# Lipschitz Spaces and Fractional Integral Operators Associated with Nonhomogeneous Metric Measure Spaces

Authors: Jiang Zhou; Dinghuai Wang
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The fractional operator on nonhomogeneous metric measure spaces is introduced, which is a bounded operator fromL
p
μ into the space L
q
,
∞
μ. Moreover, the Lipschitz spaces on nonhomogeneous metric measure spaces are also introduced, which contain the classical Lipschitz spaces. The authors establish some equivalent characterizations for the Lipschitz spaces, and some results of the boundedness of fractional operator in Lipschitz spaces are also presented.


---

## Body

## 1. Introduction

As we know, the theory on spaces of homogeneous type is needed to assume that measureμ of metric spaces (
𝒳
,
d
,
μ
) satisfies the doubling measure condition, which means that there exists a constant C, such that, for every ball B
(
x
,
r
) of center x and radius r, μ
(
B
(
x
,
2
r
)
)
≤
C
μ
(
B
(
x
,
r
)
). In recent years, many classical theories have been proved still valid without the assumption of doubling measure condition; see [1–12]. Recall that a Radon measure μ on R
d is said to only satisfy the polynomial growth condition, if there exists a positive constant c such that, for all x
∈
ℝ
d and r
>
0, μ
(
B
(
x
,
r
)
)
≤
c
r
n, where n is some fixed number in (
0
,
d
] and B
(
x
,
r
)
=
{
y
∈
ℝ
d
:
|
x
-
y
|
<
r
}. The analysis associated with such nondoubling measures μ is proved to play a striking role in solving the long-standing open Painlevé’s problem by Tolsa [13]. Obviously, the nondoubling measure μ with the polynomial growth condition may not satisfy the well-known doubling condition, which is a key assumption in harmonic analysis on spaces of homogeneous type. In 2010, Hytönen [14] introduced a new class of metric measure spaces satisfying both the so-called geometrically doubling and the upper doubling conditions (see the definition below), which are called nonhomogeneous spaces. Recently, many classical results have been proved still valid if the underlying spaces are replaced by the nonhomogeneous spaces of Hytönen (see [4–6, 9–12]).Let(
𝒳
,
d
,
μ
) be a nonhomogeneous metric measure space in the sense of Hytönen [14]. In this paper, we establish the definition of fractional operator on nonhomogeneous metric measure spaces, which contains the classical fractional integral operator introduced by García-Cuerva and Gatto [7], and similar to the definition introduced by Fu et al. [11], then we get the (
L
p
(
μ
)
,
L
q
,
∞
(
μ
)
)-boundedness for fractional integral operator on nonhomogeneous metric measure spaces. In Section 3, we also establish the definition of Lipschitz spaces on nonhomogeneous metric measure spaces, which contains the classical Lipschitz spaces. We establish some equivalent characterizations for the Lipschitz spaces. In Section 4, we present some results of the boundedness of fractional operator in Lipschitz spaces.To state the main results of this paper, we first recall some necessary notions and remarks.Definition 1 (see [15]).
A metric space(
𝒳
,
d
,
μ
) is said to be geometrically doubling if there exists some N
0
∈
ℕ such that, for any ball B
(
x
,
r
)
⊂
𝒳, there exist a finite ball covering {
B
(
x
i
,
r
/
2
)
}
i of B
(
x
,
r
) such that the cardinality of this covering is at most N
0.Definition 2 (see [14]).
A metric measure space(
𝒳
,
d
,
μ
) is said to be upper doubling if μ is a Borel measure on 𝒳 and there exist a dominating function λ
:
𝒳
×
(
0
,
∞
)
→
(
0
,
∞
) and a positive constant c
λ such that, for each x
∈
𝒳, r
→
λ
(
x
,
r
) is nondecreasing and
(1)
μ
(
B
(
x
,
r
)
)
≤
λ
(
x
,
r
)
≤
c
λ
λ
(
x
,
r
2
)
∀
x
∈
𝒳
,
r
>
0
.
A metric measure space(
𝒳
,
d
,
μ
) is called a nonhomogeneous metric measure space if (
𝒳
,
d
,
μ
) is geometrically doubling and (
𝒳
,
d
,
μ
) is upper doubling.Remark 3.
(i) Obviously, a space of homogeneous type is a special case of upper doubling spaces, where we take the dominating functionλ
(
x
,
r
)
:
=
μ
(
B
(
x
,
r
)
). On the other hand, the Euclidean space ℝ
d with any Radon measure μ as in (1) is also an upper doubling space by taking the dominating function λ
(
x
,
r
)
:
=
C
r
k.
(ii) Let(
𝒳
,
d
,
μ
) be upper doubling with λ being the dominating function on 𝒳
×
(
0
,
∞
) as in Definition 2. It was proved in [6] that there exists another dominating function λ
~ such that λ
~
≤
λ and, for all x
,
y
∈
𝒳 with d
(
x
,
y
)
≤
r,
(2)
λ
~
(
x
,
r
)
≤
C
λ
~
λ
~
(
y
,
r
)
.
Thus, in this paper, we always suppose that λ satisfies (2).Definition 4 (see [14]).
Letα
,
β
α
∈
(
1
,
∞
). A ball B
⊂
X is called (
α
,
β
)-doubling if μ
(
α
B
)
≤
β
α
μ
(
B
).
As stated in lemma of [4], there exist plenty of doubling balls with small radii and with large radii. In the rest of the paper, unless α and β
α are specified otherwise, by an (
α
,
β
α
)-doubling ball we mean a (
6
,
β
6
)-doubling with a fixed number β
6
>
max
⁡
⁡
{
C
λ
3
log
⁡
2
⁡
6
,
6
n
}, where n
=
log
⁡
2
⁡
N
0 is viewed as a geometric dimension of the spaces.Definition 5 (see [11]).
Letϵ
∈
(
0
,
∞
). A dominating function λ is satisfying the ϵ-weak reverse doubling condition if, for all r
∈
(
0,2
diam
⁡
(
𝒳
)
) and a
∈
(
1,2
diam
⁡
(
𝒳
)
/
r
), there exists a number C
(
a
)
∈
[
1
,
∞
), depending only on a and 𝒳, such that, for all x
∈
𝒳,
(3)
λ
(
x
,
a
r
)
≥
C
(
a
)
λ
(
x
,
r
)
and, moreover,
(4)
∑
k
=
1
∞
‍
1
[
C
(
a
k
)
]
ϵ
<
∞
.Remark 6.
(i) It is easy to see that ifϵ
1
<
ϵ
2 and λ satisfies the ϵ
1-weak reverse doubling condition, then λ also satisfies the ϵ
2-weak reverse doubling condition.
(ii) Assume thatdiam
⁡
(
𝒳
)
=
∞. For any fixed x
∈
𝒳, we know that
(5)
lim
⁡
r
→
0
λ
(
x
,
r
)
=
0
,
lim
⁡
r
→
∞
λ
(
x
,
r
)
=
∞
.
(ii) It is easy to see that theϵ-weak reverse doubling condition is much weaker than the assumption introduced by Bui and Duong in [4, Subsection 7.3]: there exists m
∈
(
0
,
∞
) such that, for all x
∈
𝒳 and a
,
r
∈
(
0
,
∞
), λ
(
x
,
a
r
)
=
a
m
λ
(
x
,
r
).Definition 7 (see [14]).
For any two ballsB
⊂
S, define
(6)
K
B
,
S
=
1
+
∫
2
S
∖
B
1
λ
(
c
B
,
d
(
x
,
c
B
)
)
d
μ
(
x
)
,
where c
B is the center of the ball B.Remark 8.
The following discrete version,K
~
B
,
S, of K
B
,
S defined in Definition 7, was first introduced by Bui and Duong [4] in nonhomogeneous metric measure spaces, which is more close to the quantity K
Q
,
R introduced by Tolsa [1] in the setting of nondoubling measures. For any two balls B
⊂
S, let K
~
B
,
S be defined by
(7)
K
~
B
,
S
∶
=
1
+
∑
k
=
1
N
B
,
S
‍
μ
(
6
k
B
)
λ
(
c
B
,
6
r
k
r
B
)
,
where r
B and r
S, respectively, denote the radii of the balls B and S, and N
B
,
S denotes the smallest integer satisfying 6
N
B
,
S
r
B
≥
r
S. Obviously, K
B
,
S
≤
C
K
~
B
,
S. As was pointed by Bui and Duong [4], in general, it is not true that K
B
,
S
~
C
K
~
B
,
S.Definition 9 (see [14]).
Letρ
∈
(
1
,
∞
). A function f
∈
L
loc
⁡
1
(
μ
) is said to be in the space RBMO(
μ
) if there exist a positive constant C, and for any ball B
⊂
𝒳, a number f
B such that
(8)
1
μ
(
ρ
B
)
∫
B
‍
|
f
(
x
)
-
f
B
|
d
μ
(
x
)
≤
C
,
for any two balls B and B
1 such that B
⊂
B
1,
(9)
|
f
B
-
f
B
1
|
≤
C
K
B
,
B
1
.
The infimum of the positive constantsC satisfying above two inequalities is defined to be the RBMO(
μ
) norm of f and denoted by ∥
f
∥
RBMO
(
μ
).
From [14, Lemma 4.6], it follows that the space RBMO
(
μ
) is independent of ρ
∈
(
1
,
∞
).
In this paper, we consider a variant of the fractional integrals from [7, Definition 4.1].Definition 10.
Let0
<
α
<
n and 0
<
δ
≤
1. A function K
α
∈
L
loc
⁡
1
(
𝒳
×
𝒳
∖
{
(
x
,
y
)
:
x
=
y
}
) is said to be a fractional kernel of order α and regularity δ if it satisfies the following two conditions:(i)
for allx
,
y
∈
𝒳 with x
≠
y,
(10)
|
K
α
(
x
,
y
)
|
≤
C
1
[
λ
(
x
,
d
(
x
,
y
)
)
]
1
-
α
/
n
;
(ii)
for allx
,
x
~
,
y
∈
𝒳 with λ
(
x
,
d
(
x
,
y
)
)
≥
2
λ
(
x
,
d
(
x
,
x
~
)
),
(11)
|
K
α
(
x
,
y
)
-
K
α
(
x
~
,
y
)
|
+
|
K
α
(
y
,
x
)
-
K
α
(
y
,
x
~
)
|
≤
C
[
λ
(
x
,
d
(
x
,
x
~
)
)
]
δ
/
n
[
λ
(
x
,
d
(
x
,
y
)
)
]
1
-
α
/
n
+
δ
/
n
.
A linear operatorI
α is called fractional integral operator with K
α satisfying (10) and (11),
(12)
I
α
f
(
x
)
∶
=
∫
𝒳
‍
K
α
(
x
,
y
)
f
(
y
)
d
μ
(
y
)
.Remark 11.
By takingλ
(
x
,
d
(
x
,
y
)
)
=
d
(
x
,
y
)
n, it is easy to see that Definition 10 in this paper contains Definition 4.1 introduced by García-Cuerva and Gatto in [7], and Definition 10 is similar to Definition 1.9 introduced by Fu et al. in [11].Definition 12.
LetK
α be a fractional kernel of order α and regularity δ, f
∈
L
p
(
μ
) and 0
<
α
-
n
/
p
<
δ. We define
(13)
I
~
α
f
(
x
)
=
∫
𝒳
‍
{
K
α
(
x
,
y
)
-
K
α
(
x
0
,
y
)
}
f
(
y
)
d
μ
(
y
)
,
where x
0 is some fixed point of 𝒳.
We observe that the integral in (13) converges both locally and at ∞ as a consequence of (10), (11), and Hölder’s inequality. Of course the function just defined depends on the election of x
0, but the difference between any two functions obtained in (13) for different elections of x
0 is just a constant.From now on, we will assume thatμ
(
𝒳
)
=
∞. The results below are also true when μ
(
𝒳
)
<
∞.Now we state the first main theorem of this paper.Theorem 13.
Let1
≤
p
<
n
/
α and 1
/
q
=
1
/
p
-
α
/
n. If λ satisfy the ϵ-weak reverse doubling condition with ϵ
∈
(
0
,
min
⁡
⁡
{
α
/
n
,
(
1
/
p
-
α
/
n
)
p
′
}
), then
(14)
μ
(
{
x
∈
𝒳
:
|
I
α
f
(
x
)
|
>
ν
}
)
≤
(
C
∥
f
∥
L
p
(
μ
)
ν
)
q
;
that is, I
α is a bounded operator from L
p
(
μ
) into the space L
q
,
∞
(
μ
).Next, let us introduce Lipschitz spaces on nonhomogeneous metric measure spaces.Definition 14.
Given thatβ
∈
(
0,1
], we say that the function f
:
𝒳
→
ℂ satisfies a Lipschitz condition of order β provided that
(15)
|
f
(
x
)
-
f
(
y
)
|
≤
C
[
λ
(
x
,
d
(
x
,
y
)
)
]
β
/
n
for
every
x
,
y
∈
𝒳
and the smallest constant in inequality (15) will be denoted by ∥
f
∥
Lip
(
β
). It is easy to see that the linear space with the norm ∥
·
∥
Lip
(
β
) is a Banach space, and we will call it Lip
(
β
).Remark 15.
Lipschitz condition can also be defined by(
15
)
′
|
f
(
x
)
-
f
(
y
)
|
≤
C
[
λ
(
y
,
d
(
x
,
y
)
)
]
β
/
n
for
every
x
,
y
∈
𝒳
;
by (2), it is easy to see that (15) and (
15
)
′ are equivalent.The second main result of this paper is the following some equivalent characterizations for the Lipschitz spaces.Theorem 16.
For a functionf
∈
L
l
o
c
1
(
μ
), the conditions (
A
), (
B
), and (
C
) are equivalent as follows.(A)
There exist some constantC
1 and a collection of numbers of f
B, one for each B, such that these two properties hold: for any all B with radius r
(16)
1
μ
(
2
B
)
∫
B
‍
|
f
(
x
)
-
f
B
|
d
μ
(
x
)
≤
C
1
λ
(
x
,
r
)
β
/
n
,
and for any ball U such that B
∈
U and radius (
U
)
≤
2
r,
(17)
|
f
B
-
f
U
|
≤
C
1
λ
(
x
,
r
)
β
/
n
.
(B)
There is a constantC
2 such that
(18)
|
f
(
x
)
-
f
(
y
)
|
≤
C
2
λ
(
x
,
d
(
x
,
y
)
)
β
/
n
,
for μ-almost every x and y in the support of μ.
(C)
For any givenp, 1
≤
p
≤
∞, there is a constant C
(
p
), such that for every ball B of radius r, one has
(19)
(
1
μ
(
B
)
∫
B
‍
|
f
(
x
)
-
m
B
(
f
)
|
p
d
μ
(
x
)
)
1
/
p
≤
C
(
p
)
λ
(
x
,
r
)
β
/
n
,
where m
B
=
(
1
/
μ
(
B
)
)
∫
B
‍
f
(
y
)
d
μ
(
y
) and also for any ball U such that B
⊂
U and radius (
U
)
≤
2
r
(20)
|
m
B
(
f
)
-
m
U
(
f
)
|
≤
C
(
p
)
λ
(
x
,
r
)
β
/
n
.
In addition, the quantitiesinf
⁡
⁡
C
1, inf
⁡
⁡
C
2, and inf
⁡
⁡
C
(
p
) with a fixed p are equivalent.Now we state the third main result of this paper.Theorem 17.
LetK
α be a fractional kernel. n
/
α
<
p
≤
∞ and α
-
n
/
p
<
δ. If λ satisfy the ϵ-weak reverse doubling condition with ϵ
∈
(
0
,
min
⁡
⁡
{
(
1
-
(
(
α
-
δ
)
/
n
)
)
p
′
,
(
α
/
n
-
1
/
p
)
p
′
}
), then I
~
α maps L
p
(
μ
) boundedly into Lip
(
α
-
n
/
p
).Theorem 18.
LetK
α be a fractional kernel and α
+
β
<
δ; if λ satisfy the ϵ-weak reverse doubling condition with ϵ
∈
(
0
,
min
⁡
{
(
α
+
β
)
/
n
,
(
δ
-
α
-
β
)
/
n
}
), then I
~
α maps Lip
(
β
) boundedly into Lip
(
α
+
β
) if and only if I
~
α
(
1
)
=
0.Finally we present a result which can be viewed either as an extension of the casep
=
∞ of the Theorem 17 or as extension of the case β
=
0 of Theorem 18.Theorem 19.
LetK
α be a fractional kernel and 0
<
α
<
δ. If λ satisfy the ϵ-weak reverse doubling condition with ϵ
∈
(
0
,
min
⁡
⁡
{
α
/
n
,
(
δ
-
α
)
/
n
}
), then I
~
α maps RBMO(
μ
) boundedly into Lip
(
α
) if and only if I
~
α
(
1
)
=
0.Finally, we make some conventions on notation. Throughout the whole paper,C stands for a positive constant, which is independent of the main parameters, but it may vary from line to line.


## 2. Proof of Theorem13

Proof of Theorem13.
We are going to adapt to our context of the proof given by García-Cuerva and Gatto [7]. Consider
(21)
|
I
α
f
(
x
)
|
≤
∫
𝒳
‍
|
f
(
y
)
|
[
λ
(
x
,
d
(
x
,
y
)
)
]
1
-
α
/
n
d
μ
(
y
)
≤
∫
B
(
x
,
r
)
‍
|
f
(
y
)
|
[
λ
(
x
,
d
(
x
,
y
)
)
]
1
-
α
/
n
d
μ
(
y
)
+
∫
𝒳
∖
B
(
x
,
r
)
‍
|
f
(
y
)
|
[
λ
(
x
,
d
(
x
,
y
)
)
]
1
-
α
/
n
d
μ
(
y
)
∶
=
I
1
+
I
2
.
By Hölder’s inequality, if p
>
1, then 1
-
(
1
-
α
/
n
)
p
′
<
0;
(22)
|
I
2
|
≤
∥
f
∥
L
p
(
μ
)
×
(
∫
𝒳
∖
B
(
x
,
r
)
‍
1
[
λ
(
x
,
d
(
x
,
y
)
)
]
(
1
-
α
/
n
)
p
′
d
μ
(
y
)
)
1
/
p
′
≤
∥
f
∥
L
p
(
μ
)
×
(
∑
j
=
1
∞
∫
2
j
B
∖
2
j
-
1
B
‍
1
[
λ
(
x
,
d
(
x
,
y
)
)
]
(
1
-
α
/
n
)
p
′
d
μ
(
y
)
)
1
/
p
′
≤
∥
f
∥
L
p
(
μ
)
×
(
∑
j
=
1
∞
‍
μ
(
B
(
x
,
2
j
r
)
)
[
λ
(
x
,
2
j
-
1
r
)
]
(
1
-
α
/
n
)
p
′
)
1
/
p
′
≤
∥
f
∥
L
p
(
μ
)
×
(
∑
j
=
1
∞
‍
λ
(
x
,
2
j
r
)
[
λ
(
x
,
2
j
-
1
r
)
]
(
1
-
α
/
n
)
p
′
)
1
/
p
′
≤
C
∥
f
∥
L
p
(
μ
)
(
∑
j
=
1
∞
‍
[
λ
(
x
,
2
j
-
1
r
)
]
1
-
(
1
-
α
/
n
)
p
′
)
1
/
p
′
≤
C
∥
f
∥
L
p
(
μ
)
(
∑
j
=
1
∞
‍
1
[
C
(
2
j
)
]
(
1
-
α
/
n
)
p
′
-
1
)
1
/
p
′
×
[
λ
(
x
,
r
)
]
α
/
n
-
1
/
p
≤
C
∥
f
∥
L
p
(
μ
)
[
λ
(
x
,
r
)
]
α
/
n
-
1
/
p
,
which holds even for p
=
1. We can and assume that ∥
f
∥
L
p
(
μ
)
=
1. By (5), we can choose r
∈
(
0
,
∞
) such that C
[
λ
(
x
,
r
)
]
α
/
n
-
1
/
p
=
ν
/
2. Then
(23)
{
x
∈
𝒳
:
|
I
α
f
(
x
)
|
>
ν
}
⊂
{
x
∈
𝒳
:
|
I
1
|
>
ν
2
}
∪
{
x
∈
𝒳
:
|
I
2
|
>
ν
2
}
.
By the relation between r and ν, μ
(
{
x
∈
𝒳
:
|
I
2
|
>
ν
/
2
}
)
=
0. We use Hölder’s inequality once more to obtain
(24)
|
I
1
|
≤
(
∫
B
(
x
,
r
)
‍
d
μ
(
y
)
[
λ
(
x
,
d
(
x
,
y
)
)
]
1
-
α
/
n
)
1
/
p
′
×
(
∫
B
(
x
,
r
)
‍
|
f
(
y
)
|
p
[
λ
(
x
,
d
(
x
,
y
)
)
]
1
-
α
/
n
d
μ
(
y
)
)
1
/
p
≤
C
(
∑
j
=
0
∞
‍
μ
(
B
(
x
,
2
-
j
r
)
)
[
λ
(
x
,
2
-
j
-
1
r
)
]
1
-
α
/
n
)
1
/
p
′
×
(
∫
B
(
x
,
r
)
‍
|
f
(
y
)
|
p
[
λ
(
x
,
d
(
x
,
y
)
)
]
1
-
α
/
n
d
μ
(
y
)
)
1
/
p
≤
C
(
∑
j
=
0
∞
‍
λ
(
x
,
2
j
r
)
[
λ
(
x
,
2
-
j
-
1
r
)
]
1
-
α
/
n
)
1
/
p
′
×
(
∫
B
(
x
,
r
)
‍
|
f
(
y
)
|
p
[
λ
(
x
,
d
(
x
,
y
)
)
]
1
-
α
/
n
d
μ
(
y
)
)
1
/
p
≤
C
(
∑
j
=
0
∞
‍
[
λ
(
x
,
2
-
j
-
1
r
)
]
α
/
n
)
1
/
p
′
×
(
∫
B
(
x
,
r
)
‍
|
f
(
y
)
|
p
[
λ
(
x
,
d
(
x
,
y
)
)
]
1
-
α
/
n
d
μ
(
y
)
)
1
/
p
≤
C
(
∑
j
=
0
∞
‍
1
[
C
(
2
j
)
]
α
/
n
[
λ
(
x
,
r
)
]
α
/
n
)
1
/
p
′
×
(
∫
B
(
x
,
r
)
‍
|
f
(
y
)
|
p
[
λ
(
x
,
d
(
x
,
y
)
)
]
1
-
α
/
n
d
μ
(
y
)
)
1
/
p
≤
C
[
λ
(
x
,
r
)
]
α
/
p
′
n
×
(
∫
B
(
x
,
r
)
‍
|
f
(
y
)
|
p
[
λ
(
x
,
d
(
x
,
y
)
)
]
1
-
α
/
n
d
μ
(
y
)
)
1
/
p
.
Then, by applying Tchebichev’s inequality, we have(25)
μ
(
{
x
∈
𝒳
:
|
I
α
f
(
x
)
|
>
ν
}
)
≤
μ
(
{
x
∈
𝒳
:
|
I
1
|
>
ν
2
}
)
≤
C
[
λ
(
x
,
r
)
]
α
p
/
p
′
n
ν
-
p
×
∫
𝒳
‍
∫
B
(
x
,
r
)
‍
|
f
(
y
)
|
p
[
λ
(
x
,
d
(
x
,
y
)
)
]
1
-
α
/
n
d
μ
(
y
)
d
μ
(
x
)
=
C
[
λ
(
x
,
r
)
]
α
p
/
p
′
n
ν
-
p
×
∫
𝒳
‍
∫
B
(
y
,
r
)
‍
d
μ
(
x
)
[
λ
(
x
,
d
(
x
,
y
)
)
]
1
-
α
/
n
|
f
(
y
)
|
p
d
μ
(
y
)
≤
C
[
λ
(
x
,
r
)
]
α
p
/
p
′
n
ν
-
p
[
λ
(
x
,
r
)
]
α
/
n
∥
f
∥
L
p
(
μ
)
=
C
[
λ
(
x
,
r
)
]
=
C
ν
-
q
.
This completes the proof of Theorem 13.Corollary 20.
Let1
<
p
<
n
/
α and 1
/
q
=
1
/
p
-
α
/
n. If λ satisfy the ϵ-weak reverse doubling condition with ϵ
∈
(
0
,
min
⁡
⁡
{
α
/
n
,
(
1
/
p
-
α
/
n
)
p
′
}
), then
(26)
∥
I
α
f
∥
L
q
(
μ
)
≤
C
∥
f
∥
L
p
(
μ
)
.Proof.
It suffices to apply Marcinkiewicz’s interpolation theorem with indices slightly bigger and slightly smaller thanp.


## 3. Proof of Theorem16

Before we give the proof of Theorem16, we first introduce a technical lemma from [8, Lemma 3.2].Lemma 21.
Letf
∈
L
l
o
c
1
(
μ
). If β
2
>
2
d, then, for almost every x with respect to μ, there exists a sequence of (
2
,
β
2
)-doubling balls B
j
=
B
(
x
,
r
j
) with r
j
→
0, such that
(27)
lim
⁡
j
→
∞
1
μ
(
B
j
)
∫
B
j
‍
f
(
y
)
d
μ
(
y
)
=
f
(
x
)
.Proof of Theorem16.
(
A
)
⇒
(
B
). Consider x as in the lemma and let B
j
=
B
(
x
,
r
j
), j
≥
1, a sequence of (
2
,
β
2
)-doubling balls with r
j
→
0. Consider
(28)
|
m
B
j
(
f
)
-
f
B
j
|
≤
1
μ
(
B
j
)
∫
B
j
‍
|
f
(
y
)
-
f
B
j
|
d
μ
(
y
)
≤
μ
(
2
B
j
)
μ
(
B
j
)
1
μ
(
2
B
j
)
∫
B
j
‍
|
f
(
y
)
-
f
B
j
|
d
μ
(
y
)
≤
β
C
1
[
λ
(
x
,
r
j
)
]
β
/
n
,
and by (5), Lemma 21, we obtain that
(29)
lim
⁡
j
→
∞
f
B
j
=
f
(
x
)
.
Let x and y be two points as in the lemma; take B
=
B
(
x
,
r
) any ball with r
≤
d
(
x
,
y
) and let U
=
B
(
x
,
2
d
(
x
,
y
)
). Now define B
k
=
B
(
x
,
2
k
r
), for 0
≤
k
≤
k
~, where k
~ is the first integer such that 2
k
~
r
≥
d
(
x
,
y
). Then
(30)
|
f
B
-
f
U
|
≤
∑
k
=
0
k
~
-
1
‍
|
f
B
k
-
f
B
k
+
1
|
+
|
f
B
k
~
-
f
U
|
≤
C
1
∑
k
=
0
k
~
‍
[
λ
(
x
,
2
k
r
)
]
β
/
n
≤
C
1
∑
k
=
0
k
~
‍
c
λ
β
k
/
n
[
λ
(
x
,
r
)
]
β
/
n
≤
C
′
C
1
λ
(
x
,
d
(
x
,
y
)
)
β
/
n
,
where C
′ is independent of x and d
(
x
,
y
).
A similar argument can be made for the pointy with any ball B
′
=
B
(
y
,
s
) such that s
≤
d
(
x
,
y
) and V
=
B
(
y
,
3
d
(
x
,
y
)
). Therefore
(31)
|
f
B
-
f
B
′
|
≤
|
f
B
-
f
U
|
+
|
f
U
-
f
V
|
+
|
f
V
-
f
B
′
|
≤
C
′′
C
1
λ
(
x
,
d
(
x
,
y
)
)
β
/
n
.
Take two sequences of (
2
,
β
2
)-doubling balls B
j
=
B
(
x
,
r
j
) and B
′
=
B
(
y
,
s
j
) with r
j
→
0 and s
j
→
0. We have
(32)
|
f
(
x
)
-
f
(
y
)
|
=
lim
⁡
j
→
0
|
f
B
j
-
f
B
j
′
|
≤
C
′′
C
1
λ
(
x
,
d
(
x
,
y
)
)
β
/
n
.
(
B
)
⇒
(
C
). For x
0
∈
B
=
B
(
x
,
r
), by the properties of function λ and Hölder’s inequality, we obtain
(33)
(
1
μ
(
B
)
∫
B
‍
|
f
(
x
)
-
m
B
(
f
)
|
p
d
μ
(
x
)
)
1
/
p
≤
(
1
μ
(
B
)
∫
B
‍
|
1
μ
(
B
)
∫
B
‍
(
f
(
x
)
-
f
(
y
)
)
d
μ
(
y
)
|
p
d
μ
(
x
)
)
1
/
p
≤
(
1
μ
(
B
)
∫
B
‍
[
C
2
λ
(
x
,
r
)
]
β
p
/
n
d
μ
(
x
)
)
1
/
p
≤
(
1
μ
(
B
)
∫
B
‍
[
C
2
C
λ
λ
(
x
0
,
r
)
]
β
p
/
n
d
μ
(
x
)
)
1
/
p
≤
C
2
C
λ
[
λ
(
x
0
,
r
)
]
β
/
n
≤
C
2
C
λ
2
[
λ
(
x
,
r
)
]
β
/
n
.
By the similar argument, for any ball U such that B
⊂
U and radius U
≤
2
r,
(34)
|
m
B
(
f
)
-
m
U
(
f
)
|
≤
C
2
C
λ
2
[
λ
(
x
,
2
r
)
]
β
/
n
≤
c
λ
β
/
n
C
2
C
λ
2
[
λ
(
x
,
r
)
]
β
/
n
.
(
C
)
⇒
(
A
). Define first f
B
=
m
B
(
f
). Then (17) is exactly (20). To prove (16), we write
(35)
1
μ
(
2
B
)
∫
B
‍
|
f
(
x
)
-
f
B
|
d
μ
(
x
)
≤
1
μ
(
2
B
)
(
∫
B
‍
|
f
(
x
)
-
f
B
|
p
d
μ
(
x
)
)
1
/
p
μ
(
B
)
1
/
p
′
≤
μ
(
B
)
μ
(
2
B
)
(
1
μ
(
B
)
∫
B
‍
|
f
(
x
)
-
f
B
|
p
d
μ
(
x
)
)
1
/
p
≤
C
(
p
)
λ
(
x
,
r
)
β
/
n
.
This concludes the proof of the theorem.Remark 22.
Theorem16 is also true if the number 2 in condition (
A
) is replaced by any fixed ρ
>
1. In that case, the proof uses (
ρ
,
β
ρ
)-doubling balls, that is, balls satisfying μ
(
ρ
B
)
≤
β
ρ
μ
(
B
).


## 4. Proofs of Theorems17–19

Proof of Theorem17.
Without loss of generality, we assume thatp
<
∞. Consider x
≠
y and let B be the ball with center x and radius r
=
d
(
x
,
y
). Then, we have
(36)
|
I
~
α
f
(
x
)
-
I
~
α
f
(
y
)
|
≤
∫
2
B
‍
|
K
α
(
x
,
z
)
|
|
f
(
z
)
|
d
μ
(
z
)
+
∫
2
B
‍
|
K
α
(
y
,
z
)
|
|
f
(
z
)
|
d
μ
(
z
)
+
∫
𝒳
∖
2
B
‍
|
K
α
(
x
,
z
)
-
K
α
(
y
,
z
)
|
k
k
k
k
k
k
×
|
f
(
z
)
|
d
μ
(
z
)
∶
=
J
1
+
J
2
+
J
3
.
For the first term, by n
/
α
<
p, then 1
-
(
1
-
α
/
n
)
p
′
>
0,
(37)
J
1
≤
∫
2
B
‍
|
f
(
z
)
|
[
λ
(
x
,
d
(
x
,
y
)
)
]
1
-
α
/
n
d
μ
(
z
)
≤
∥
f
∥
L
p
(
μ
)
(
∫
2
B
‍
d
μ
(
z
)
[
λ
(
x
,
d
(
x
,
y
)
)
]
(
1
-
α
/
n
)
p
′
)
1
/
p
′
≤
∥
f
∥
L
p
(
μ
)
(
∑
j
=
0
∞
‍
μ
(
B
(
x
,
2
-
j
+
1
r
)
)
[
λ
(
x
,
d
(
x
,
2
-
j
r
)
)
]
(
1
-
α
/
n
)
p
′
)
1
/
p
′
≤
∥
f
∥
L
p
(
μ
)
(
∑
j
=
0
∞
‍
λ
(
x
,
2
-
j
+
1
r
)
[
λ
(
x
,
d
(
x
,
2
-
j
r
)
)
]
(
1
-
α
/
n
)
p
′
)
1
/
p
′
≤
C
∥
f
∥
L
p
(
μ
)
(
∑
j
=
0
∞
‍
[
λ
(
x
,
d
(
x
,
2
-
j
r
)
)
]
1
-
(
1
-
α
/
n
)
p
′
)
1
/
p
′
≤
C
∥
f
∥
L
p
(
μ
)
(
∑
j
=
0
∞
‍
1
[
C
(
2
j
)
]
(
α
/
n
-
1
/
p
)
p
′
)
1
/
p
′
×
[
λ
(
x
,
r
)
]
α
/
n
-
1
/
p
≤
C
∥
f
∥
L
p
(
μ
)
[
λ
(
x
,
r
)
]
α
/
n
-
1
/
p
.
The second term is estimated in a similar way after noting that 2
B
⊂
B
(
y
,
3
r
).
Next, by using (2), Hölder’s inequality, and α
-
n
/
p
<
δ, we get
(38)
J
3
≤
C
∫
𝒳
∖
2
B
‍
[
λ
(
x
,
d
(
x
,
y
)
)
]
δ
/
n
[
λ
(
x
,
d
(
x
,
z
)
)
]
1
-
α
/
n
+
δ
/
n
|
f
(
z
)
|
d
μ
(
z
)
≤
C
[
λ
(
x
,
d
(
x
,
y
)
)
]
δ
/
n
∥
f
∥
L
p
(
μ
)
×
(
∫
𝒳
∖
2
B
‍
1
[
λ
(
x
,
d
(
x
,
z
)
)
]
(
1
-
(
α
-
δ
)
/
n
)
p
′
d
μ
(
z
)
)
1
/
p
′
≤
C
∥
f
∥
L
p
(
μ
)
[
λ
(
x
,
d
(
x
,
y
)
)
]
δ
/
n
×
(
∫
𝒳
∖
2
B
‍
1
[
λ
(
x
,
d
(
x
,
z
)
)
]
(
1
-
(
α
-
δ
)
/
n
)
p
′
d
μ
(
z
)
)
1
/
p
′
≤
C
∥
f
∥
L
p
(
μ
)
[
λ
(
x
,
d
(
x
,
y
)
)
]
δ
/
n
[
λ
(
x
,
d
(
x
,
y
)
)
]
(
α
-
δ
)
/
n
-
1
/
p
≤
C
∥
f
∥
L
p
(
μ
)
[
λ
(
x
,
d
(
x
,
y
)
)
]
α
/
n
-
1
/
p
.
Putting together the three estimates,
(39)
|
I
~
α
f
(
x
)
-
I
~
α
f
(
y
)
|
≤
C
∥
f
∥
L
p
(
μ
)
[
λ
(
x
,
d
(
x
,
y
)
)
]
α
/
n
-
1
/
p
,
then I
~
α maps L
p
(
μ
) boundedly into Lip
(
α
-
n
/
p
).Proof of Theorem18.
IfI
~
α
∈
Lip
(
β
), then by the continuity of the operator I
~
α implies that I
~
α
(
1
) must be constant; that is, I
~
α
(
1
)
(
x
)
=
I
~
α
(
1
)
(
x
0
)
=
0.
On the other hand, we can observe that(40)
I
~
α
(
1
)
=
0
⟺
I
~
α
(
1
)
(
x
)
-
I
~
α
(
1
)
(
y
)
=
0
;
this implies that
(41)
∫
𝒳
‍
{
K
α
(
x
,
z
)
-
K
α
(
y
,
z
)
}
d
μ
(
z
)
=
0
.
Thus we can write(42)
I
~
α
(
f
)
(
x
)
-
I
~
α
(
f
)
(
y
)
=
∫
𝒳
‍
{
K
α
(
x
,
z
)
-
K
α
(
y
,
z
)
}
(
f
(
z
)
-
f
(
x
)
)
d
μ
(
z
)
=
∫
2
B
(
x
,
r
)
‍
K
α
(
x
,
z
)
(
f
(
z
)
-
f
(
x
)
)
d
μ
(
z
)
+
∫
2
B
(
x
,
r
)
‍
-
K
α
(
y
,
z
)
(
f
(
z
)
-
f
(
x
)
)
d
μ
(
z
)
+
∫
𝒳
∖
2
B
(
x
,
r
)
‍
{
K
α
(
x
,
z
)
-
K
α
(
y
,
z
)
}
k
k
k
k
k
k
k
k
k
k
k
k
×
(
f
(
z
)
-
f
(
x
)
)
d
μ
(
z
)
∶
=
M
1
+
M
2
+
M
3
,
where r
=
d
(
x
,
y
). For M
1,
(43)
|
M
1
|
≤
∫
2
B
(
x
,
r
)
‍
|
f
(
z
)
-
f
(
x
)
|
[
λ
(
x
,
d
(
x
,
z
)
)
]
1
-
α
/
n
d
μ
(
z
)
≤
C
∫
2
B
(
x
,
r
)
‍
[
λ
(
x
,
d
(
x
,
z
)
)
]
β
/
n
[
λ
(
x
,
d
(
x
,
z
)
)
]
1
-
α
/
n
d
μ
(
z
)
≤
C
[
λ
(
x
,
d
(
x
,
y
)
)
]
(
α
+
β
)
/
n
.
Similarly, we know that 2
B
(
x
,
r
)
⊂
3
B
(
y
,
r
); using Hölder’s inequality (
1
/
t
+
1
/
t
′
=
1
), and letting t
>
n
/
α, then we have
(44)
|
M
2
|
≤
∫
2
B
(
x
,
r
)
‍
|
f
(
z
)
-
f
(
x
)
|
[
λ
(
y
,
d
(
y
,
z
)
)
]
1
-
α
/
n
d
μ
(
z
)
≤
C
∫
2
B
(
x
,
r
)
‍
[
λ
(
x
,
d
(
x
,
z
)
)
]
β
/
n
[
λ
(
y
,
d
(
y
,
z
)
)
]
1
-
α
/
n
d
μ
(
z
)
≤
C
(
∫
2
B
(
x
,
r
)
‍
[
λ
(
x
,
d
(
x
,
z
)
)
]
β
t
/
n
d
μ
(
z
)
)
1
/
t
×
(
∫
2
B
(
x
,
r
)
‍
1
[
λ
(
y
,
d
(
y
,
z
)
)
]
(
1
-
α
/
n
)
t
′
d
μ
(
z
)
)
1
/
t
′
≤
C
(
∑
j
=
0
∞
‍
∫
2
-
j
+
1
B
(
x
,
r
)
∖
2
-
j
B
(
x
,
r
)
‍
[
d
(
x
,
2
-
j
+
1
r
)
)
λ
(
(
x
,
2
-
j
+
1
r
)
x
,
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
d
(
x
,
2
-
j
+
1
r
)
)
]
β
t
/
n
d
μ
(
z
)
∑
j
=
0
∞
‍


)
1
/
t
×
(
∫
3
B
(
y
,
r
)
‍
1
[
λ
(
y
,
d
(
y
,
z
)
)
]
(
1
-
α
/
n
)
t
′
d
μ
(
z
)
)
1
/
t
′
≤
C
[
λ
(
x
,
d
(
x
,
y
)
)
]
α
/
n
[
λ
(
y
,
d
(
x
,
y
)
)
]
β
/
n
≤
C
[
λ
(
x
,
d
(
x
,
y
)
)
]
(
α
+
β
)
/
n
.
In order to estimate M
3, we use (2) to obtain
(45)
|
M
3
|
≤
C
∫
𝒳
∖
2
B
(
x
,
r
)
[
λ
(
x
,
d
(
x
,
y
)
)
]
δ
/
n
[
λ
(
x
,
d
(
x
,
z
)
)
]
β
/
n
[
λ
(
x
,
d
(
x
,
z
)
)
]
1
-
(
α
-
δ
)
/
n
d
μ
(
z
)
≤
C
[
λ
(
x
,
d
(
x
,
y
)
)
]
δ
/
n
×
∫
𝒳
∖
2
B
(
x
,
r
)
1
[
λ
(
x
,
d
(
x
,
z
)
)
]
1
-
(
α
+
β
-
δ
)
/
n
d
μ
(
z
)
≤
C
[
λ
(
x
,
d
(
x
,
y
)
)
]
δ
/
n
[
λ
(
x
,
d
(
x
,
y
)
)
]
(
α
+
β
-
δ
)
/
n
≤
C
[
λ
(
x
,
d
(
x
,
y
)
)
]
(
α
+
β
)
/
n
.
Combining the estimates for M
1, M
2, and M
3, we obtain
(46)
|
I
~
α
f
(
x
)
-
I
~
α
f
(
y
)
|
≤
C
[
λ
(
x
,
d
(
x
,
y
)
)
]
(
α
+
β
)
/
n
;
this finishes the proof.Proof of Theorem19.
I
~
α
(
f
)
∈
Lip
(
α
) implies that I
~
α
(
1
)
=
0 and, equivalently, that
(47)
∫
𝒳
‍
(
K
α
(
x
,
y
)
-
K
α
(
x
0
,
y
)
)
d
μ
(
y
)
=
0
for all x.
Take two pointsx
≠
y and let B
=
B
(
x
,
r
) with r
=
d
(
x
,
y
). Then
(48)
|
I
~
α
(
f
)
(
x
)
-
I
~
α
(
f
)
(
y
)
|
=
|
∫
𝒳
‍
{
K
α
(
x
,
z
)
-
K
α
(
y
,
z
)
}
(
f
(
z
)
-
f
2
B
)
d
μ
(
z
)
|
≤
|
∫
2
B
‍
K
α
(
x
,
z
)
(
f
(
z
)
-
f
2
B
)
d
μ
(
z
)
|
+
|
∫
2
B
‍
K
α
(
y
,
z
)
(
f
(
z
)
-
f
2
B
)
d
μ
(
z
)
|
+
|
∫
𝒳
∖
2
B
‍
{
K
α
(
x
,
z
)
-
K
α
(
y
,
z
)
}
(
f
(
z
)
-
f
2
B
)
d
μ
(
z
)
|
∶
=
N
1
+
N
2
+
N
3
.
For the first term, by Hölder’s inequality with some p
>
n
/
α,
(49)
N
1
≤
∫
2
B
‍
1
[
λ
(
x
,
d
(
x
,
z
)
)
]
1
-
α
/
n
|
f
(
z
)
-
f
2
B
|
d
μ
(
z
)
≤
(
∫
2
B
‍
d
μ
(
z
)
[
λ
(
x
,
d
(
x
,
z
)
)
]
(
1
-
α
/
n
)
p
′
)
1
/
p
′
×
(
∫
2
B
‍
|
f
(
z
)
-
f
2
B
|
p
d
μ
(
z
)
)
1
/
p
≤
C
[
λ
(
x
,
r
)
]
α
/
n
-
1
/
p
μ
(
ρ
B
)
1
/
p
∥
f
∥
RBMO
(
μ
)
≤
C
[
λ
(
x
,
r
)
]
α
/
n
∥
f
∥
RBMO
(
μ
)
.
Using 2
B
⊂
B
(
y
,
3
r
), the second term can be dealt with in the same way as the N
1; then N
2
≤
C
[
λ
(
x
,
r
)
]
α
/
n
∥
f
∥
RBMO
(
μ
). It is easy to see that |
f
2
k
+
1
B
-
f
2
B
|
≤
∥
f
∥
RBMO
(
μ
)
K
2
B
,
2
k
+
1
B
≤
k
∥
f
∥
RBMO
(
μ
); then
(50)
N
3
≤
∫
𝒳
∖
2
B
[
λ
(
x
,
d
(
x
,
y
)
)
]
δ
/
n
[
λ
(
x
,
d
(
x
,
z
)
)
]
1
-
α
/
n
+
δ
/
n
|
f
(
z
)
-
f
2
B
|
d
μ
(
z
)
≤
C
[
λ
(
x
,
d
(
x
,
y
)
)
]
δ
/
n
×
∑
k
=
1
∞
(
|
f
2
k
+
1
B
-
f
2
B
|
[
λ
(
x
,
d
(
x
,
2
k
r
)
)
]
1
-
α
/
n
+
δ
/
n
∫
2
k
+
1
B
∖
2
k
B
|
f
(
z
)
-
f
2
k
+
1
B
|
[
λ
(
x
,
d
(
x
,
z
)
)
]
1
-
α
/
n
+
δ
/
n
d
μ
(
z
)
kkkkkkkk
+
|
f
2
k
+
1
B
-
f
2
B
|
[
λ
(
x
,
d
(
x
,
2
k
r
)
)
]
1
-
α
/
n
+
δ
/
n
μ
(
2
k
+
1
B
)
)
≤
C
[
λ
(
x
,
d
(
x
,
y
)
)
]
δ
/
n
∥
f
∥
RBMO
(
μ
)
×
(
∑
k
=
1
∞
μ
(
ρ
2
k
+
1
B
)
[
λ
(
x
,
d
(
x
,
2
k
r
)
)
]
1
-
α
/
n
+
δ
/
n
kkkkk
+
∑
k
=
1
∞
k
μ
(
2
k
+
1
B
)
[
λ
(
x
,
d
(
x
,
2
k
r
)
)
]
1
-
α
/
n
+
δ
/
n
)
≤
C
[
λ
(
x
,
d
(
x
,
y
)
)
]
δ
/
n
∥
f
∥
RBMO
(
μ
)
×
(
∑
k
=
0
∞
(
k
+
1
)
[
λ
(
x
,
d
(
x
,
2
k
r
)
)
]
α
/
n
-
δ
/
n
)
≤
C
[
λ
(
x
,
d
(
x
,
y
)
)
]
α
/
n
∥
f
∥
RBMO
(
μ
)
.
Thus the proof of Theorem 19 is completed.



---

*Source: 174010-2014-04-17.xml*