# The Boundedness of Marcinkiewicz Integral Associated with Schrödinger Operator and Its Commutator

Authors: Dongxiang Chen; Dan Zou
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

The authors prove that Marcinkiewicz integral operator is not only are bounded onL
p, for 1
<
P
<
∞, but also a bounded map from L
1
(
R
n
) to weak L
1
(
R
n
). Meanwhile, the B
M
O
L-boundedness and (
H
L
1
,
L
1
)-boundedness are also obtained. Finally, the L
p-boundedness and (
L
∞
,
B
M
O
L
)-boundedness for the commutator of Marcinkiewicz integral of schrödinger type are established.


---

## Body

## 1. Introduction and Notation

Let us consider the Schrödinger operator(1)
L
=
-
Δ
+
V
in R
d
,
d
≥
3. The function V is nonnegative, V
≠
0, and belongs to a reverse Hölder class R
H
q, for some exponent q
>
d
/
2; that is, there exists a constant C such that
(2)
(
1
|
B
|
∫
B
‍
V
(
y
)
q
d
y
)
1
/
q
≤
C
|
B
|
∫
B
‍
V
(
y
)
d
y
,
for every ball B
⊂
R
d.We introduce the definition of the reverse Hölder index ofV as q
0
=
sup
⁡
{
q
:
V
∈
R
H
q
}. It is known that V
∈
R
H
q implies V
∈
R
H
q
+
ϵ, for some ϵ
>
0. Therefore, under the assumption V
∈
R
H
d
/
2, we may conclude q
0
>
d
/
2.The classical Marcinkiewicz integral operatorμ is defined by
(3)
μ
f
(
x
)
=
(
∫
0
∞
‍
|
∫
|
x
-
y
|
≤
t
‍
Ω
(
x
-
y
)
|
x
-
y
|
n
-
1
f
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
.The above operator was introduced by Stein in [1] as an extension of the notion of Marcinkiewicz function from one dimension to higher dimension. Meanwhile, Stein [1] showed that if Ω
∈
Li
p
α
(
S
n
-
1
), for some 0
<
α
≤
1, then μ is a bounded operator on L
p
(
R
n
), for 1
<
p
≤
2, and is a bounded map from L
1
(
R
n
) to weak L
1
(
R
n
). Benedek et al. [2] showed that if Ω is continuously differentiable in x
≠
0, then μ is a bounded operator on L
p
(
R
n
), for 1
<
p
≤
∞. Ding et al. [3] proved that the Marcinkiewicz function μ is bounded from H
1
(
R
n
) to L
1
(
R
n
) with Ω satisfying cancelation condition on S
n
-
1 and L
1-Dini condition.Similar to the classical Marcinkiewicz functionμ, we define the Marcinkiewicz integral μ
j
L associated with the Schrödinger operator L by
(4)
μ
j
L
f
(
x
)
=
(
∫
0
∞
‍
|
∫
|
x
-
y
|
≤
t
‍
K
j
L
(
x
,
y
)
f
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
,
where K
j
L
(
x
,
y
)
=
K
j
L
~
(
x
,
y
)
|
x
-
y
| and K
J
L
~
(
x
,
y
) is the kernel of R
j
L
=
(
∂
/
∂
x
j
)
L
-
1
/
2
,
j
=
1
,
…
,
d. In particular, when V
=
0
,
K
j
Δ
(
x
,
y
)
=
K
j
Δ
~
(
x
,
y
)
|
x
-
y
|
=
(
(
x
-
y
)
j
/
|
x
-
y
|
)
/
|
x
-
y
|
d
-
1 and K
j
Δ
~
(
x
,
y
) is the kernel of R
j
=
(
∂
/
∂
x
j
)
Δ
-
1
/
2
,
j
=
1
,
…
,
d.We also give the definition of the commutator generalized byμ
L
j and b by
(5)
[
b
,
μ
L
j
]
(
f
)
(
x
)
=

(

∫
0
∞
‍
|
∫
|
x
-
y
|
≤
t
‍
K
j
L
(
x
,
y
)

(
b
(
x
)
-
b
(
y
)
)
f
(
y
)
d
y
|
2
d
t
t
3

)
1
/
2
.
In this paper, we write K
j
(
x
,
y
)
=
K
j
Δ
(
x
,
y
) and
(6)
μ
j
f
(
x
)
=
(
∫
0
∞
‍
|
∫
|
x
-
y
|
≤
t
‍
K
j
(
x
,
y
)
f
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
.For a given potentialV
∈
R
H
q, with q
>
d
/
2, we introduce the auxiliary function
(7)
ρ
(
x
)
=
sup
⁡
{
r
>
0
:
1
r
d
-
2
∫
B
(
x
,
r
)
‍
V
≤
1
}
,
x
∈
R
d
.
The above assumptions ρ
(
x
) are finite, for all x
∈
R
d.Proposition 1 (see [4]).
There existC and k
0
≥
1 such that
(8)
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
)
,
for all x
,
y
∈
R
d.In particular,ρ
(
x
)
⋍
ρ
(
y
), if |
x
-
y
|
<
C
ρ
(
x
). A ball B
(
x
,
ρ
(
x
)
) is called critical.Proposition 2 (see [5]).
There exist a sequence of points{
x
k
}
k
=
1
∞ in R
d, so that the family B
k
=
B
(
x
k
,
ρ
(
x
k
)
)
,
k
≥
1,satisfies the following:
(1)
⋃
k
B
k
=
R
d
;
(2)
there exists
N
such that, for every
k
∈
ℕ
,
card
{
j
:
4
B
j
⋂
4
B
k
≠
Ø
}
≤
N.Functionf is an element of BM
O
L, if ∃
C
>
0 such that, for every ball B
=
B
(
x
,
r
),
(9)
∫
B
‍
|
f
-
f
B
|
≤
C
|
B
|
,
if r
<
ρ
(
x
), and
(10)
∫
B
‍
|
f
|
≤
C
|
B
|
,
if r
≥
ρ
(
x
), where f
B
=
(
1
/
|
B
|
)
∫
B
‍
f
(
x
)
d
x. Let ∥
f
∥
BM
O
L
(
R
d
) be the smallest C in the inequalities above. It is easy to verify that ∥
f
∥
BMO
(
R
d
)
≤
2
∥
f
∥
BM
O
L
(
R
d
), for all f
∈
BM
O
L
(
R
d
).Lemma 3 (see [6]).
For anyl
>
0, there exists C
l
>
0 such that
(11)
|
K
j
L
(
x
,
y
)
|
≤
C
l
(
1
+
(
|
x
-
y
|
/
ρ
(
y
)
)
)
l
1
|
x
-
y
|
d
-
1
,
|
K
j
L
(
x
,
y
)
-
K
j
(
x
,
y
)
|
≤
C
ρ
(
x
)
|
x
-
y
|
d
-
2
.Tang and Dong [6] have shown that Marcinkiewicz integral μ
j
L is bounded on L
p
(
R
n
), for 1
<
p
<
∞, and are bounded from L
1
(
R
n
) to weak L
1
(
R
n
). Meanwhile, they also proved that μ
j
L are bounded on BM
O
L
(
R
n
) and are also mapped from H
L
1
(
R
n
) to L
1
(
R
n
) under the assumption that K
j
L satisfy the condition in Lemma 3.Now, we introduce a newBMO-type space of schrödinger operator. Let θ
>
0; we define the class BM
O
θ
(
ρ
) of locally integrable functions b such that
(12)
1
|
B
(
x
,
r
)
|
∫
B
(
x
,
r
)
‍
|
b
(
y
)
-
b
B
|
d
y
≤
C
(
1
+
r
ρ
(
x
)
)
θ
,
for all x
∈
R
d and r
>
0, where b
B
=
(
1
/
|
B
|
)
∫
B
‍
b
(
y
)
d
y. A norm for b
∈
BM
O
θ
(
ρ
), denoted by [
b
]
θ, is given by the infimum of the constants in the inequalities above. Notice that if we let θ
=
0, we obtain the John-Nirenberg space BMO.With the above definition, we defineBM
O
∞
(
ρ
)
=
⋃
θ
>
0
BM
O
θ
(
ρ
). Clearly, BMO
⊂
BM
O
θ
(
ρ
)
⊂
BM
O
θ
′
(
ρ
), for 0
<
θ
<
θ
′, and hence BMO
⊂
BM
O
∞
(
ρ
). Moreover, it is in general a larger class.Forθ
>
0, we denote by BM
O
θ
log
⁡
(
ρ
) the set of functions b such that
(13)
1
|
B
(
x
,
r
)
|
∫
B
(
x
,
r
)
‍
|
b
(
y
)
-
b
B
|
d
y
≤
C
(
1
+
r
/
ρ
(
x
)
)
θ
1
+
log
⁡
+
(
ρ
(
x
)
/
r
)
,
for all x
∈
R
d and r
>
0. A norm for b
∈
BM
O
θ
log
⁡
(
ρ
), denoted by ∥
b
∥
*
log
⁡, is given by the infimum of the constants in the inequalities above. Correspondingly, we define BM
O
∞
log
⁡
(
ρ
)
=
⋃
θ
>
0
BM
O
θ
log
⁡
(
ρ
).Next, we give some information on the Hardy space associated with Schrödinger operatorL. We say that a function f
∈
L
1 is said to belong to H
L
1, if the semigroup maximal operator T
L
* is bounded on L
1. The H
L
1- norm of f is given by ∥
f
∥
H
L
1
=
∥
T
L
*
f
∥
L
1, where T
L
*
f
(
x
)
=
sup
⁡
t
>
0
|
T
t
L
f
(
x
)
|
,
{
T
t
L
}
=
{
e
-
t
L
} is a semigroup generated by the Schrödinger operator L (see [7]).Shen [4] gave the following kernel estimate that we needed.Lemma 4.
IfV
∈
R
H
d, then,(i)
for everyN, there exists a constant C such that
(14)
|
K
j
L
(
x
,
z
)
|
≤
C
(
1
+
(
|
x
-
z
|
/
ρ
(
x
)
)
)
-
N
|
x
-
z
|
d
-
1
,
(ii)
for everyN and 0
<
δ
<
min
⁡
{
1,1
-
d
/
q
0
}, there exists a constant C such that
(15)
|
K
j
L
(
x
,
z
)
-
K
j
L
(
y
,
z
)
|
≤
C
|
x
-
y
|
δ
(
1
+
(
|
x
-
z
|
/
ρ
(
x
)
)
)
-
N
|
x
-
z
|
d
-
1
+
δ
,
where |
x
-
y
|
<
(
2
/
3
)
|
x
-
z
|,
(iii)
ifK denotes the R
d vector valued kernel of the classical Riesz operator, for every 0
<
δ
<
2
-
d
/
q
0, we have
(16)
|
K
j
L
(
x
,
z
)
-
K
j
(
x
,
z
)
|
≤
C
|
x
-
z
|
d
-
1
(
|
x
-
z
|
ρ
(
z
)
)
δ
,
where K
j
(
x
,
z
)
=
K
(
x
,
z
)
|
x
-
z
|.Inspired by [6], we consider the same boundedness of Marcinkiewicz integral whose kernel satisfies Lemma 4 and L
p-boundedness of its commutator for 1
<
p
<
∞ and (
L
∞
,
B
M
O
L
)-boundedness.Theorem 5.
The operatorsμ
j
L are bounded on L
p
(
R
d
), for 1
<
p
<
∞, and are bounded from L
1
(
R
d
) to weak L
1
(
R
d
). On the other hand, μ
j
L are bounded on B
M
O
L
(
R
d
) and are bounded from H
L
1
(
R
d
) to L
1
(
R
d
).In the note, we devote ourselves to establish the following boundedness of the commutators of Marcinkiewicz integral of schrodinger operator type.Theorem 6.
LetV
∈
R
H
d, b
∈
B
M
O
∞
(
ρ
), and p
0 such that 1
/
p
0 = (
1
/
q
0
-
1
/
d
)
+, where q
0 is the reverse Hölder index of V; then, for all f
∈
L
p, we have
(17)
∥
[
b
,
μ
j
L
]
f
∥
p
≤
C
[
b
]
θ
∥
f
∥
p
,
where 1
<
p
<
p
0.Theorem 7.
LetV
∈
R
H
d and b
∈
B
M
O
∞
(
ρ
); if b satisfies the stronger condition b
∈
B
M
O
∞
log
⁡
(
ρ
), then [
b
,
μ
j
L
] are bounded from L
∞
(
R
d
) to B
M
O
L
(
R
d
).Throughout this paper,C denotes the constants that are independent of the main parameters involved but whose value may differ from line to line. By A
⋍
B, we mean that there exists a constant C
>
1 such that 1
/
C
≤
A
/
B
≤
C. We use the symbol A
≲
B to denote that there exists a positive constant C such that A
≤
C
B.


## 2. Notation and Preliminaries

Denote the following maximal functions forg
∈
L
l
o
c
1
(
R
d
), α
>
0, and x
∈
R
d:
(18)
M
ρ
,
α
g
(
x
)
=
sup
⁡
x
∈
B
∈
B
ρ
,
α
1
|
B
|
∫
B
‍
|
g
|
,
M
ρ
,
α
♯
g
(
x
)
=
sup
⁡
x
∈
B
∈
B
ρ
,
α
1
|
B
|
∫
B
‍
|
g
-
g
B
|
,
where B
ρ
,
α
=
{
B
(
y
,
r
)
:
y
∈
R
d and r
≤
α
ρ
(
y
)
}.Lemma 8 (Fefferman-Stein type inequality [8]).
For1
<
p
<
∞, there exist β and γ such that if {
Q
k
}
k
=
1
∞ is a sequence of balls as that in Proposition 2, then
(19)
∫
R
d
‍
|
M
ρ
,
β
(
g
)
|
p
≲
∫
R
d
‍
|
M
ρ
,
γ
♯
(
g
)
|
p
+
∑
k
‍
|
Q
k
|
(
1
|
Q
k
|
∫
2
Q
k
‍
|
g
|
)
p
,
for all g
∈
L
l
o
c
1
(
R
d
).Lemma 9 (see [9]).
Letθ
>
0 and 1
≤
s
<
∞. If b
∈
B
M
O
θ
(
ρ
), then
(20)
(
1
|
B
|
∫
B
‍
|
b
-
b
B
|
s
)
1
/
s
≲
[
b
]
θ
(
1
+
r
ρ
(
x
)
)
θ
′
,
for all B
=
B
(
x
,
r
), with x
∈
R
d and r
>
0, where θ
′
=
(
k
0
+
1
)
θ and k
0 is the constant appearing in Proposition 1.Lemma 10 (see [9]).
Letb
∈
B
M
O
θ
(
ρ
), B
=
B
(
x
0
,
r
), and s
≥
1; then
(21)
(
1
|
2
k
B
|
∫
2
k
B
‍
|
b
-
b
B
|
s
)
1
/
s
≲
[
b
]
θ
k
(
1
+
2
k
r
ρ
(
x
0
)
)
θ
′
,
for all k
∈
N, with θ
′ as in Lemma 9.Next, we introduce our important Lemmas.We denoted byM the Hardy-Littlewood maximal function and, for s
>
1, by M
s, the operator is defined as M
s
=
(
M
(
f
s
)
)
1
/
s.Lemma 11.
LetV
∈
R
H
d
,
1
/
p
0
=
(
1
/
q
0
-
1
/
d
)
+, and b
∈
B
M
O
θ
(
ρ
). Then, for any 1
<
s
<
p
0, there exists a constant C such that
(22)
1
|
Q
|
∫
Q
‍
|
[
b
,
μ
j
L
]
f
|
≤
C
[
b
]
θ
inf
⁡
y
∈
Q
M
s
f
(
y
)
,
for all f
∈
L
l
o
c
s
(
R
d
) and every ball Q
=
B
(
x
0
,
ρ
(
x
0
)
).Proof.
Letf
∈
L
p
(
R
d
) and Q
=
B
(
x
0
,
ρ
(
x
0
)
). We first observe
(23)
[
b
,
μ
j
L
]
f
≤
|
b
-
b
Q
|
μ
j
L
f
+
μ
j
L
(
f
(
b
-
b
Q
)
)
,
and so we have to deal with the average on Q of each term.
Thanks to Hölder's inequality with1
<
s
<
p
0 and Lemma 10, one has
(24)
1
|
Q
|
∫
Q
‍
|
(
b
-
b
Q
)
μ
j
L
f
|
≤
(
1
|
Q
|
∫
Q
‍
|
b
-
b
Q
|
s
′
)
1
/
s
′
(
1
|
Q
|
∫
Q
‍
|
μ
j
L
f
|
s
)
1
/
s
≲
[
b
]
θ
(
1
|
Q
|
∫
Q
‍
|
μ
j
L
f
|
s
)
1
/
s
.
Letf
=
f
1
+
f
2 with f
1
=
f
χ
2
Q; then L
s
(
R
d
)-boundedness of μ
j
L, for 1
<
s
<
p
0, says that
(25)
(
1
|
Q
|
∫
Q
‍
|
μ
j
L
f
1
|
s
)
1
/
s
≲
(
1
|
Q
|
∫
2
Q
‍
|
f
|
s
)
1
/
s
≲
inf
⁡
y
∈
Q
M
s
f
(
y
)
.
Now, for x
∈
Q, ρ
(
x
)
⋍
ρ
(
x
0
) and using Lemma 4 and Minkowski's inequality, we get
(26)
|
μ
j
L
f
2
(
x
)
|
=
(
∫
0
∞
‍
|
∫
|
x
-
y
|
≤
t
‍
K
j
L
(
x
,
y
)
f
2
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
≲
(
|
f
2
(
y
)
|
d
y
∫
|
x
-
y
|
≤
t
‍
(
1
+
(
|
x
-
y
|
/
ρ
(
x
)
)
)
-
N
|
x
-
y
|
d
-
1
|
2
∫
0
∞
‍
|
∫
|
x
-
y
|
≤
t
‍
(
1
+
(
|
x
-
y
|
/
ρ
(
x
)
)
)
-
N
|
x
-
y
|
d
-
1
w
w
w
w
w
w
w
w
w
×
|
f
2
(
y
)
|
d
y
∫
|
x
-
y
|
≤
t
‍
(
1
+
(
|
x
-
y
|
/
ρ
(
x
)
)
)
-
N
|
x
-
y
|
d
-
1
|
2
d
t
t
3
∫
0
∞
‍
)
1
/
2
≲
∫
R
d
‍
|
f
2
(
y
)
|
ρ
(
x
0
)
N
|
x
-
y
|
d
-
1
+
N
(
∫
|
x
-
y
|
≤
t
‍
d
t
t
3
)
1
/
2
d
y
≲
ρ
(
x
0
)
N
∫
R
d
‍
|
f
2
(
y
)
|
|
x
-
y
|
d
+
N
d
y
≲
ρ
(
x
0
)
N
∫
(
2
Q
)
c
‍
|
f
(
y
)
|
|
x
-
y
|
d
+
N
d
y
≲
ρ
(
x
0
)
N
∑
k
=
1
∞
‍
∫
2
k
+
1
Q
∖
2
k
Q
‍
|
f
(
y
)
|
|
x
-
y
|
d
+
N
d
y
≲
ρ
(
x
0
)
N
∑
k
=
1
∞
‍
(
2
k
ρ
(
x
0
)
)
-
N
|
2
k
+
1
Q
|
∫
2
k
+
1
Q
‍
|
f
(
y
)
|
d
y
≲
inf
⁡
y
∈
Q
M
s
f
(
y
)
.
For the second term, we split againf
=
f
1
+
f
2. By choosing 1
<
s
~
<
s
<
p
0 and denoting ν
=
s
s
~
/
(
s
-
s
~
), by the boundedness of μ
j
L on L
s
~
(
R
d
) and Hölder's inequality, we obtain
(27)
1
|
Q
|
∫
Q
‍
|
μ
j
L
(
f
1
(
b
-
b
B
)
)
|
≤
(
1
|
Q
|
∫
Q
‍
|
μ
j
L
(
f
1
(
b
-
b
B
)
)
|
s
~
)
1
/
s
~
≲
(
1
|
Q
|
∫
2
Q
‍
|
f
(
b
-
b
B
)
|
s
~
)
1
/
s
~
≲
(
1
|
Q
|
∫
2
Q
‍
|
f
|
s
)
1
/
s
(
1
|
Q
|
∫
2
Q
‍
|
b
-
b
B
|
ν
)
1
/
ν
≲
[
b
]
θ
inf
⁡
y
∈
Q
M
s
f
(
y
)
,
where, in the last inequality, we have used Lemma 9.
For the remaining term, let1
=
1
/
s
+
1
/
s
~, ρ
(
x
)
⋍
ρ
(
x
0
) using Lemma 9 , Lemma 10, and Minkowski's inequality, we arrive to
(28)
|
μ
j
L
(
f
2
(
b
-
b
Q
)
)
(
x
)
|
=
(
∫
0
∞
‍
|
∫
|
x
-
y
|
≤
t
‍
K
j
L
(
x
,
y
)
f
2
(
y
)
(
b
(
y
)
-
b
Q
)
d
y
|
2
d
t
t
3
)
1
/
2
≲

(

∫
0
∞
‍

|
∫
|
x
-
y
|
≤
t
‍

|
f
2
(
y
)
|
|
b
(
y
)
-
b
Q
|
|
x
-
y
|
d
-
1

(
1
+
(
|
x
-
y
|
/
ρ
(
x
)
)
)
N
d
y
|
2

d
t
t
3

)
1
/
2
≲
∫
R
d
‍
|
f
2
(
y
)
|
|
b
(
y
)
-
b
Q
|
|
x
-
y
|
d
-
1
(
1
+
(
|
x
-
y
|
/
ρ
(
x
)
)
)
N
w
w
w
×
(
∫
|
x
-
y
|
≤
t
‍
d
t
t
3
)
1
/
2
d
y
≲
ρ
(
x
0
)
N
∫
(
2
Q
)
c
‍
|
f
(
y
)
|
|
b
(
y
)
-
b
Q
|
|
x
-
y
|
d
+
N
d
y
≲
ρ
(
x
0
)
N
∑
k
=
1
∞
‍
(
2
k
ρ
(
x
0
)
)
-
N
|
2
k
+
1
Q
|
×
∫
2
k
+
1
Q
∖
2
k
Q
‍
|
f
(
y
)
|
|
b
(
y
)
-
b
Q
|
d
y
≲
∑
k
=
1
∞
‍
2
-
k
N
|
2
k
+
1
Q
|
∫
2
k
+
1
Q
‍
|
f
(
y
)
|
|
b
(
y
)
-
b
Q
|
d
y
≲
∑
k
=
1
∞
‍
2
-
k
N
(
1
|
2
k
+
1
Q
|
∫
2
k
+
1
Q
‍
|
b
(
y
)
-
b
Q
|
s
′
d
y
)
1
/
s
′
×
(
1
|
2
k
+
1
Q
|
∫
2
k
+
1
Q
‍
|
f
(
y
)
|
s
d
y
)
1
/
s
≲
∑
k
=
1
∞
‍
2
-
k
N
k
(
1
+
2
k
ρ
(
x
0
)
ρ
(
x
0
)
)
θ
′
[
b
]
θ
inf
⁡
y
∈
Q
M
s
f
(
y
)
≲
∑
k
=
1
∞
‍
k
2
k
(
-
N
+
θ
′
)
[
b
]
θ
inf
⁡
y
∈
Q
M
s
f
(
y
)
≲
[
b
]
θ
inf
⁡
y
∈
Q
M
s
f
(
y
)
.
Since N can be chosen large enough, the last series converges. Thus, we finished the proof.Remark 12.
It is easy to check that if the critical ballQ is replaced by 2
Q, the above lemma also holds.Lemma 13.
LetV
∈
R
H
d and b
∈
B
M
O
∞
(
ρ
), then, for any 1
<
s
<
p
0 and γ
≥
1, there exists a constant C such that
(29)
|
μ
j
L
(
f
2
(
b
-
b
B
)
)
(
u
)
-
μ
j
L
(
f
2
(
b
-
b
B
)
)
(
z
)
|
≲
[
b
]
θ
inf
⁡
x
∈
B
M
s
f
(
x
)
,
for all f
,
u
,
z
∈
B
=
B
(
x
0
,
r
), with r
<
γ
ρ
(
x
0
) and f
2 as in Lemma 11.Proof.
We write(30)
|
μ
j
L
(
f
2
(
b
-
b
B
)
)
(
u
)
-
μ
j
L
(
f
2
(
b
-
b
B
)
)
(
z
)
|
≤
(
∫
0
∞
‍
|
∫
|
u
-
y
|
≤
t
<
|
z
-
y
|
‍
|
K
j
L
(
u
,
y
)
w
w
w
w
w
w
w
w
w
w
w
w
i
×
f
2
(
y
)
(
b
(
y
)
-
b
B
)
K
j
L
|
d
y
∫
|
u
-
y
|
≤
t
<
|
z
-
y
|
‍
|
2
d
t
t
3
)
1
/
2
+
(
∫
0
∞
‍
|
∫
|
z
-
y
|
≤
t
<
|
u
-
y
|
‍
|
K
j
L
(
z
,
y
)
w
w
w
w
w
w
w
w
w
w
w
w
w
i
×
f
2
(
y
)
(
b
(
y
)

-

b
B
)
K
j
L
|
d
y
∫
|
u
-
y
|
≤
t
<
|
z
-
y
|
‍
|
2
d
t
t
3
)
1
/
2
+
(
∫
0
∞
‍
|
∫
{
|
u
-
y
|
≤
t
,
|
z
-
y
|
≤
t
}
‍
|
K
j
L
(
u
,
y
)
-
K
j
L
(
z
,
y
)
|
w
w
w
w
w
w
w
w
w
w
w
w
w
w
a
×

|
f
2
(
y
)
(
b
(
y
)
-
b
B
)
|
d
y
∫
{
|
u
-
y
|
≤
t
,
|
z
-
y
|
≤
t
}
‍
|
2

d
t
t
3

)
1
/
2
=
I
1
+
I
2
+
I
3
.
Since the estimates for I
1 and I
2 follow along similar lines, we only consider I
1. By denoting Q
=
B
(
x
0
,
γ
ρ
(
x
0
)
), since in our situation ρ
(
u
)
⋍
ρ
(
x
0
) and |
u
-
y
|
⋍
|
z
-
y
|, by Minkowski's inequality and Lemma 4, we have
(31)
I
1
≲
∫
R
d
‍
|
f
2
(
y
)
|
|
b
(
y
)
-
b
B
|
|
K
j
L
(
u
,
y
)
|
w
w
w
×
(
∫
|
u
-
y
|
≤
t
<
|
z
-
y
|
‍
d
t
t
3
)
1
/
2
d
y
≲
r
1
/
2
∫
(
2
B
)
c
‍
|
f
(
y
)
|
|
b
(
y
)
-
b
B
|
|
K
j
L
(
u
,
y
)
|
|
u
-
y
|
3
/
2
d
y
≲
r
1
/
2
∫
Q
∖
2
B
‍
|
f
(
y
)
|
|
b
(
y
)
-
b
B
|
|
u
-
y
|
d
+
1
/
2
d
y
+
r
1
/
2
ρ
(
x
0
)
N
∫
Q
c
‍
|
f
(
y
)
|
|
b
(
y
)
-
b
B
|
|
u
-
y
|
d
+
1
/
2
+
N
d
y
=
I
11
+
I
12
.
Splitting into annuli, we have(32)
I
11
≲
1
r
d
∑
j
=
2
j
0
‍
2
-
j
(
d
+
1
/
2
)
∫
2
j
B
‍
|
f
(
y
)
|
|
b
(
y
)
-
b
B
|
d
y
,
where j
0 is the least integer such that 2
j
0
≥
γ
ρ
(
x
0
)
/
r.
By Hölder's inequality and Lemma10, we obtain, for j
≤
j
0,
(33)
∫
2
j
B
‍
|
f
(
y
)
|
|
b
(
y
)
-
b
B
|
d
y
≤
j
[
b
]
θ
|
2
j
B
|
inf
⁡
x
∈
B
M
s
f
(
x
)
.
Then,
(34)
I
11
≲
[
b
]
θ
inf
⁡
x
∈
B
M
s
f
(
x
)
∑
j
=
2
j
0
‍
j
2
-
j
/
2
≲
[
b
]
θ
inf
⁡
x
∈
B
M
s
f
(
x
)
.
To deal withI
12, by using Lemma 10 and choosing N
>
θ
′, we have
(35)
I
12
≲
ρ
(
x
0
)
N
r
d
+
N
∑
j
=
j
0
-
1
∞
‍
2
-
j
(
d
+
1
/
2
+
N
)
∫
2
j
B
‍
|
f
(
y
)
|
|
b
(
y
)
-
b
B
|
d
y
≲
[
b
]
θ
inf
⁡
x
∈
B
M
s
f
(
x
)
(
ρ
(
x
0
)
r
)
N
-
θ
′
∑
j
=
j
0
-
1
∞
‍
j
2
-
j
(
1
/
2
+
N
-
θ
′
)
≲
[
b
]
θ
inf
⁡
x
∈
B
M
s
f
(
x
)
∑
j
=
j
0
-
1
∞
‍
j
2
-
j
/
2
≲
[
b
]
θ
inf
⁡
x
∈
B
M
s
f
(
x
)
.
ForI
3, again by Minkowski's inequality and in our situation ρ
(
u
)
⋍
ρ
(
x
0
) and |
u
-
y
|
⋍
|
z
-
y
|, we get
(36)
I
3
≲
∫
R
d
‍
|
f
2
(
y
)
|
|
b
(
y
)
-
b
B
|
|
K
j
L
(
u
,
y
)
-
K
j
L
(
z
,
y
)
|
w
w
w
×
(
∫
{
|
u
-
y
|
≤
t
,
|
z
-
y
|
≤
t
}
‍
d
t
t
3
)
1
/
2
d
y
≲
∫
(
2
B
)
c
‍
|
f
(
y
)
|
|
b
(
y
)
-
b
B
|
|
K
j
L
(
u
,
y
)
-
K
j
L
(
z
,
y
)
|
|
u
-
y
|
d
y
≲
r
δ
∫
Q
∖
2
B
‍
|
f
(
y
)
|
|
b
(
y
)
-
b
B
|
|
u
-
y
|
d
+
δ
d
y
+
r
δ
ρ
(
x
0
)
N
∫
Q
c
‍
|
f
(
y
)
|
|
b
(
y
)
-
b
B
|
|
u
-
y
|
d
+
δ
+
N
d
y
=
I
31
+
I
32
.
Thanks to Hölder's inequality and Lemma10, we have
(37)
I
31
≲
1
r
d
∑
j
=
2
j
0
‍
2
-
j
(
d
+
δ
)
∫
2
j
B
‍
|
f
(
y
)
|
|
b
(
y
)
-
b
B
|
d
y
≲
1
r
d
∑
j
=
2
j
0
‍
2
-
j
(
d
+
δ
)
j
[
b
]
θ
|
2
j
B
|
inf
⁡
x
∈
B
M
s
f
(
x
)
≲
[
b
]
θ
inf
⁡
x
∈
B
M
s
f
(
x
)
∑
j
=
2
j
0
‍
j
2
-
j
δ
≲
[
b
]
θ
inf
⁡
x
∈
B
M
s
f
(
x
)
.
To deal withI
22, similarly as I
12, we have
(38)
I
32
≲
ρ
(
x
0
)
N
r
d
+
N
∑
j
=
j
0
-
1
∞
‍
2
-
j
(
d
+
δ
+
N
)
∫
2
j
B
‍
|
f
(
y
)
|
|
b
(
y
)
-
b
B
|
d
y
≲
[
b
]
θ
inf
⁡
x
∈
B
M
s
f
(
x
)
(
ρ
(
x
0
)
r
)
N
-
θ
′
∑
j
=
j
0
-
1
∞
‍
j
2
-
j
(
δ
+
N
-
θ
′
)
≲
[
b
]
θ
inf
⁡
x
∈
B
M
s
f
(
x
)
∑
j
=
j
0
-
1
∞
‍
j
2
-
j
δ
≲
[
b
]
θ
inf
⁡
x
∈
B
M
s
f
(
x
)
.
We have completed the proof of the lemma.


## 3. The Boundedness of Marcikiewicz Integral and Its Commutator

In this section, we first employ the same technique in [6] to prove Theorem 5.Proof.
Similarly as [6], it suffices to prove the following pointwise estimate:
(39)
μ
j
L
f
(
x
)
≤
μ
j
f
(
x
)
+
C
M
f
(
x
)
,
a
.
e
.
x
∈
R
d
,
where M denotes the Hardy-Littlewood operator.
Fixx
∈
R
d and let r
=
ρ
(
x
). Notice that
(40)
μ
j
L
f
(
x
)
≤
(
∫
0
r
‍
|
∫
|
x
-
y
|
≤
t
‍
K
j
L
(
x
,
y
)
f
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
+
(
∫
r
∞
‍
|
∫
|
x
-
y
|
≤
r
‍
K
j
L
(
x
,
y
)
f
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
+
(
∫
r
∞
‍
|
∫
r
<
|
x
-
y
|
≤
t
‍
K
j
L
(
x
,
y
)
f
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
≤

(

∫
0
r
‍

|

∫
|
x
-
y
|
≤
t
‍

[
K
j
L
(
x
,
y
)

-

K
j
(
x
,
y
)
]

f
(
y
)
d
y

|
2

d
t
t
3
)
1
/
2
+
(
∫
0
r
‍
|
∫
|
x
-
y
|
≤
t
‍
K
j
(
x
,
y
)
f
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
+
(
∫
r
∞
‍
|
∫
|
x
-
y
|
≤
r
‍
K
j
L
(
x
,
y
)
f
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
+
(
∫
r
∞
‍
|
∫
r
<
|
x
-
y
|
≤
t
‍
K
j
L
(
x
,
y
)
f
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
=
E
1
+
E
2
+
E
3
+
E
4
.
ForE
1, by Lemma 4, we have
(41)
E
1
≲
(
∫
0
r
‍
|
∫
|
x
-
y
|
≤
t
‍
1
|
x
-
y
|
d
-
1
(
|
x
-
y
|
ρ
(
x
)
)
δ
f
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
≲
r
-
σ
(
∫
0
r
‍
|
∑
k
=
-
∞
0
‍
1
(
2
k
-
1
t
)
d
-
δ
-
1
∫
|
x
-
y
|
≤
2
k
t
‍
|
f
(
y
)
|
d
y
|
2
d
t
t
3
)
1
/
2
≲
r
-
σ
(
∫
0
r
‍
|
∑
k
=
-
∞
0
‍
(
2
k
)
δ
+
1
t
δ
+
1
(
2
k
t
)
d
∫
|
x
-
y
|
≤
2
k
t
‍
|
f
(
y
)
|
d
y
|
2
d
t
t
3
)
1
/
2
≲
r
-
σ
(
∫
0
r
‍
|
∑
k
=
-
∞
0
‍
(
2
k
)
δ
+
1
t
δ
+
1
M
f
(
x
)
|
2
d
t
t
3
)
1
/
2
≲
r
-
σ
(
∫
0
r
‍
t
2
δ
-
1
d
t
)
1
/
2
M
f
(
x
)
≲
M
f
(
x
)
.
Obviously,(42)
E
2
≤
μ
j
f
(
x
)
.
ForE
3, using Lemma 4 again, we get
(43)
E
3
≲
(
∫
r
∞
‍
|
∫
|
x
-
y
|
≤
r
‍
1
|
x
-
y
|
d
-
1
f
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
≲
(
∫
r
∞
‍
|
∑
k
=
-
∞
0
‍
1
(
2
k
-
1
r
)
d
-
1
∫
|
x
-
y
|
≤
2
k
r
‍
f
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
≲
(
∫
r
∞
‍
|
∑
k
=
-
∞
0
‍
2
k
r
(
2
k
r
)
d
∫
|
x
-
y
|
≤
2
k
r
‍
f
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
≲
(
∫
r
∞
‍
|
∑
k
=
-
∞
0
‍
2
k
r
M
f
(
x
)
|
2
d
t
t
3
)
1
/
2
≲
r
(
∫
r
∞
‍
d
t
t
3
)
1
/
2
M
f
(
x
)
≲
M
f
(
x
)
.
It remains to estimateE
4. From Lemma 4, we obtain
(44)
E
4
≲
(
∫
r
∞
‍
|
r
∫
r
<
|
x
-
y
|
≤
t
‍
|
f
(
y
)
|
|
x
-
y
|
n
d
y
|
2
d
t
t
3
)
1
/
2
≲
r
(
∫
r
∞
‍
|
∑
k
=
0
[
log
⁡
2
t
/
r
]
+
1
‍
(
2
k
r
)
-
n
∫
|
x
-
y
|
≤
2
k
r
‍
f
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
≲
r
(
∫
r
∞
‍
|
(
[
log
⁡
2
t
r
]
+
1
)
M
f
(
x
)
|
2
d
t
t
3
)
1
/
2
≲
r
(
∫
r
∞
‍
t
r
M
f
(
x
)
2
d
t
t
3
)
1
/
2
≲
M
f
(
x
)
.
With the help of theL
p-boundedness and weak (
1,1
)-boundedness of μ
j and M, we can get the same boundedness of μ
j
L.
For theBM
O
L-boundedness and (
H
L
1
,
L
1
)-boundedness of μ
j
L, we only make similar modification in the procedure of the same estimate in [6]. Here, we omit it.Next we will establish some boundedness for the commutator of Marcinkiewicz integral of Schrödinger operator type.We start with the proof of Theorem6.Proof.
For any functionf
∈
L
p
(
R
d
) with 1
<
p
<
p
0, we notice that, due to Lemma 11, we have [
b
,
μ
j
L
]
f
∈
L
loc
⁡
1
(
R
d
).
By using Lemma8 and Lemma 11 with 1
<
p
<
p
0, and Remark 12, we have
(45)
∥
[
b
,
μ
j
L
]
f
∥
p
p
≤
∫
R
d
‍
|
M
ρ
,
β
(
[
b
,
μ
j
L
]
f
)
|
p
≲
∫
R
d
‍
|
M
ρ
,
γ
♯
(
[
b
,
μ
j
L
]
f
)
|
p
+
∑
k
‍
|
Q
k
|
(
1
|
Q
k
|
∫
2
Q
k
‍
|
[
b
,
μ
j
L
]
f
|
)
p
≲
∫
R
d
‍
|
M
ρ
,
γ
♯
(
[
b
,
μ
j
L
]
f
)
|
p
+
[
b
]
θ
p
∑
k
‍
∫
2
Q
k
‍
|
M
s
f
|
p
.
By the finite overlapping property given by Proposition2 and the boundedness of M
s in L
p
(
R
d
), the second term is controlled by [
b
]
θ
p
∥
f
∥
p
p. Thus, we have to take care of the first term.
Our goal is to find a point-wise estimate ofM
ρ
,
γ
♯
(
[
b
,
μ
j
L
]
f
). Let x
∈
R
d and B
=
B
(
x
0
,
r
), with r
<
γ
ρ
(
x
0
) such that x
∈
B. If f
=
f
1
+
f
2, with f
1
=
f
χ
2
B, then we write
(46)
1
|
B
|
∫
B
‍
[
b
,
μ
j
L
]
f
(
x
)
d
x
≤
1
|
B
|
∫
B
‍
|
b
-
b
B
|
μ
j
L
f
(
x
)
d
x
+
1
|
B
|
∫
B
‍
μ
j
L
(
f
1
(
b
-
b
B
)
)
(
x
)
d
x
+
1
|
B
|
∫
B
‍
μ
j
L
(
f
2
(
b
-
b
B
)
)
(
x
)
d
x
=
U
1
+
U
2
+
U
3
.
Let1
<
s
<
p
0; an application of Hölder's inequality and Lemma 10 gives
(47)
U
1
=
1
|
B
|
∫
B
‍
|
|
b
-
b
B
|
μ
j
L
f
-
1
|
B
|
∫
B
‍
|
b
-
b
B
|
μ
j
L
f
|
≤
2
|
B
|
∫
B
‍
|
(
b
-
b
B
)
μ
j
L
f
|
≤
(
1
|
B
|
∫
B
‍
|
b
-
b
B
|
s
′
)
1
/
s
′
(
1
|
B
|
∫
B
‍
|
μ
j
L
f
|
s
)
1
/
s
≲
[
b
]
θ
M
s
μ
j
L
f
(
x
)
,
since r
/
ρ
(
x
0
)
<
γ.
To estimateU
2, let 1
<
s
~
<
s
<
p
0 and ν
=
s
~
s
/
(
s
-
s
~
). Then,
(48)
U
2
=
1
|
B
|
∫
B
‍
|
μ
j
L
(
(
b
-
b
B
)
f
1
)
-
1
|
B
|
∫
B
‍
μ
j
L
(
(
b
-
b
B
)
f
1
)
|
≤
2
|
B
|
∫
B
‍
|
μ
j
L
(
(
b
-
b
B
)
f
1
)
|
≲
(
1
|
B
|
∫
B
‍
|
μ
j
L
(
(
b
-
b
B
)
f
1
)
|
s
~
)
1
/
s
~
≲
(
1
|
B
|
∫
2
B
‍
|
(
b
-
b
B
)
f
|
s
~
)
1
/
s
~
≲
(
1
|
B
|
∫
2
B
‍
|
b
-
b
B
|
ν
)
1
/
ν
(
1
|
B
|
∫
2
B
‍
|
f
|
s
)
1
/
s
≲
[
b
]
θ
M
s
f
(
x
)
.
ForU
3, using Lemma 13, we get
(49)
U
3
=
1
|
B
|
∫
B
‍
|
∫
B
‍
μ
j
L
(
f
2
(
b
-
b
B
)
)
(
u
)
w
w
w
w
a
-
1
|
B
|
∫
B
‍
μ
j
L
(
f
2
(
b
-
b
B
)
)
(
z
)
|
d
u
d
z
≲
1
|
B
|
2
∬
B
|
μ
j
L
(
f
2
(
b
-
b
B
)
)
(
u
)
w
w
w
w
w
w
i
-
μ
j
L
(
f
2
(
b
-
b
B
)
)
(
z
)
|
d
u
d
z
≲
1
|
B
|
2
∬
B
[
b
]
θ
M
s
f
(
x
)
d
u
d
z
≲
[
b
]
θ
M
s
f
(
x
)
.
Therefore, we have proved that(50)
|
M
ρ
,
γ
♯
(
[
b
,
μ
j
L
]
f
)
|
≲
[
b
]
θ
[
M
s
μ
j
L
f
(
x
)
+
M
s
f
(
x
)
]
.
Sinces
<
p, we obtain the desired result.Finally, we will prove Theorem7.Proof.
Letf
∈
L
∞
(
R
d
) and Q
=
B
(
x
0
,
γ
ρ
(
x
0
)
). From Proposition 2, we only consider averages over critical ball. Thanks to Lemma 11, one gets
(51)
1
|
Q
|
∫
Q
‍
|
[
b
,
μ
j
L
]
f
|
≲
[
b
]
θ
inf
⁡
y
∈
Q
M
s
f
(
y
)
≲
[
b
]
θ
∥
f
∥
∞
.
Next, we deal with the oscillations; letB
=
B
(
x
0
,
r
) with r
<
γ
ρ
(
x
0
). Lemma 11 states that the function [
b
,
μ
j
L
]
f belongs to L
loc
⁡
1
(
R
d
).
We write(52)
[
b
,
μ
j
L
]
f
≤
|
b
-
b
B
|
μ
j
L
f
+
μ
j
L
(
f
1
(
b
-
b
B
)
)
+
μ
j
L
(
f
2
(
b
-
b
B
)
)
,
and its mean oscillations on B as E
1, E
2, and E
3.
The estimates for the termsE
2 and E
3 are the same with U
2 and U
3 in Theorem 6. Due to the boundedness of μ
j
L in L
s
~
(
R
d
), for s
~
>
1, and Lemma 13, we get that E
2 and E
3 are bounded by [
b
]
θ
∥
f
∥
∞.
To deal withE
1, we fix u
∈
B and write
(53)
|
b
-
b
B
|
μ
j
L
f
≤
|
b
-
b
B
|
μ
j
L
f
1
+
|
b
-
b
B
|
(
μ
j
L
f
2
-
μ
j
L
f
2
(
u
)
)
+
μ
j
L
f
21
(
u
)
|
b
-
b
B
|
+
μ
j
L
f
22
(
u
)
|
b
-
b
B
|
,
where f
2
=
f
21
+
f
22, with f
22
=
f
χ
4
Q
∖
2
B and Q
=
B
(
x
0
,
γ
ρ
(
x
0
)
). We denoted each oscillation E
11, E
12, E
13, and E
14.
We observe thatμ
j
L
f
21
(
u
) and μ
j
L
f
22
(
u
) are finite, for any u
∈
B, since f
∈
L
∞ and
(54)
∫
(
2
B
)
c
‍
|
K
j
L
(
u
,
z
)
|
d
z
<
∞
.
We will see thatE
11, E
12, and E
13 are bounded under the condition b
∈
BM
O
∞
(
ρ
).
ForE
11, by choosing s such that μ
j
L is bounded on L
s
(
R
d
), we have
(55)
E
11
≤
2
|
B
|
∫
B
‍
|
(
b
-
b
B
)
μ
j
L
f
1
|
≲
(
1
|
B
|
∫
B
‍
|
b
-
b
B
|
s
′
)
1
/
s
′
(
1
|
B
|
∫
B
‍
|
μ
j
L
f
1
|
s
)
1
/
s
≲
(
1
|
B
|
∫
B
‍
|
b
-
b
B
|
s
′
)
1
/
s
′
(
1
|
B
|
∫
2
B
‍
|
f
|
s
)
1
/
s
≲
[
b
]
θ
∥
f
∥
∞
.
To deal withE
12, we claim
(56)
|
μ
j
L
f
2
(
x
)
-
μ
j
L
f
2
(
u
)
|
≲
∥
f
∥
∞
,
for any x and u in B.
It is easy to see that(57)
|
μ
j
L
f
2
(
x
)
-
μ
j
L
f
2
(
u
)
|
≤
(
∫
0
∞
‍
|
∫
|
x
-
y
|
≤
t
<
|
u
-
y
|
‍
|
K
j
L
(
x
,
y
)
f
2
(
y
)
|
d
y
|
2
d
t
t
3
)
1
/
2
+
(
∫
0
∞
‍
|
∫
|
u
-
y
|
≤
t
<
|
x
-
y
|
‍
|
K
j
L
(
u
,
y
)
f
2
(
y
)
|
d
y
|
2
d
t
t
3
)
1
/
2
+
(
∫
0
∞
‍
|
∫
{
|
x
-
y
|
≤
t
,
|
u
-
y
|
≤
t
}
‍
|
K
j
L
(
x
,
y
)
-
K
j
L
(
u
,
y
)
|
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
a
×
|
f
2
(
y
)
|
d
y
∫
0
∞
‍
|
2
d
t
t
3
)
1
/
2
=
I
1
+
I
2
+
I
3
.
Since the estimates forI
1 and I
2 follow along similar lines, we only consider I
1. Since |
x
-
y
|
⋍
|
u
-
y
|, by Minkowski's inequality and Lemma 4, we have
(58)
I
1
≲
∫
R
n
‍
|
f
2
(
y
)
|
|
K
j
L
(
x
,
y
)
|
(
∫
|
x
-
y
|
≤
t
<
|
u
-
y
|
‍
d
t
t
3
)
1
/
2
d
y
≲
r
1
/
2
∫
(
2
B
)
c
‍
|
f
(
y
)
|
|
K
j
L
(
x
,
y
)
|
|
x
-
y
|
3
/
2
d
y
≲
r
1
/
2
∫
(
2
B
)
c
‍
|
f
(
y
)
|
|
x
-
y
|
n
+
1
/
2
d
y
≲
1
r
n
∑
j
=
1
∞
‍
2
-
j
(
n
+
1
/
2
)
∫
2
j
+
1
B
‍
|
f
(
y
)
|
d
y
≲
∥
f
∥
∞
.
ForI
3, again thanks to the Minkowski's inequality, Lemma 4, and |
x
-
y
|
⋍
|
u
-
y
|, we obtain
(59)
I
3
≲
∫
R
n
‍
|
f
2
(
y
)
|
|
K
j
L
(
x
,
y
)
-
K
j
L
(
u
,
y
)
|
w
w
w
×
(
∫
{
|
x
-
y
|
≤
t
,
|
u
-
y
|
≤
t
}
‍
d
t
t
3
)
1
/
2
d
y
≲
∫
(
2
B
)
c
‍
|
f
(
y
)
|
|
K
j
L
(
x
,
y
)
-
K
j
L
(
u
,
y
)
|
|
x
-
y
|
d
y
≲
r
δ
∫
(
2
B
)
c
‍
|
f
(
y
)
|
|
x
-
y
|
n
+
δ
d
y
≲
1
r
n
∑
j
=
1
∞
‍
2
-
j
(
n
+
δ
)
∫
2
j
+
1
B
‍
|
f
(
y
)
|
d
y
≲
∥
f
∥
∞
.
Thus, the claim is completed.
Then,(60)
E
12
≤
2
|
B
|
∫
B
‍
|
b
-
b
B
|
|
μ
j
L
f
2
(
x
)
-
μ
j
L
f
2
(
u
)
|
d
x
≲
[
b
]
θ
∥
f
∥
∞
.
ThatE
13
≲
[
b
]
θ
∥
f
∥
∞ is a consequence of (54).
Therefore, the theorem will follow, if there exists a constantC such that, for any B
∈
B
ρ
,
γ and u
∈
B,
(61)
(
1
|
B
|
∫
B
‍
|
b
(
y
)
-
b
B
|
d
y
)
|
μ
j
L
f
22
(
u
)
|
≤
C
∥
b
∥
*
log
⁡
∥
f
∥
∞
.
Using Minkowski's inequality,(62)
|
μ
j
L
f
22
(
u
)
|
=
(
∫
0
∞
‍
|
∫
|
x
-
y
|
≤
t
‍
K
j
L
(
x
,
y
)
f
22
(
y
)
d
y
|
2
d
t
t
3
)
1
/
2
≲
∫
R
d
‍
|
f
22
(
y
)
|
u
-
y
|
d
-
1
|
(
∫
|
u
-
y
|
≤
t
‍
d
t
t
3
)
1
/
2
d
y
≲
∫
4
Q
∖
2
B
‍
|
f
(
y
)
|
|
u
-
y
|
d
d
y
≲
∥
f
∥
∞
log
⁡
ρ
(
x
0
)
r
.
Sinceb
∈
BM
O
∞
log
⁡
(
ρ
), we conclude that (61) holds proving the boundedness of [
b
,
μ
j
L
].



---

*Source: 402713-2014-05-08.xml*