# Weighted Endpoint Estimates for Commutators of Singular Integral Operators on Orlicz-Morrey Spaces

Authors: Jinyun Qi; Hongxia Shi; Wenming Li
Journal: Journal of Function Spaces
 (2019)

---

## Abstract

In this paper, we obtain the weighted endpoint estimates for the commutators of the singular integral operators with theB
M
O functions and the associated maximal operators on Orlicz-Morrey Spaces. We also get the similar results for the commutators of the fractional integral operators with the B
M
O functions and the associated maximal operators.


---

## Body

## 1. Introduction and Main Results

The Morrey spaces were introduced by Morrey in [1] to investigate the local behavior of solutions to second-order elliptic partial differential equations. Chiarenza and Frasca [2] showed the boundedness of the Hardy-Littlewood maximal operator, singular integral operators, and the fractional integral operators on the Morrey spaces. Komori and Shirai [3] introduced the weighted Morrey spaces and proved that, for 1
<
p
<
∞ and w
∈
A
p, T and [
b
,
T
] are bounded on L
p
,
κ
(
w
), and if p
=
1 and w
∈
A
1, then for all t
>
0 and any cube Q, (1)
w
x
∈
Q
:
T
f
x
>
t
≤
C
t
f
L
1
,
κ
w
w
Q
κ
.In this paper, we obtain the weighted endpoint estimates for the commutators of the singular integral operators withB
M
O functions and associated maximal operators. We also obtain the similar results for the commutators of the fractional integral operators with B
M
O functions and associated maximal operators.Letf be a measurable function on R
n and 1
≤
p
<
∞, 0
≤
κ
<
1, for two weights w and u, and the weighted Morrey space is defined by (2)
L
p
,
κ
w
,
u
=
f
∈
L
l
o
c
p
w
:
f
L
p
,
κ
w
,
u
<
∞
,where(3)
f
L
p
,
κ
w
,
u
=
s
u
p
Q
1
u
Q
κ
∫
Q
f
x
p
w
x
d
x
1
/
p
,and the supremum is taken over all cubes Q in R
n. When w
=
u, we write L
p
,
κ
(
w
,
u
) as L
p
,
κ
(
w
).We say thatT is a singular integral operator if there exists a function K which satisfies the following conditions: (4)
T
f
x
=
p
.
v
.
∫
R
n
K
x
-
y
f
y
d
y
,
K
x
≤
C
x
n
,
∇
K
x
≤
C
x
n
+
1
,
x
≠
0
.TheB
M
O
(
R
n
) space is defined by (5)
B
M
O
R
n
=
b
∈
L
l
o
c
R
n
:
b
B
M
O
=
s
u
p
Q
1
Q
∫
Q
b
x
-
b
Q
d
x
<
∞
,where b
Q
=
1
/
|
Q
|
∫
Q
‍
b
(
y
)
d
y.For the singular integral operatorT and b
∈
B
M
O, the commutator [
b
,
T
] is defined by(6)
b
,
T
f
x
=
∫
R
n
b
x
-
b
y
K
x
-
y
f
y
d
y
.In order to state our results, we need to recall some notations and facts about the Young functions and Orlicz spaces; for further information, see [4]. A function Φ
:
[
0
,
∞
)
→
[
0
,
∞
) is a Young function if it is convex and increasing, and if Φ
(
0
)
=
0 and Φ
(
t
)
→
∞ as t
→
∞.LetΦ be a Young function, 0
<
κ
<
1 and two weights w and u, and the weighted Orlicz-Morrey Class L
Φ
,
κ
(
w
,
u
) is defined as (7)
L
Φ
,
κ
w
,
u
=
f
:
f
L
Φ
,
κ
w
,
u
<
∞
,where (8)
f
L
Φ
,
κ
w
,
u
=
s
u
p
Q
1
u
Q
κ
∫
Q
Φ
f
x
w
x
d
x
.When w
=
u, we write L
Φ
,
κ
(
w
,
u
) as L
Φ
,
κ
(
w
).Given a locally integrable functionf and a Young function Φ, define the mean Luxemburg norm of f on a cube Q by(9)
f
Φ
,
Q
=
inf
⁡
λ
>
0
:
1
Q
∫
Q
Φ
f
x
λ
d
x
≤
1
.Forα
,
0
≤
α
<
n, and a Young function Φ, we define Orlicz maximal operator (10)
M
α
,
Φ
f
x
=
s
u
p
Q
∋
x
Q
α
/
n
f
Φ
,
Q
.Ifα
=
0, we write M
α
,
Φ simply as M
Φ. If α
=
0 and Φ
(
t
)
=
t, M
α
,
Φ is the Hardy-Littlewood maximal operator M. If Φ
ε
t
=
t
log
⁡
(
e
+
t
)
ε
,
ε
≥
0, we write M
Φ
ε simply as M
L
(
log
⁡
L
)
ε.If0
<
α
<
n and Φ
(
t
)
=
t, M
α
,
Φ is the fractional maximal operator of order α and we write it as M
α. If Φ
ε
t
=
t
log
⁡
(
e
+
t
)
ε, we write M
α
,
Φ simply as M
α
,
L
(
log
⁡
L
)
ε.Takew
∈
A
1, which means M
w
(
x
)
≤
C
w
(
x
) for a.e. x
∈
R
n.Givenα
,
0
<
α
<
n, for an appropriate function f on R
n, the fractional integral operator (or the Riesz potential) of order α is defined by(11)
I
α
f
x
=
∫
R
n
f
y
x
-
y
n
-
α
d
y
.Forb
∈
B
M
O
(
R
n
), we define the commutators of the operator I
α and b by (12)
b
,
I
α
f
x
=
∫
R
n
b
x
-
b
y
f
y
x
-
y
n
-
α
d
y
.The following theorems are our main results.Theorem 1.
Letw
∈
A
1 and Φ
t
=
t
log
⁡
(
e
+
t
), then there exists a positive constant C such that, for any cube Q and any t
>
0, (13)
w
x
∈
Q
:
M
L
log
⁡
L
f
x
>
t
≤
C
f
t
L
Φ
,
κ
w
w
Q
κ
.Theorem 2.
LetT be any singular integral operator, w
∈
A
1, Φ
t
=
t
log
⁡
(
e
+
t
), and b
∈
B
M
O. Then there exists a positive constant C such that, for any cube Q and any t
>
0, (14)
w
x
∈
Q
:
b
,
T
f
x
>
t
≤
C
f
t
L
Φ
,
κ
w
w
Q
κ
.Theorem 3.
Let0
<
α
<
n, w
∈
A
1, 1
/
q
=
1
-
α
/
n, 0
<
κ
<
1
/
q, Φ
t
=
t
log
⁡
(
e
+
t
), Ψ
(
t
)
=
t
1
/
q
log
⁡
(
e
+
t
)
-
1, and Θ
(
t
)
=
t
1
/
q
log
⁡
(
e
+
t
-
1
). Then there exists a positive constant C such that, for any cube Q and any t
>
0, (15)
Ψ
w
x
∈
Q
:
M
α
,
L
log
⁡
L
f
x
>
t
≤
C
f
t
L
Φ
,
κ
w
,
Θ
w
w
Q
κ
.Theorem 4.
Let0
<
α
<
n, w
∈
A
1, b
∈
B
M
O, 1
/
q
=
1
-
α
/
n, 0
<
κ
<
1
/
q, Φ
(
t
)
=
t
log
⁡
(
e
+
t
), Ψ
(
t
)
=
t
1
/
q
l
o
g
(
e
+
t
)
-
1, and Θ
(
t
)
=
t
1
/
q
l
o
g
(
e
+
t
-
1
). Then there exists a positive constant C such that, for any cube Q and any t
>
0, (16)
Ψ
w
x
∈
Q
:
b
,
I
α
f
x
>
t
≤
C
f
t
L
Φ
,
κ
w
,
Θ
w
w
Q
κ
.


## 2. Proof of Theorems1 and 2

Lemma 5 (see [5]).
LetΦ
t
=
t
log
⁡
(
e
+
t
), then there exists a positive constant C such that, for any weight w and all t
>
0, (17)
w
x
∈
R
n
:
M
L
log
⁡
L
f
x
>
t
≤
C
∫
R
n
Φ
f
x
t
M
w
x
d
xfor every locally integrable function f.Lemma 6 (see [6]).
Letw
∈
A
1, then there exist a constant C
>
0 and η
>
0 such that, for any cube Q and a measurable subset E
⊂
Q, (18)
w
E
w
Q
≤
C
E
Q
η
.Proof of Theorem1.
Fix a cubeQ centered at x
0. By Lemma 5, we have (19)
w
x
∈
Q
:
M
L
log
⁡
L
f
x
>
t
=
∫
x
∈
R
n
:
M
L
log
⁡
L
f
x
>
t
χ
Q
w
x
d
x
≤
C
∫
R
n
Φ
f
x
t
M
χ
Q
w
x
d
x
≤
C
∫
3
Q
+
∫
3
Q
c
Φ
f
x
t
M
χ
Q
w
x
d
x
≤
I
+
I
I
.
To estimate term I, sincew
∈
A
1, we have (20)
I
≤
C
∫
3
Q
Φ
f
x
t
w
x
d
x
≤
C
f
t
L
Φ
,
κ
w
w
Q
κ
.
For term II, observe that, forx
∈
(
3
Q
)
c, x
∈
R and R
∩
Q
≠
∅. We have (21)
1
R
∫
R
χ
Q
y
w
y
d
y
=
1
R
∫
R
∩
Q
w
y
d
y
≤
C
x
-
x
0
n
∫
Q
w
y
d
y
=
C
x
-
x
0
n
w
Q
.Therefore we obtain (22)
M
χ
Q
w
x
≤
C
x
-
x
0
-
n
w
Q
.Since w
∈
A
1, using Lemma 6, we get (23)
I
I
≤
C
∫
3
Q
c
Φ
f
x
t
x
-
x
0
-
n
w
Q
d
x
≤
C
w
Q
∑
j
=
1
∞
∫
3
j
+
1
Q
∖
3
j
Q
Φ
f
x
t
x
-
x
0
-
n
d
x
≤
C
w
Q
∑
j
=
1
∞
1
3
j
Q
∫
3
j
+
1
Q
Φ
f
x
t
d
x
≤
C
w
Q
κ
∑
j
=
1
∞
w
Q
1
-
κ
w
3
j
+
1
Q
1
-
κ
1
w
3
j
+
1
Q
κ
∫
3
j
+
1
Q
Φ
f
x
t
w
x
d
x
≤
C
w
Q
κ
f
t
L
Φ
,
κ
w
∑
j
=
1
∞
1
3
j
n
η
1
-
κ
≤
C
w
Q
κ
f
t
L
Φ
,
κ
w
.This ends the proof.Lemma 7 (see [7]).
LetT be any Calderón-Zygmund singular integral operator, Φ
t
=
t
log
⁡
(
e
+
t
), ε
>
0, and b
∈
B
M
O. Then there exists a positive constant C such that, for all weights w, (24)
w
x
∈
R
n
:
b
,
T
f
x
>
t
≤
C
∫
R
n
Φ
f
x
t
M
L
log
⁡
L
1
+
ε
w
x
d
x
.Lemma 8 (see [6]).
Letw
∈
A
1, then there exist a constant C
>
0 and θ
>
0 such that, for any cube Q, (25)
1
Q
∫
Q
w
y
1
+
θ
d
y
1
/
1
+
θ
≤
C
1
Q
∫
Q
w
y
d
y
.Proof of Theorem2.
Fix a cubeQ centered at x
0. By Lemma 7, we have (26)
w
x
∈
Q
:
b
,
T
f
x
>
t
=
∫
x
∈
R
n
:
b
,
T
f
x
>
t
w
x
χ
Q
x
d
x
≤
C
∫
R
n
Φ
f
x
t
M
L
log
⁡
L
1
+
ε
w
χ
Q
x
d
x
≤
C
∫
3
Q
+
∫
3
Q
c
Φ
f
x
t
M
L
log
⁡
L
1
+
ε
w
χ
Q
x
d
x
≤
I
+
I
I
.
To estimate term I, sincew
∈
A
1, it is easy to prove that M
L
(
log
⁡
L
)
1
+
ε
(
w
χ
Q
)
(
x
)
≤
C
w
(
x
), x
∈
3
Q, and we have (27)
I
≤
C
∫
3
Q
Φ
f
x
t
w
x
d
x
≤
C
f
t
L
Φ
,
κ
w
w
Q
κ
.
For term II, observe that, forx
∈
(
3
Q
)
c, x
∈
R, R is a cube, and R
∩
Q
≠
∅, by Lemma 8, for any δ
:
0
<
δ
≤
θ, we have (28)
1
R
∫
R
w
y
χ
Q
y
1
+
δ
d
y
1
/
1
+
δ
≤
1
R
∫
Q
w
y
1
+
δ
d
y
1
/
1
+
δ
=
Q
R
1
/
1
+
δ
1
Q
∫
Q
w
y
1
+
δ
d
y
1
/
1
+
δ
≤
C
Q
R
1
/
1
+
δ
1
Q
∫
Q
w
y
d
y
≤
C
Q
R
1
/
1
+
δ
w
Q
Q
.Noticing the definition of the maximal function M, we obtain (29)
M
L
log
⁡
L
1
+
ε
w
χ
Q
x
≤
M
w
1
+
δ
χ
Q
x
1
/
1
+
δ
≤
C
Q
x
-
x
0
n
1
/
1
+
δ
w
Q
Q
.By Lemma 6, we get (30)
I
I
≤
C
∫
3
Q
c
Φ
f
x
t
Q
x
-
x
0
n
1
/
1
+
δ
w
Q
Q
d
x
≤
C
∑
j
=
1
∞
∫
3
j
+
1
Q
∖
3
j
Q
Φ
f
x
t
Q
x
-
x
0
n
1
/
1
+
δ
w
Q
Q
d
x
≤
C
w
Q
κ
∑
j
=
1
∞
w
Q
w
3
j
+
1
Q
1
-
κ
Q
3
j
+
1
Q
-
δ
/
1
+
δ
1
w
3
j
+
1
Q
κ
∫
3
j
+
1
Q
Φ
f
x
t
w
x
d
x
≤
C
w
Q
κ
∑
j
=
1
∞
Q
3
j
+
1
Q
η
1
-
κ
-
δ
/
1
+
δ
f
t
L
Φ
,
κ
w
≤
C
w
Q
κ
f
t
L
Φ
,
κ
w
,in which we take δ
>
0 small enough such that η
(
1
-
κ
)
-
δ
/
(
1
+
δ
)
>
0. This ends the proof.


## 3. Proof of Theorems3 and 4

Given an increasing functionφ
:
[
0
,
∞
)
→
[
0
,
∞
), as in [8], we define the function h
φ by (31)
h
φ
s
=
s
u
p
t
>
0
φ
s
t
φ
t
,
0
≤
s
<
∞
.Ifφ is submultiplicative, then h
φ
≈
φ. Also, for all s
,
t
>
0, φ
(
s
t
)
≤
h
φ
(
s
)
φ
(
t
).In this section, we setΦ
t
=
t
log
⁡
(
e
+
t
), it is submultiplicative, and so h
Φ
≈
Φ. Let 0
<
α
<
n, and q be a number 1
/
q
=
1
-
α
/
n. Denote (32)
Ψ
t
=
0
,
t
=
0
,
t
Φ
t
α
/
n
,
t
>
0
.So(33)
Ψ
t
≈
t
1
/
q
log
⁡
e
+
t
-
1
.The function Ψ is invertible with (34)
Ψ
-
1
t
≈
Γ
t
=
t
log
⁡
e
+
t
q
=
Φ
t
q
.Lemma 9 (see [8]).
Ifφ
(
t
)
/
t is decreasing, then, for any positive sequence {
t
j
}, (35)
φ
∑
j
t
j
≤
∑
j
φ
t
j
.Lemma 10.
Let0
<
α
<
n, 1
/
q
=
1
-
α
/
n. Then there exists a constant C
>
0 such that, for any t
>
0, for any weight w, we have (36)
Ψ
w
x
∈
R
n
:
M
α
,
L
log
⁡
L
f
x
>
t
≤
C
∫
R
n
Φ
f
y
t
h
Ψ
M
w
y
d
y
.Proof.
By homogeneity, we may assume thatt
=
1. Define the set (37)
Ω
=
x
∈
R
n
:
M
α
,
L
log
⁡
L
f
x
>
1
.It is easy to see that Ω is open and we may assume that it is not empty. To estimate the size of Ω, it is enough to estimate the size of every compact set F contained in Ω. We can cover F by a finite family of cubes {
Q
j
} for which (38)
Q
j
α
/
n
f
L
log
⁡
L
,
Q
j
>
1
.Using Vitali’s covering lemma, we can extract a subfamily of disjoint cubes {
Q
k
} such that (39)
F
⊂
⋃
k
3
Q
k
.For each k, by homogeneity and the properties of the norm ·
Φ
,
Q, we have (40)
1
<
1
Q
k
∫
Q
k
Φ
f
y
Q
k
α
/
n
d
y
≤
C
Φ
Q
k
α
/
n
Q
k
∫
Q
k
Φ
f
y
d
y
≤
C
Ψ
Q
k
∫
Q
k
Φ
f
y
d
y
.
For eachk, we have (41)
Ψ
w
Q
k
≤
C
Ψ
w
Q
k
Ψ
Q
k
∫
Q
k
Φ
f
y
d
y
≤
C
h
Ψ
w
Q
k
Q
k
∫
Q
k
Φ
f
y
d
y
≤
C
∫
Q
k
Φ
f
y
h
Ψ
M
w
y
d
y
.It is easy to see that Ψ
(
t
)
/
t is decreasing; by Lemma 9, we have (42)
Ψ
w
F
≤
∑
k
Ψ
w
Q
k
≤
C
∑
k
∫
Q
k
Φ
f
y
h
Ψ
M
w
y
d
y
≤
C
∫
R
n
Φ
f
y
h
Ψ
M
w
y
d
y
.This ends the proof.Proof of Theorem3.
Fix a cubeQ centered at x
0. By Lemma 10, we have (43)
Ψ
w
x
∈
Q
:
M
α
,
L
log
⁡
L
f
x
>
t
=
Ψ
∫
x
∈
R
n
:
M
α
,
L
log
⁡
L
f
x
>
t
w
x
χ
Q
x
d
x
≤
C
∫
R
n
Φ
f
x
t
h
Ψ
M
w
χ
Q
x
d
x
≤
C
∫
3
Q
+
∫
3
Q
c
Φ
f
x
t
h
Ψ
M
w
χ
Q
x
d
x
≤
I
+
I
I
.
Now we estimate term I. Noticing that, fors
>
0, we have (44)
h
Ψ
s
=
s
u
p
t
>
0
Ψ
s
t
Ψ
t
=
s
s
u
p
t
>
0
Φ
t
α
/
n
Φ
(
s
t
α
/
n
≤
C
Θ
s
.Since w
∈
A
1, we get (45)
I
≤
C
∫
3
Q
Φ
f
x
t
h
Ψ
w
x
d
x
≤
C
∫
3
Q
Φ
f
x
t
Θ
w
x
d
x
≤
C
f
t
L
Φ
,
κ
w
,
Θ
w
w
Q
κ
.
For term II, observe that, forx
∈
(
3
Q
)
c, x
∈
R and R
∩
Q
≠
∅. As in the proof of Theorem 1, we have (46)
M
χ
Q
w
x
≤
C
x
-
x
0
-
n
w
Q
.Since w
∈
A
1, Θ is submultiplicative, and using Lemma 6, we get (47)
I
I
≤
C
∫
3
Q
c
Φ
f
x
t
h
Ψ
x
-
x
0
-
n
w
Q
d
x
≤
C
∑
j
=
1
∞
∫
3
j
+
1
Q
∖
3
j
Q
Φ
f
x
t
Θ
w
Q
3
j
+
1
Q
d
x
≤
C
∑
j
=
1
∞
∫
3
j
+
1
Q
Φ
f
x
t
Θ
w
3
j
+
1
Q
3
j
+
1
Q
w
Q
w
3
j
+
1
Q
d
x
≤
C
∑
j
=
1
∞
∫
3
j
+
1
Q
Φ
f
x
t
Θ
w
x
w
Q
w
3
j
+
1
Q
d
x
≤
C
∑
j
=
1
∞
w
Q
1
/
q
w
3
j
+
1
Q
1
/
q
log
⁡
e
+
w
3
j
+
1
Q
w
Q
∫
3
j
+
1
Q
Φ
f
x
t
Θ
w
x
d
x
≤
C
w
Q
κ
f
t
L
Φ
,
κ
w
,
Θ
w
∑
j
=
1
∞
1
3
j
n
η
1
/
q
-
κ
log
⁡
e
+
3
j
n
η
≤
C
w
Q
κ
f
t
L
Φ
,
κ
w
,
Θ
w
.This ends the proof.Lemma 11 (see [9]).
Let0
<
α
<
n, 1
/
q
=
1
-
α
/
n, w
∈
A
1, and b
∈
B
M
O. Then there exists a constant C
>
0 such that, for any t
>
0, (48)
Ψ
w
x
∈
R
n
:
b
,
I
α
f
x
>
t
≤
C
∫
R
n
Φ
f
y
t
Θ
w
y
d
y
.Lemma 12 (see [6]).
Letf
(
x
)
≥
0, f
∈
L
l
o
c
1
(
R
n
), and 0
<
δ
<
1, then M
(
f
)
δ
∈
A
1.Proof of Theorem4.
Fix a cubeQ centered at x
0, for any w
∈
A
1 and δ
:
0
<
δ
≤
θ, and by Lemma 12, we have M
(
w
1
+
δ
χ
Q
)
1
/
(
1
+
δ
)
∈
A
1. By Lemma 11, we obtain (49)
Ψ
w
x
∈
Q
:
b
,
I
α
f
x
>
t
=
Ψ
∫
x
∈
R
n
:
b
,
I
α
f
x
>
t
w
x
χ
Q
x
d
x
≤
C
Ψ
∫
x
∈
R
n
:
b
,
I
α
f
x
>
t
M
w
χ
Q
x
d
x
≤
C
Ψ
∫
x
∈
R
n
:
b
,
I
α
f
x
>
t
M
w
1
+
δ
χ
Q
x
1
/
1
+
δ
d
x
≤
C
∫
R
n
Φ
f
x
t
Θ
M
w
1
+
δ
χ
Q
x
1
/
1
+
δ
d
x
≤
C
∫
3
Q
+
∫
3
Q
c
Φ
f
x
t
Θ
M
w
1
+
δ
χ
Q
x
1
/
1
+
δ
d
x
≤
I
+
I
I
.
Now we estimate term I. Noticing thatw
∈
A
1, Lemma 8, we have Θ
(
(
M
(
w
1
+
δ
χ
Q
)
(
x
)
)
1
/
(
1
+
δ
)
≤
C
Θ
(
M
w
(
x
)
)
≤
C
Θ
(
w
(
x
)
). Then (50)
I
≤
C
∫
3
Q
Φ
f
x
t
Θ
w
x
d
x
≤
C
f
t
L
Φ
,
κ
(
w
,
Θ
w
w
Q
κ
.
For term II, as the proof of Theorem2, for x
∈
(
3
Q
)
c, (51)
M
w
1
+
δ
χ
Q
x
1
/
1
+
δ
≤
C
Q
x
-
x
0
n
1
/
1
+
δ
w
Q
Q
.By Lemma 6, we get (52)
I
I
≤
C
∫
3
Q
c
Φ
f
x
t
Θ
Q
x
-
x
0
n
1
/
1
+
δ
w
Q
Q
d
x
≤
C
∑
j
=
1
∞
∫
3
j
+
1
Q
∖
3
j
Q
Φ
f
x
t
Θ
Q
3
j
+
1
Q
η
-
δ
/
1
+
δ
w
x
d
x
≤
C
w
Q
κ
∑
j
=
1
∞
Q
3
j
+
1
Q
η
1
/
q
-
κ
-
δ
/
q
1
+
δ
log
⁡
e
+
3
j
+
1
Q
Q
η
-
δ
/
1
+
δ
·
1
w
3
j
+
1
Q
κ
∫
3
j
+
1
Q
Φ
f
x
t
Θ
w
x
d
x
≤
C
w
Q
κ
f
t
L
Φ
,
κ
w
,
Θ
w
∑
j
=
1
∞
1
3
j
n
η
1
/
q
-
κ
-
δ
/
q
1
+
δ
log
⁡
e
+
3
j
n
η
-
δ
/
1
+
δ
≤
C
w
Q
κ
f
t
L
Φ
,
κ
w
,
Θ
w
,in which we take δ
>
0 small enough such that η
(
1
/
q
-
κ
)
-
δ
/
q
(
1
+
δ
)
>
0 and η
-
δ
/
(
1
+
δ
)
>
0. This ends the proof.



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*Source: 5458101-2019-03-04.xml*