
ω
2
.
Wheni
≤
k
-
2
,
k
-
1
≤
j
≤
k
+
1
. Noting that ν
(
B
k
)
~
ν
(
B
j
)
,
ω
2
(
B
k
)
~
ω
2
(
B
j
), then(43)
ν
B
k
α
/
n
T
Π
b
f
χ
i
,
g
χ
j
χ
k
L
q
ν
ω
→
≤
C
b
1
∗
k
-
i
·
2
k
-
i
α
1
r
+
n
r
1
/
q
1
-
n
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
×
b
2
∗
·
ν
B
j
α
2
/
n
g
χ
j
L
q
2
ω
2
.Similar to the estimates (43), we can verify (26) in the following case: k
-
1
≤
i
≤
k
+
1
,
j
≤
k
-
2
;
k
-
1
≤
i
≤
k
+
1
,
j
>
k
+
1 and i
>
k
+
1
,
k
-
1
≤
j
≤
k
+
1
. Thus we obtain(44)
ν
B
k
α
/
n
L
1
χ
k
L
q
ν
ω
→
≤
C
b
1
∗
D
1
k
,
i
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
×
b
2
∗
D
2
k
,
j
ν
B
j
α
2
/
n
g
χ
j
L
q
2
ω
2
.
Applying (29) and Hölder’s inequality, (45)
L
2
x
=
b
1
x
-
λ
1
i
T
f
χ
i
,
b
2
-
λ
2
j
g
χ
j
≤
C
2
-
2
max
⁡
i
,
j
,
k
n
b
1
x
-
λ
1
i
f
χ
i
L
1
·
b
2
-
λ
2
j
g
χ
j
L
1
≤
C
2
-
2
max
⁡
i
,
j
,
k
n
b
1
x
-
λ
1
i
·
f
χ
i
L
q
1
ω
1
∫
B
i
ω
1
x
1
-
q
1
′
d
x
1
/
q
1
′
×
g
χ
j
L
q
2
ω
2
·
∫
B
j
b
2
x
-
λ
2
j
q
2
′
ω
2
x
1
-
q
2
′
d
x
1
/
q
2
′
.By Lemma 2 we know ω
2
1
-
q
2
′
∈
A
q
2
′. Then, by Lemma 3, we have(46)
∫
B
j
b
2
x
-
λ
2
j
q
2
′
ω
2
x
1
-
q
2
′
d
x
1
/
q
2
′
≤
C
b
2
∗
∫
B
j
ω
2
x
1
-
q
2
′
d
x
1
/
q
2
′
.Then, by (46) and (16), we have(47)
L
2
x
≤
C
b
2
∗
2
-
2
max
⁡
i
,
j
,
k
n
b
1
x
-
λ
1
i
·
f
χ
i
L
q
1
ω
1
∫
B
i
ω
1
x
1
-
q
1
′
d
x
1
/
q
1
′
×
g
χ
j
L
q
2
ω
2
·
∫
B
j
ω
2
x
1
-
q
2
′
d
x
1
/
q
2
′
≤
C
2
-
max
⁡
i
,
k
n
+
i
n
b
1
x
-
λ
1
i
f
χ
i
L
q
1
ω
1
·
ω
1
B
i
-
1
/
q
1
×
b
2
∗
k
-
j
·
2
-
m
a
x
j
,
k
n
+
j
n
g
χ
j
L
q
2
ω
2
ω
2
B
j
-
1
/
q
2
.Note that, for 0
<
q
<
∞,(48)
∫
B
k
b
1
x
-
λ
1
i
q
ν
ω
→
x
d
x
1
/
q
≤
C
b
1
∗
k
-
i
ν
ω
→
B
k
1
/
q
≤
C
b
1
∗
k
-
i
ω
1
B
k
1
/
q
1
ω
2
B
k
1
/
q
2
.Thus,(49)
ν
B
k
α
/
n
L
2
χ
k
L
q
ν
ω
→
≤
C
b
1
∗
k
-
i
·
2
-
m
a
x
i
,
k
n
+
i
n
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
ν
B
k
ν
B
i
α
1
/
n
·
ω
1
B
k
ω
1
B
i
1
/
q
1
×
b
2
∗
k
-
j
·
2
-
m
a
x
j
,
k
n
+
j
n
ν
B
j
α
2
/
n
g
χ
j
L
q
2
ω
2
ν
B
k
ν
B
j
α
2
/
n
·
ω
2
B
k
ω
2
B
j
1
/
q
2
.Similar to the estimates of (44), we get(50)
ν
B
k
α
/
n
L
2
χ
k
L
q
ν
ω
→
≤
C
b
1
∗
D
1
k
,
i
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
×
b
2
∗
D
2
k
,
j
ν
B
j
α
2
/
n
g
χ
j
L
q
2
ω
2
.
By symmetry, we also get(51)
ν
B
k
α
/
n
L
3
χ
k
L
q
ν
ω
→
≤
C
b
1
∗
D
1
k
,
i
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
×
b
2
∗
D
2
k
,
j
ν
B
j
α
2
/
n
g
χ
j
L
q
2
ω
2
.
Finally, it remains to estimateJ
4. Applying (29) and Hölder’s inequality,(52)
L
4
x
=
T
b
1
-
λ
1
i
f
χ
i
,
b
2
-
λ
2
j
g
χ
j
≤
C
2
-
2
m
a
x
i
,
j
,
k
n
b
1
-
λ
1
i
f
χ
i
L
1
·
b
2
-
λ
2
j
g
χ
j
L
1
≤
C
2
-
2
m
a
x
i
,
j
,
k
n
f
χ
i
L
q
1
ω
1
·
∫
B
i
b
1
x
-
λ
1
i
q
1
′
ω
1
x
1
-
q
1
′
d
x
1
/
q
1
′
×
g
χ
j
L
q
2
ω
2
·
∫
B
j
b
2
x
-
λ
2
j
q
2
′
ω
2
x
1
-
q
2
′
d
x
1
/
q
2
′
.By (46) and (16),(53)
L
4
x
≤
C
b
1
∗
2
-
max
⁡
i
,
k
n
f
χ
i
L
q
1
ω
1
·
∫
B
i
ω
1
x
1
-
q
1
′
d
x
1
/
q
1
′
×
b
2
∗
·
2
-
max
⁡
j
,
k
n
g
χ
j
L
q
2
ω
2
∫
B
j
ω
2
x
1
-
q
2
′
d
x
1
/
q
2
′
≤
C
b
1
∗
2
-
max
⁡
i
,
k
n
+
i
n
f
χ
i
L
q
1
ω
1
ω
1
B
i
-
1
/
q
1
×
b
2
∗
2
-
max
⁡
j
,
k
n
+
j
n
g
χ
j
L
q
2
ω
2
ω
2
B
j
-
1
/
q
2
.Then(54)
ν
B
k
α
/
n
L
4
χ
k
L
q
ν
ω
→
≤
C
b
1
∗
·
2
-
max
⁡
i
,
k
n
+
i
n
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
ν
B
k
ν
B
i
α
1
/
n
·
ω
1
B
k
ω
1
B
i
1
/
q
1
×
b
2
∗
2
-
max
⁡
j
,
k
n
+
j
n
ν
B
j
α
2
/
n
·
g
χ
j
L
q
2
ω
2
ν
B
k
ν
B
j
α
2
/
n
ω
2
B
k
ω
2
B
j
1
/
q
2
≤
C
b
1
∗
k
-
i
2
-
max
⁡
i
,
k
n
+
i
n
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
·
ν
B
k
ν
B
i
α
1
/
n
ω
1
B
k
ω
1
B
i
1
/
q
1
×
b
2
∗
k
-
j
·
2
-
m
a
x
j
,
k
n
+
j
n
ν
B
j
α
2
/
n
g
χ
j
L
q
2
ω
2
ν
B
k
ν
B
j
α
2
/
n
·
ω
2
B
k
ω
2
B
j
1
/
q
2
.Thus (55)
ν
B
k
α
/
n
L
4
χ
k
L
q
ν
ω
→
≤
C
b
1
∗
D
1
k
,
i
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
×
b
2
∗
D
2
k
,
j
ν
B
j
α
2
/
n
g
χ
j
L
q
2
ω
2
.
Now, we givethe proof of Theorem1.
Ifq
>
1, by Minkowski’s inequality and Theorem 5, we have(56)
ν
B
k
α
/
n
T
Π
b
f
,
g
χ
k
L
q
ν
ω
→
≤
C
ν
B
k
α
/
n
∑
i
=
-
∞
∞
∑
j
=
-
∞
∞
T
Π
b
f
χ
i
,
g
χ
j
χ
k
L
q
ν
ω
→
≤
C
∑
i
=
-
∞
∞
∑
j
=
-
∞
∞
ν
B
k
α
/
n
T
Π
b
f
χ
i
,
g
χ
j
χ
k
L
q
ν
ω
→
≤
C
b
1
∗
b
2
∗
∑
i
=
-
∞
∞
D
1
k
,
i
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
×
∑
j
=
-
∞
∞
D
2
k
,
j
ν
B
j
α
2
/
n
g
χ
j
L
q
2
ω
2
≤
C
b
1
∗
b
2
∗
∑
i
=
-
∞
∞
D
1
k
,
i
1
-
ϵ
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
×
∑
j
=
-
∞
∞
D
2
k
,
j
1
-
ϵ
ν
B
j
α
2
/
n
g
χ
j
L
q
2
ω
2for any 0
<
ϵ
<
1, since D
1
(
k
,
i
)
+
D
2
(
k
,
j
)
→
0 whenever i
,
j
→
±
∞.
If0
<
q
≤
1, note the fact that (57)
∑
i
=
-
∞
∞
D
1
k
,
i
q
ϵ
/
1
-
q
<
∞
,
∑
j
=
-
∞
∞
D
1
k
,
j
q
ϵ
/
1
-
q
<
∞
,and then by Theorem 5, the inequality (
∑
|
a
i
|
)
q
≤
∑
|
a
i
|
q, and Hölder’s inequality, we have (58)
ν
B
k
α
/
n
T
Π
b
f
,
g
χ
k
L
q
ν
ω
→
≤
C
∑
i
=
-
∞
∞
∑
j
=
-
∞
∞
ν
B
k
α
/
n
T
Π
b
f
χ
i
,
g
χ
j
χ
k
L
q
ν
ω
→
q
1
/
q
≤
C
b
1
∗
b
2
∗
∑
i
=
-
∞
∞
D
1
k
,
i
q
ν
B
i
α
1
q
/
n
f
χ
i
L
q
1
ω
1
q
×
∑
j
=
-
∞
∞
D
2
k
,
j
q
ν
B
j
α
2
q
/
n
g
χ
j
L
q
2
ω
2
q
1
/
q
.
≤
C
b
1
∗
·
b
2
∗
∑
i
=
-
∞
∞
D
1
k
,
i
1
-
ϵ
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
·
∑
j
=
-
∞
∞
D
1
k
,
j
q
ϵ
/
1
-
q
1
-
q
/
q
×
∑
j
=
-
∞
∞
D
2
k
,
j
1
-
ϵ
ν
B
j
α
2
/
n
g
χ
j
L
q
2
ω
2
·
∑
j
=
-
∞
∞
D
2
k
,
i
q
ϵ
/
1
-
q
1
-
q
/
q
≤
C
b
1
∗
b
2
∗
·
∑
i
=
-
∞
∞
D
1
k
,
i
1
-
ϵ
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
×
∑
j
=
-
∞
∞
D
2
k
,
j
1
-
ϵ
ν
B
j
α
2
/
n
g
χ
j
L
q
2
ω
2
.
Since0
<
p
<
∞, by (56), (58) and Hölder’s inequality,(59)
T
Π
b
f
,
g
K
˙
q
α
,
p
ν
,
ν
ω
→
R
n
≤
∑
k
=
-
∞
∞
ν
B
k
α
p
/
n
·
T
Π
b
f
,
g
χ
k
L
q
ν
ω
→
p
1
/
p
≤
C
b
1
∗
b
2
∗
·
∑
k
=
-
∞
∞
∑
i
=
-
∞
∞
D
1
k
,
i
1
-
ϵ
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
×
∑
j
=
-
∞
∞
D
2
k
,
j
1
-
ϵ
ν
B
j
α
2
/
n
g
χ
j
L
q
2
ω
2
p
1
/
p
≤
C
b
1
∗
·
b
2
∗
·
∑
k
=
-
∞
∞
∑
i
=
-
∞
∞
D
1
k
,
i
1
-
ϵ
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
p
1
1
/
p
1
×
∑
k
=
-
∞
∞
∑
j
=
-
∞
∞
D
2
k
,
j
1
-
ϵ
ν
B
j
α
2
/
n
g
χ
j
L
q
2
ω
2
p
2
1
/
p
2
=
C
b
1
∗
b
2
∗
N
1
×
N
2
.It is enough to show that N
1
≤
C
f
K
˙
q
1
α
1
,
p
1
(
ν
,
ω
1
)
(
R
n
) and N
2
≤
C
g
K
˙
q
2
α
2
,
p
2
(
ν
,
ω
2
)
(
R
n
)
. By the symmetry, we only give the estimate for N
1
. Consider(60)
N
1
p
1
≤
∑
k
=
-
∞
∞
∑
i
=
-
∞
k
-
2
D
1
k
,
i
1
-
ϵ
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
p
1
+
∑
k
=
-
∞
∞
∑
i
=
k
-
1
k
+
1
D
1
k
,
i
1
-
ϵ
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
p
1
+
∑
k
=
-
∞
∞
∑
i
=
k
+
2
∞
D
1
k
,
i
1
-
ϵ
ν
B
i
α
1
/
n
f
χ
i
L
q
1
ω
1
p
1
=
N
11
+
N
12
+
N
13
.
If0
<
p
1
≤
1, then, by the inequality (
∑
|
a
i
|
)
p
1
≤
∑
|
a
i
|
p
1 and the fact that (61)
∑
k
=
i
+
2
∞
D
1
k
,
i
1
-
ϵ
p
1
=
∑
k
=
i
+
2
∞
k
-
i
1
-
ϵ
p
1
2
1
-
ϵ
p
1
k
-
i
α
1
r
+
n
r
1
/
q
1
-
n
<
∞
,we get(62)
N
11
≤
∑
i
=
-
∞
∞
ν
B
i
α
1
p
1
/
n
f
χ
i
L
q
1
ω
1
p
1
∑
k
=
i
+
2
∞
D
1
k
,
i
1
-
ϵ
p
1
≤
C
∑
i
=
-
∞
∞
ν
B
i
α
1
p
1
/
n
f
χ
i
L
q
1
ω
1
p
1
≤
C
f
K
˙
q
1
α
1
,
p
1
ν
,
ω
1
R
n
p
1
.Note that(63)
∑
i
=
-
∞
k
-
2
D
1
k
,
i
1
-
ϵ
=
∑
k
=
i
+
2
∞
D
1
k
,
i
1
-
ϵ
≤
C
∑
k
=
i
+
2
∞
k
-
i
1
-
ϵ
2
1
-
ϵ
k
-
i
α
1
r
+
n
r
1
/
q
1
-
n
<
∞
.Then, for p
1
>
1, we have(64)
N
11
≤
C
∑
k
=
-
∞
∞
∑
i
=
-
∞
k
-
2
D
1
k
,
i
1
-
ϵ
ν
B
i
α
1
p
1
/
n
·
f
χ
i
L
q
1
ω
1
p
1
∑
i
=
-
∞
k
-
2
D
1
k
,
i
1
-
ϵ
p
1
/
p
1
′
≤
C
∑
k
=
-
∞
∞
∑
i
=
-
∞
k
-
2
D
1
k
,
i
1
-
ϵ
ν
B
i
α
1
p
1
/
n
f
χ
i
L
q
1
ω
1
p
1
≤
C
∑
i
=
-
∞
∞
ν
B
i
α
1
p
1
/
n
f
χ
i
L
q
1
ω
1
p
1
∑
k
=
i
+
2
∞
D
1
k
,
i
1
-
ϵ
≤
C
f
K
˙
q
1
α
1
,
p
1
ν
,
ω
1
R
n
p
1
.
SinceD
1
(
k
,
i
)
=
1 for i
=
k
-
1
,
k
,
k
+
1, then(65)
N
12
≤
C
∑
k
=
-
∞
∞
∑
i
=
k
-
1
k
+
1
ν
B
i
α
1
p
1
/
n
f
χ
i
L
q
1
ω
1
p
1
≤
C
∑
i
=
-
∞
∞
∑
k
=
i
-
1
i
+
1
ν
B
i
α
1
p
1
/
n
f
χ
i
L
q
1
ω
1
p
1
≤
C
∑
i
=
-
∞
∞
ν
B
i
α
1
p
1
/
n
f
χ
i
L
q
1
ω
1
p
1
≤
C
f
K
˙
q
1
α
1
,
p
1
ν
,
ω
1
R
n
p
1
.Thus, we have obtained that N
12
≤
C
f
K
˙
q
1
α
1
,
p
1
(
ν
,
ω
1
)
(
R
n
)
p
1 holds for any 0
<
p
1
<
∞
.
Let us now turn to estimate the last termN
13. If 0
<
p
1
≤
1, then(66)
N
13
≤
C
∑
k
=
-
∞
∞
∑
i
=
k
+
2
∞
D
1
k
,
i
1
-
ϵ
p
1
ν
B
i
α
1
p
1
/
n
f
χ
i
L
q
1
ω
1
p
1
≤
C
∑
i
=
-
∞
∞
ν
B
i
α
1
p
1
/
n
f
χ
i
L
q
1
ω
1
p
1
∑
k
=
-
∞
i
-
2
D
1
k
,
i
1
-
ϵ
p
1
.But(67)
∑
k
=
-
∞
i
-
2
D
1
k
,
i
1
-
ϵ
p
1
=
∑
k
=
-
∞
i
-
2
i
-
k
1
-
ϵ
p
1
2
1
-
ϵ
p
1
k
-
i
δ
α
1
/
n
+
δ
1
/
q
1
<
∞
.Then(68)
N
13
≤
C
∑
i
=
-
∞
∞
ν
B
i
α
1
p
1
/
n
f
χ
i
L
q
1
ω
1
p
1
≤
C
f
K
˙
q
1
α
1
,
p
1
ν
,
ω
1
R
n
p
1
.Note that(69)
∑
k
=
-
∞
i
-
2
D
1
k
,
i
1
-
ϵ
=
∑
i
=
k
+
2
∞
D
1
k
,
i
1
-
ϵ
≤
C
∑
i
=
k
+
2
∞
i
-
k
1
-
ϵ
2
1
-
ϵ
k
-
i
δ
α
1
/
n
+
δ
1
/
q
1
<
∞
.Then, for p
>
1, we have(70)
N
13
≤
C
∑
k
=
-
∞
∞
∑
i
=
k
+
2
∞
D
1
k
,
i
1
-
ϵ
ν
B
i
α
1
p
1
/
n
·
f
χ
i
L
q
1
ω
1
p
1
∑
i
=
k
+
2
∞
D
1
k
,
i
1
-
ϵ
p
1
/
p
1
′
≤
C
∑
k
=
-
∞
∞
∑
i
=
k
+
2
∞
D
1
k
,
i
1
-
ϵ
ν
B
i
α
1
p
1
/
n
f
χ
i
L
q
1
ω
1
p
1
≤
C
∑
i
=
-
∞
∞
ν
B
i
α
1
p
1
/
n
f
χ
i
L
q
1
ω
1
p
1
∑
k
=
-
∞
i
-
2
D
1
k
,
i
1
-
ϵ
≤
C
f
K
˙
q
1
α
1
,
p
1
ν
,
ω
1
R
n
p
1
.
This completes the proof of Theorem1.Remark 6.
Using Theorem5 and the methods in [18], we can easily get the boundedness of the iterated commutator T
Π
b on weighted Herz-Morrey spaces.



---

*Source: 967390-2015-06-23.xml*