# Local Good-λ Estimate for the Sharp Maximal Function and Weighted Morrey Space

Authors: Yasuo Komori-Furuya
Journal: Journal of Function Spaces
 (2015)

---

## Abstract

We give a characterization of weighted Morrey space by using Fefferman and Stein’s sharp maximal function. For this purpose, we consider a local good-λ inequality.


---

## Body

## 1. Introduction

The Fefferman-Stein sharp maximal functionM
♯
f (see Definition 2) plays an important role in harmonic analysis. For example, we know f
L
p
≈
M
♯
f
L
p, where 1
<
p
<
∞. We also know f
L
p
(
w
)
≈
M
♯
f
L
p
(
w
), where L
p
(
w
) is the weighted L
p space (for further details, see [1, 2]).On the other hand, many studies have been done for Morrey spaces:(1)
L
p
,
k
R
n
=
f
;
f
L
p
,
k
=
s
u
p
Q
1
Q
k
∫
Q
f
x
p
d
x
1
/
p
<
∞
.Komori and Shirai [3] introduced weighted Morrey spaces L
p
,
k
(
w
)
(
R
n
) (for the precise definition, see Section 2) and proved the boundedness of the Hardy-Littlewood maximal operator M and singular integral operators.Di Fazio and Ragusa [4] proved (2)
M
f
L
p
,
k
≤
C
M
♯
f
L
p
,
k
f
o
r
f
∈
L
p
,
k
R
n
.From this inequality they obtained (3)
f
L
p
,
k
≈
M
♯
f
L
p
,
k
i
f
f
∈
L
p
,
k
R
n
.However, the condition f
∈
L
p
,
k
(
R
n
) is strong; that is, the left side of inequality (2) is assumed to be finite. In applications (3) is more important than (2). In this paper, we will prove (3) under weaker condition without using (2). Our proof is different from the one in [4]. We use another method, that is, local good-λ estimate. We think this itself is interesting (see Section 4, Lemma 16), and we also consider weighted estimates.This paper is organized as follows. In Section2 we make some definitions for maximal functions, functions spaces, and weights. In Section 3 we state known results and our theorem. In Section 4 we prove our theorems.


## 2. Preliminaries

The following notation is used: for a setE
⊂
R
n, we denote the Lebesgue measure of E by |
E
|, and for a nonnegative locally integrable function w, we write w
(
E
)
=
∫
E
‍
w
(
x
)
d
x. We denote the characteristic function of E by χ
E. Throughout this paper, all cubes are assumed to have their sides parallel to the coordinate axes.


### 2.1. Maximal Functions

First we define some maximal functions.Definition 1 (the Hardy-Littlewood maximal function).
Consider(4)
M
f
x
=
s
u
p
Q
∋
x
1
Q
∫
Q
f
y
d
y
,where the supremum is taken over all cubes containing x.Definition 2 (the sharp maximal function).
Consider(5)
M
♯
f
x
=
s
u
p
Q
∋
x
1
Q
∫
Q
f
y
-
f
Q
d
y
,where f
Q
=
(
1
/
|
Q
|
)
∫
Q
‍
f
(
y
)
d
y.
Next we define the dyadic maximal function. A dyadic cube is a cube of the form(6)
∏
j
=
1
n
m
j
2
l
,
m
j
+
1
2
l
for some integers
m
1
,
m
2
,
…
,
m
n
,
l
.Definition 3 (the dyadic maximal function).
Consider(7)
M
d
f
x
=
s
u
p
Q
∋
x
1
Q
∫
Q
f
y
d
y
,where the supremum is taken over all dyadic cubes containing x.


### 2.2. Weights

Next we define Muckenhoupt’s weight classes (see [1, 2]).Definition 4 (A
q).
Let1
<
q
<
∞. For a nonnegative locally integrable function w, one says that w
∈
A
q if (8)
1
Q
∫
Q
w
x
d
x
1
Q
∫
Q
w
x
-
1
/
q
-
1
d
x
q
-
1
≤
C
,where C is a positive constant independent of cubes Q.Definition 5 (A
∞).
One writes(9)
A
∞
=
⋃
1
<
q
<
∞
A
q
.Definition 6 (doubling condition).
One says thatw
∈
D, if (10)
w
2
Q
≤
C
w
Q
,for any cubes, where 2
Q is the cube that has the same center as Q with double the side length.Definition 7 (reverse doubling condition).
One says thatw
∈
R
D, if there exists C
>
1 such that (11)
w
2
Q
≥
C
w
Q
f
o
r
a
n
y
c
u
b
e
s
Q
,and one denotes the supremum of C in (11) by [
w
]
R
D.Definition 8 (δ-reverse doubling condition).
One says thatw
∈
R
D
(
δ
), where 0
<
δ
<
1, if (12)
w
S
w
Q
≤
C
S
Q
δ
,for any cubes Q and subsets S
⊂
Q, and one denotes the infimum of C in (12) by [
w
]
R
D
(
δ
).The notation in Definitions6, 7, and 8 may not be standard. However, we use these notions frequently; therefore, we use these symbols for short.The following proposition is well-known (see, e.g., [1, 2]).Proposition 9.
Consider the following:(13)
I
f
w
∈
A
∞
,
t
h
e
n
w
∈
D
,
w
∈
R
D
,
(14)
I
f
w
∈
A
∞
,
t
h
e
n
t
h
e
r
e
e
x
i
s
t
s
0
<
δ
<
1
s
u
c
h
t
h
a
t
w
∈
R
D
δ
.


### 2.3. Morrey Spaces

Ordinary Morrey spaces are defined as follows (see [4–6]):Definition 10.
Let1
≤
p
<
∞ and 0
≤
k
≤
1. One defines (15)
L
p
,
k
R
n
=
f
∈
L
l
o
c
p
;
f
L
p
,
k
=
s
u
p
Q
1
Q
k
∫
Q
f
x
p
d
x
1
/
p
<
∞
,where the supremum is taken over all cubes Q.Next we define weighted Morrey spaces (see [3]).Definition 11.
Let1
≤
p
<
∞ and 0
≤
k
≤
1. One defines (16)
L
p
,
k
R
n
,
w
=
L
p
,
k
w
=
f
∈
L
l
o
c
p
w
;
f
L
p
,
k
w
=
s
u
p
Q
1
w
Q
k
∫
Q
f
x
p
w
x
d
x
1
/
p
<
∞
,where the supremum is taken over all cubes Q.


## 2.1. Maximal Functions

First we define some maximal functions.Definition 1 (the Hardy-Littlewood maximal function).
Consider(4)
M
f
x
=
s
u
p
Q
∋
x
1
Q
∫
Q
f
y
d
y
,where the supremum is taken over all cubes containing x.Definition 2 (the sharp maximal function).
Consider(5)
M
♯
f
x
=
s
u
p
Q
∋
x
1
Q
∫
Q
f
y
-
f
Q
d
y
,where f
Q
=
(
1
/
|
Q
|
)
∫
Q
‍
f
(
y
)
d
y.
Next we define the dyadic maximal function. A dyadic cube is a cube of the form(6)
∏
j
=
1
n
m
j
2
l
,
m
j
+
1
2
l
for some integers
m
1
,
m
2
,
…
,
m
n
,
l
.Definition 3 (the dyadic maximal function).
Consider(7)
M
d
f
x
=
s
u
p
Q
∋
x
1
Q
∫
Q
f
y
d
y
,where the supremum is taken over all dyadic cubes containing x.


## 2.2. Weights

Next we define Muckenhoupt’s weight classes (see [1, 2]).Definition 4 (A
q).
Let1
<
q
<
∞. For a nonnegative locally integrable function w, one says that w
∈
A
q if (8)
1
Q
∫
Q
w
x
d
x
1
Q
∫
Q
w
x
-
1
/
q
-
1
d
x
q
-
1
≤
C
,where C is a positive constant independent of cubes Q.Definition 5 (A
∞).
One writes(9)
A
∞
=
⋃
1
<
q
<
∞
A
q
.Definition 6 (doubling condition).
One says thatw
∈
D, if (10)
w
2
Q
≤
C
w
Q
,for any cubes, where 2
Q is the cube that has the same center as Q with double the side length.Definition 7 (reverse doubling condition).
One says thatw
∈
R
D, if there exists C
>
1 such that (11)
w
2
Q
≥
C
w
Q
f
o
r
a
n
y
c
u
b
e
s
Q
,and one denotes the supremum of C in (11) by [
w
]
R
D.Definition 8 (δ-reverse doubling condition).
One says thatw
∈
R
D
(
δ
), where 0
<
δ
<
1, if (12)
w
S
w
Q
≤
C
S
Q
δ
,for any cubes Q and subsets S
⊂
Q, and one denotes the infimum of C in (12) by [
w
]
R
D
(
δ
).The notation in Definitions6, 7, and 8 may not be standard. However, we use these notions frequently; therefore, we use these symbols for short.The following proposition is well-known (see, e.g., [1, 2]).Proposition 9.
Consider the following:(13)
I
f
w
∈
A
∞
,
t
h
e
n
w
∈
D
,
w
∈
R
D
,
(14)
I
f
w
∈
A
∞
,
t
h
e
n
t
h
e
r
e
e
x
i
s
t
s
0
<
δ
<
1
s
u
c
h
t
h
a
t
w
∈
R
D
δ
.


## 2.3. Morrey Spaces

Ordinary Morrey spaces are defined as follows (see [4–6]):Definition 10.
Let1
≤
p
<
∞ and 0
≤
k
≤
1. One defines (15)
L
p
,
k
R
n
=
f
∈
L
l
o
c
p
;
f
L
p
,
k
=
s
u
p
Q
1
Q
k
∫
Q
f
x
p
d
x
1
/
p
<
∞
,where the supremum is taken over all cubes Q.Next we define weighted Morrey spaces (see [3]).Definition 11.
Let1
≤
p
<
∞ and 0
≤
k
≤
1. One defines (16)
L
p
,
k
R
n
,
w
=
L
p
,
k
w
=
f
∈
L
l
o
c
p
w
;
f
L
p
,
k
w
=
s
u
p
Q
1
w
Q
k
∫
Q
f
x
p
w
x
d
x
1
/
p
<
∞
,where the supremum is taken over all cubes Q.


## 3. Theorems

The following three theorems are well-known.Theorem A (see [1, page 161]).
Let1
<
p
0
≤
p
<
∞. If f
∈
L
p
0
(
R
n
), then (17)
f
L
p
≈
M
♯
f
L
p
.Theorem B (see [1, page 410]).
Let1
<
p
0
≤
p
<
∞. One assumes that w
∈
A
∞. If M
d
f
∈
L
p
0
(
w
)
(
R
n
), then (18)
M
d
f
L
p
w
≤
C
M
♯
f
L
p
w
.Theorem C (see [4]).
Let1
<
p
<
∞ and 0
<
k
<
1. If f
∈
L
p
,
k
(
R
n
), then (19)
M
d
f
L
p
,
k
≤
C
M
♯
f
L
p
,
k
.As a corollary of this theorem we know the following.Corollary C.
Let1
<
p
<
∞ and 0
<
k
<
1. If f
∈
L
p
,
k
(
R
n
), then (20)
f
L
p
,
k
≈
M
♯
f
L
p
,
k
.Our result is the following.Theorem 12.
Let1
<
p
0
≤
p
<
∞ and 0
<
k
<
1. One assumes that w
∈
A
∞. If M
d
f
∈
L
p
0
,
k
(
w
), then (21)
f
L
p
,
k
w
≤
C
p
,
n
,
k
,
w
M
♯
f
L
p
,
k
w
,where C
p
,
n
,
k
,
w is a positive constant depending only on p, n, k, and weight w.Ifw
∈
A
p
0, then M
d is bounded on L
p
0
,
k
(
w
) (see [3]). Therefore, we obtain the following corollary.Corollary 13.
Let1
<
p
0
≤
p
<
∞ and 0
<
k
<
1. One assumes that w
∈
A
p
0. If f
∈
L
p
0
,
k
(
w
)
(
R
n
), then (22)
f
L
p
,
k
w
≈
M
♯
f
L
p
,
k
w
.Furthermore, we obtain the following (compare with Corollary C).Corollary 14.
Let1
<
p
0
≤
p
<
∞ and 0
<
k
<
1. If f
∈
L
p
0
,
k
(
R
n
), then (23)
f
L
p
,
k
≈
M
♯
f
L
p
,
k
.


## 4. Proof of Theorem

First we note that(24)
f
L
p
,
k
w
≤
C
s
u
p
Q
1
w
Q
k
∫
Q
f
x
p
w
x
d
x
1
/
p
,where the supremum is taken over all dyadic cubes. Because every cube can be covered by at most 2
n dyadic cubes of comparable size, and w satisfies the doubling condition.We follow the argument in [2, page 148]. Let a dyadic cube Q be fixed. By (14), we may assume w
∈
R
D
(
δ
) for some 0
<
δ
<
1.Here we define another maximal function.Definition 15.
One defines the local dyadic maximal function with respect toQ as follows: (25)
M
d
Q
f
x
=
s
u
p
x
∈
R
⊆
Q
1
R
∫
R
f
y
d
y
,where the supremum is taken over all dyadic cubes R containing x such that R
⊆
Q.Note that|
f
(
x
)
|
≤
M
d
(
Q
)
f
(
x
) for almost all x
∈
Q.We need three lemmas.Lemma 16.
Letγ
>
0 and λ
>
|
f
|
Q
∶
=
(
1
/
Q
)
∫
Q
‍
|
f
(
y
)
|
d
y. One has (26)
w
x
∈
Q
;
M
d
Q
f
x
>
2
λ
,
M
♯
f
x
≤
γ
λ
≤
2
n
γ
δ
w
R
D
δ
w
x
∈
Q
;
M
d
Q
f
x
>
λ
.Remark 17.
When we consider a local good-λ estimate on Q, the condition λ
>
|
f
|
Q is important. See the proof of Lemma 16.Next we define new symbol.Definition 18 (dyadic double).
LetQ
(
1
) denote the “parent” of Q, that is, the dyadic cube containing Q with double the side length. When Q
(
j
) is a dyadic cube, one denotes by Q
(
j
+
1
) the parent of Q
(
j
). One defines Q
(
0
)
=
Q.Lemma 19.
Ifw
∈
A
∞, then the characteristic function χ
∪
j
Q
(
j
)
∉
L
p
,
k
(
w
)
(
R
n
).Lemma 20.
IfM
d
f
∈
L
p
0
,
k
(
w
)
(
R
n
) for some p
0
>
1, then one has (27)
l
i
m
j
→
∞
f
Q
j
=
0
,
f
Q
j
-
f
Q
j
+
1
≤
2
n
+
1
w
R
D
j
+
1
k
-
1
/
p
w
Q
k
-
1
/
p
M
♯
f
L
p
,
k
w
.Assuming these lemmas, we continue the proof of our theorem. To avoid routine limiting arguments, we assume that∫
Q
‍
(
M
d
(
Q
)
f
(
x
)
)
p
w
(
x
)
d
x
<
∞ (see [2, page 549]).LetΩ
λ
(
Q
)
=
{
x
∈
Q
;
M
d
(
Q
)
f
(
x
)
>
λ
}. We write (28)
∫
Q
M
d
Q
f
x
p
w
x
d
x
=
∫
0
2
f
Q
p
λ
p
-
1
w
Ω
λ
Q
d
λ
+
∫
2
f
Q
∞
p
λ
p
-
1
w
Ω
λ
Q
d
λ
=
I
+
I
I
.First we estimateI
I. By Lemma 16, we have (29)
I
I
=
2
p
∫
f
Q
∞
p
λ
p
-
1
w
Ω
2
λ
Q
d
λ
≤
2
p
2
n
γ
δ
w
R
D
δ
∫
0
∞
p
λ
p
-
1
w
Ω
λ
Q
d
λ
+
2
p
∫
0
∞
p
λ
p
-
1
w
x
∈
Q
;
M
♯
f
x
>
γ
λ
d
λ
=
2
p
2
n
γ
δ
w
R
D
δ
∫
Q
M
d
Q
f
x
p
w
x
d
x
+
2
p
γ
-
p
∫
Q
M
♯
f
x
p
w
x
d
x
.Next we estimateI. We have I
≤
C
(
|
f
|
Q
)
p
w
(
Q
). By (27), (30)
f
Q
≤
∑
j
=
0
∞
f
Q
j
-
f
Q
j
+
1
≤
2
n
+
1
w
Q
k
-
1
/
p
M
♯
f
L
p
,
k
w
∑
j
=
0
∞
w
R
D
j
+
1
k
-
1
/
p
.Therefore, we have (31)
I
≤
C
n
,
p
,
k
,
w
M
♯
f
L
p
,
k
w
p
w
Q
k
.By (29) and (31), we have (32)
1
w
Q
k
∫
Q
M
d
Q
f
x
p
w
x
d
x
≤
2
p
2
n
γ
δ
·
w
R
D
δ
1
w
Q
k
∫
Q
M
d
Q
f
x
p
w
x
d
x
+
2
p
γ
-
p
1
w
Q
k
∫
Q
M
♯
f
x
p
w
x
d
x
+
C
n
,
p
,
k
,
w
M
♯
f
L
p
,
k
w
p
.Taking γ sufficiently small, we obtain (33)
1
w
Q
k
∫
Q
M
d
Q
f
x
p
w
x
d
x
≤
C
n
,
p
,
k
,
w
M
♯
f
L
p
,
k
w
p
.Since |
f
(
x
)
|
≤
M
d
(
Q
)
f
(
x
)
a
.
e., this proves the theorem.


## 5. Proofs of Lemmas

Proof of Lemma16.
LetΩ
λ
(
Q
)
=
{
x
∈
Q
;
M
d
(
Q
)
f
(
x
)
>
λ
}. As in [2, page 148], for any x
∈
Ω
λ
(
Q
), there is a maximal dyadic cube Q
x
⊂
Q that contains x such that (34)
1
Q
x
∫
Q
x
f
y
d
y
>
λ
.Since λ
>
|
f
|
Q, the maximal dyadic cube Q
x satisfies Q
x
⫋
Q. Therefore, we can write (35)
Ω
λ
Q
=
⋃
i
Q
i
,where {
Q
i
}
i is a family of disjoint maximal dyadic cubes. We can show (36)
x
∈
Q
i
;
M
d
Q
f
x
>
2
λ
,
M
♯
f
x
≤
γ
λ
≤
2
n
γ
Q
i
.By δ-reverse doubling condition (Definition 8), we obtain the desired result.Proof of Lemma19.
For any dyadic cubesQ
(
j
), (37)
1
w
Q
j
k
∫
Q
j
1
·
w
x
d
x
=
w
Q
j
1
-
kand w
(
Q
(
j
)
) tends to infinity when j
→
∞ because w
∈
A
∞ (see Proposition 9).Proof of Lemma20.
Since(38)
f
Q
j
-
f
Q
j
+
1
≤
2
n
Q
j
+
1
∫
Q
j
+
1
f
y
-
f
Q
j
+
1
d
y
≤
2
n
+
1
i
n
f
x
∈
Q
j
+
1
M
♯
f
x
,we have(39)
f
Q
j
-
f
Q
j
+
1
≤
2
n
+
1
w
Q
j
+
1
∫
Q
j
+
1
M
♯
f
x
w
x
d
x
≤
2
n
+
1
w
Q
j
+
1
k
-
1
/
p
M
♯
f
L
p
,
k
w
≤
2
n
+
1
w
R
D
j
+
1
k
-
1
/
p
w
Q
k
-
1
/
p
M
♯
f
L
p
,
k
wby reverse doubling condition (Definition 7).
By (39), |
f
|
Q
(
j
) is a Cauchy sequence; therefore, |
f
|
Q
(
j
) converges to some constant C when j
→
∞. If C
>
0, then M
d
f
(
x
)
≥
C
χ
∪
j
Q
(
j
). But this contradicts the fact M
d
f
∈
L
p
0
,
k
(
w
) by Lemma 19. Therefore, we obtain l
i
m
j
→
∞
|
f
|
Q
(
j
)
=
0.



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*Source: 651825-2015-06-01.xml*