# Variable Exponent Function Spaces related to a Sublinear Expectation

Authors: Bochi Xu
Journal: Abstract and Applied Analysis
 (2020)

---

## Abstract

In this paper, variable exponent function spacesL
p
·, L
b
p
·, and L
c
p
· are introduced in the framework of sublinear expectation, and some basic and important properties of these spaces are given. A version of Kolmogorov’s criterion on variable exponent function spaces is proved for continuous modification of stochastic processes.


---

## Body

## 1. Introduction

Variable exponent spaces are extensively applied in the study of some nonlinear problems in natural science and engineering. Basic properties of the spaces are first given by Kováčik and Rákosník in [1]. Some theories of variable exponent spaces can also be found in [2, 3]. Harjulehto et al. present an overview of applications to differential equations with nonstandard growth in [4]. Diening et al. [5] summarize most of the existing literature of theory of variable exponent function spaces and applications to partial differential equations. In [6], Aoyama proves some important probability inequalities in variable exponent Lebesgue spaces.Nonlinear expectations play an important role in the research of financial markets. One of the most important application is that a coherent risk measure(the basic theory about coherent risk measure can be found in  [7]) is a sublinear expectation E
:
H
→
R defined on H, which is a linear space of finacial losses. In this paper, we are interested in behavior of sublinear expectation spaces with variable exponents. By the following representation theorem which can be found in Peng ([8], p. 4), we know that a sublinear expectation can be expressed as a supremum of linear expectations, i.e., there exists a family of linear expectations {
E
θ
}
θ
∈
Θ such that(1)
E
X
=
s
u
p
θ
∈
Θ
E
θ
X
,
X
∈
H
.Thus, we consider the upper expectation only in this paper. Some other important theories about nonlinear expectations can be found in Peng’s [9, 10].The remainder of the paper is divided as follows: in Section2, motivated by Fu [11] and Denis et al. [12], we first introduce L
p
·, L
b
p
· and L
c
p
· and give some properties of these spaces. And Each element of ·
p
·-completion of C
b
Ω has a quasi-continuous version is proved. In Section 3, applying the results of Section 2, we discuss a version of Kolmogorov’s criterion for continuous modification of stochastic processes, which are in the variable exponent function space related to a sublinear expectation, after proving the situation under a linear expectation.


## 2. Variable Exponent Function Spaces

LetΩ be a complete metric space equipped with the distance d, B
Ω the Borel σ
− algebra of Ω and M the collection of all probability measures on Ω
,
B
Ω.L
0
Ω: the space of all B
Ω-measurable real functions;
B
b
Ω: all bounded functions in L
0
Ω;
C
b
Ω: all continuous functions in B
b
Ω.For a given subsetP
⊆
M, we denote(2)
c
A
≔
s
u
p
P
∈
P
P
A
,
A
∈
B
Ω
.Definition 1.
A setA is a polar if c
A
=
0 and a property holds “quasi-surely” (q.s.) if it holds outside a polar set.The upper expectation ofP is defined as follows: for each X
∈
L
0
Ω such that E
P
X exists for each P
∈
P,(3)
E
X
≔
s
u
p
P
∈
P
E
P
X
.AboutE, we know the following properties.Theorem 2 (see Theorem 9 in [12]).
The upper expectationE
· of family P is a sublinear expectation on B
b
Ω as well as on C
b
Ω, i.e.,(1)
For all
X, Y
in
B
b
Ω, X
≥
Y, E
X
≥
E
Y.
(2)
For all
X, Y
in
B
b
Ω, E
X
+
Y
≤
E
X
+
E
Y.
(3)
For all
λ
≥
0, X
∈
B
b
Ω, E
λ
X
=
λ
E
X.
(4)
For all
c
∈
R, X
∈
B
b
Ω, E
X
+
c
=
E
X
+
c.Let aB
Ω-measurable real function p
:
Ω
→
1
,
∞, be a variable exponent. In the space L
0
Ω, the moduli ρ are defined by(4)
ρ
p
·
X
≔
E
X
p
=
sup
P
∈
E
P
X
p
=
sup
P
∈
∫
Ω
X
ω
p
ω
P
d
ω
.Definition 3.
The spaceL
p
· is the set of X
∈
L
0
Ω satisfying ρ
p
·
X
<
∞, and it is endowed with the Luxemburg norm:(5)
X
p
·
=
i
n
f
λ
>
0
:
ρ
p
·
X
λ
≤
1
.We setN
p
·
:
=
X
∈
L
0
Ω
:
ρ
X
=
0, and denote L
p
·
:
=
L
p
·
/
N
p
·. As usual, we do not take care about the distinction between classes and their representatives.Proposition 4.
If the variable exponentp satisfies 1
<
p
−
≤
p
ω
≤
p
+
<
∞, then the inequality(6)
E
X
Y
≤
1
+
1
p
−
−
1
p
+
X
p
·
Y
p
′
·
holds for everyX
∈
L
p
· and Y
∈
L
p
′
·, where 1
/
p
ω
+
1
/
p
′
ω
=
1.Proof.
By Young inequality, we have(7)
X
X
p
·
Y
Y
p
′
·
≤
1
p
ω
X
X
p
·
p
+
1
p
′
ω
Y
Y
p
′
·
p
′
.By the monotonicity, sub-additivity and positive homogeneity ofE, and the property of the norm, we have(8)
E
X
X
p
·
Y
Y
p
′
·
≤
E
1
p
ω
X
X
p
·
p
+
1
p
′
ω
Y
Y
p
′
·
p
′
≤
E
1
p
ω
X
X
p
·
p
+
E
1
p
′
ω
Y
Y
p
′
·
p
′
≤
1
p
−
E
X
X
p
·
p
+
1
−
1
p
+
E
Y
Y
p
′
·
p
′
≤
1
+
1
p
−
−
1
p
+
.Thus, the inequality follows.Proposition 5.
Suppose that the variable exponentp satisfies 1
<
p
−
≤
p
ω
≤
p
+
<
∞. If X
,
X
k
∈
L
p
·, then(1)
IfX
p
·
≥
1, then X
p
·
p
−
≤
ρ
p
·
X
≤
X
p
·
p
+.
(2)
IfX
p
·
≤
1, then X
p
·
p
+
≤
ρ
p
·
X
≤
X
p
·
p
−.
(3)
lim
k
→
∞
X
k
p
·
=
0, if and only if lim
k
→
∞
ρ
p
·
X
k
=
0
(4)
lim
k
→
∞
X
k
p
·
=
∞, if and only if lim
k
→
∞
ρ
p
·
X
k
=
∞.
In particular, the linear expectationE
P also follows this proposition.Proof.
(1) ByX
P
p
·
≥
1 and the definition of the norm,(9)
E
X
p
X
p
·
p
+
≤
E
X
X
p
·
p
≤
1
.Thus,ρ
p
·
X
≤
X
p
·
p
+. As(10)
X
p
·
p
−
/
p
≤
X
p
·
,we also have(11)
E
X
X
p
·
p
−
/
p
p
≥
1
.That is to sayX
p
·
p
−
≤
ρ
p
·
X
. The proof is completed. (2) can be easily proved in the similar method. By (1) and (2), we can easily get (3) and (4).Proposition 6.
Suppose that the variable exponentp satisfies 1
<
p
−
≤
p
ω
≤
p
+
<
∞. Let X
,
X
k
∈
L
p
·. Then for each α
>
0
(12)
c
X
>
α
≤
α
−
p
−
∨
α
−
p
+
E
X
p
.Proof.
For eachP
∈
P, by Markov inequality, we have(13)
P
X
α
p
>
1
≤
E
P
X
α
p
.Take the supremum inP both sides,(14)
c
X
α
p
>
1
≤
E
X
α
pThus,(15)
c
X
>
α
≤
E
X
α
p
≤
α
−
p
−
∨
α
−
p
+
E
X
p
.Lemma 7 (Proposition 17 in [12]).
Letp
∈
0
,
∞ and {
X
n
} be a sequence in L
p which converges to X in L
p. Then there exists a subsequence {
X
n
k
} which converges to X quasi-surely in the sense that it converges to X outside a polar set.Proposition 8.
If the variable exponentp satisfies 1
<
p
−
≤
p
ω
≤
p
+
<
∞. L
p
· is a Banach space.Proof.
Let{
X
n
} be a Cauchy sequence in L
p
·. Then, by Proposition 4,(16)
E
X
m
−
X
n
≤
C
X
m
−
X
n
p
·
,
whereC is a constant. That is to say {
X
n
} is a Cauchy sequence in L
1. By Proposition 14 in [12], L
1 is a Banach Space. Thus, {
X
n
} converges in L
1. Suppose that X
n
→
X, X
∈
L
1 and further by Lemma 7, we suppose X
n
→
X quasi-surely (subtracting a subsequence if necessary). For each 0
<
ε
<
1, there exists n
0 such that X
m
−
X
n
p
·
<
ε for m
,
n
≥
n
0. Fix n, by Fatou’s lemma(17)
E
X
n
−
X
ε
p
=
s
u
p
P
∈
P
E
P
X
n
−
X
ε
p
≤
s
u
p
P
∈
P
l
i
m
i
n
f
m
→
∞
E
P
X
n
−
X
m
ε
p
≤
l
i
m
i
n
f
m
→
∞
E
X
n
−
X
m
ε
p
≤
1
.
Thus,X
n
−
X
p
·
≤
ε, and further ρ
p
·
X
n
−
X
≤
X
n
−
X
p
·
p
−
≤
ε
p
−, that is to say, X
n
−
X
∈
L
p
·, and X
∈
L
p
·. The proof is completed.We denote byL
b
p
· the completion of B
b
Ω and by L
c
p
· the completion of C
b
Ω. By Proposition 8, we have L
c
p
·
⊂
L
b
p
·
⊂
L
p
· for 1
<
p
−
≤
p
ω
≤
p
+
<
∞.Proposition 9.
Suppose that the variable exponentp satisfies 1
<
p
−
≤
p
ω
≤
p
+
<
∞, then(18)
L
b
p
·
=
X
∈
L
p
·
:
l
i
m
n
→
∞
E
X
p
1
X
>
n
=
0
.Proof.
We denoteJ
p
·
=
{
X
∈
L
p
·
:
l
i
m
n
→
∞
E
X
p
1
{
X
>
n
}
=
0
}. For each X
∈
J
p
· let X
n
=
X
∧
n
∨
−
n
∈
B
b
Ω. We have(19)
ρ
p
·
X
−
X
n
≤
E
X
p
1
X
>
n
,sol
i
m
n
→
∞
ρ
p
·
X
−
X
n
=
0. Thus, X
∈
L
b
p
·. On the other hand, for any X
∈
L
b
p
·, we can find a sequence {
Y
n
}
⊂
B
b
Ω such that l
i
m
n
→
∞
ρ
p
·
X
−
Y
n
=
0. Let y
n
=
s
u
p
ω
∈
Ω
|
Y
n
ω
| and X
n
=
X
∧
y
n
∨
−
y
n. Since |
X
−
X
n
≤
X
−
Y
n
|, we have l
i
m
n
→
∞
ρ
p
·
X
−
X
n
=
0. This implies that for any sequence {
α
n
} such that α
n
→
∞ as n
→
∞, l
i
m
n
→
∞
ρ
p
·
X
−
X
∧
α
n
∨
−
α
n
=
0. And for all n
∈
N,(20)
E
X
p
1
X
>
n
<
2
p
+
−
1
E
X
−
n
p
1
X
>
n
+
E
2
p
−
1
n
p
1
X
>
n
.For the first term of right hand side, we have(21)
L
i
m
n
→
∞
E
X
−
n
p
1
X
>
n
=
l
i
m
n
→
∞
ρ
p
·
X
−
X
∧
n
∨
−
n
=
0
.For the second term,(22)
n
p
2
p
1
X
>
n
≤
X
−
n
2
p
1
X
>
n
≤
X
−
n
2
p
1
X
>
n
/
2
,so(23)
E
2
p
−
1
n
p
1
X
>
n
=
E
2
2
p
−
1
n
p
2
p
1
X
>
n
≤
E
2
2
p
−
1
X
−
n
2
p
1
X
>
n
/
2
≤
2
2
p
+
−
1
E
X
−
n
/
2
p
1
X
>
n
/
2
,andl
i
m
n
→
∞
E
2
p
−
1
n
p
1
{
X
>
n
}
=
0. Thus, X
∈
J
p
·.Proposition 10.
If the variable exponentp satisfies 1
<
p
−
≤
p
ω
≤
p
+
<
∞, let X
∈
L
b
p
·, then for each ε
>
0, there exists a δ
>
0, such that for all A
∈
B
Ω with c
A
≤
δ, we have E
X
p
1
A
≤
ε.Proof.
For eachε
>
0, by Proposition 9, there exists N
>
0 such that E
X
p
1
{
X
>
n
}
≤
ε
/
2. Set δ
=
ε
/
2
N
p
+, then for A
∈
B
Ω with c
A
≤
δ, we have.(24)
E
X
p
1
A
≤
E
X
p
1
A
1
X
>
N
+
E
X
p
1
A
1
X
≤
N
≤
E
X
p
1
X
>
N
+
N
p
+
c
A
≤
ε
.Definition 11.
A mappingX on Ω with values in topological space is said to be quasi-continuous (q.c.) if  ∀
ε
>
0, there exists an open set O with c
O
<
ε such that X
|
O
c is continuous.Definition 12.
We say thatX
:
Ω
→
R has a quasi-continuous version if there exists a quasi-continuous Y
:
Ω
→
R with X
=
Y q.s.Proposition 13.
If the variable exponentp satisfies 1
<
p
−
≤
p
ω
≤
p
+
<
∞. Then each element in L
c
p
· has a quasi-continuous version.Proof.
For eachX
∈
L
c
p
·, there exists {
X
n
}
⊂
C
b
Ω such that X
n
→
X in L
c
p
·. Let us choose a subsequence {
X
n
k
} such that(25)
ρ
p
·
X
n
k
+
1
−
X
n
k
≤
2
−
2
k
,
∀
k
≥
1
,and set for allk,(26)
A
k
=
⋃
i
=
k
∞
X
n
i
+
1
−
X
n
i
>
2
−
i
/
p
+
.Because of the subadditivity property and Proposition6,(27)
c
A
k
≤
∑
i
=
k
∞
c
X
n
i
+
1
−
X
n
i
>
2
−
i
/
p
+
≤
∑
i
=
k
∞
2
i
E
X
n
i
+
1
−
X
n
i
p
≤
∑
i
=
k
∞
2
−
i
=
2
−
k
+
1
.Thus,l
i
m
k
→
∞
c
A
k
=
0, so the Borel set A
=
⋂
k
=
1
∞
A
k is polar. As each X
n
k is continuous, for all k
≥
1, A
k is an open set. And for all k, {
X
n
i
} converges uniformly on A
k
c so that the limit is continuous on each A
k
c.Proposition 14.
Suppose that the variable exponentp satisfies 1
<
p
−
≤
p
ω
≤
p
+
<
∞, then(28)
L
c
p
·
=
X
∈
L
p
·
:
X
h
a
s
a
q
u
a
s
i
‐
c
o
n
t
i
n
u
o
u
s
v
e
r
s
i
o
n
,
a
n
d
l
i
m
n
→
∞
E
X
p
1
X
>
n
=
0
.Proof.
Let(29)
J
p
·
=
X
∈
L
p
·
:
X
h
a
s
a
q
u
a
s
i
‐
c
o
n
t
i
n
u
o
u
s
v
e
r
s
i
o
n
,
a
n
d
l
i
m
n
→
∞
E
X
p
1
X
>
n
=
0
.For eachX
∈
L
c
p
·, X has a quasi-continuous version by Proposition 13. Since X
∈
L
c
p
·
⊂
L
b
p
·, by Proposition 9 we have l
i
m
n
→
∞
E
X
p
1
{
X
>
n
}
=
0. Thus, X
∈
J
p
·.On the other hand, LetX
∈
J
p
· be quasi-continuous. For all n
∈
N, define(30)
Y
n
=
X
∧
n
∨
−
n
.Sincel
i
m
n
→
∞
E
X
p
1
{
X
>
n
}
=
0 and(31)
E
X
−
Y
n
p
≤
E
X
−
Y
n
p
1
X
>
n
+
E
X
−
Y
n
p
1
X
≤
n
≤
E
X
p
1
X
>
n
,we havel
i
m
n
→
∞
E
|
X
−
Y
n
|
p
=
0.For alln
∈
N, Y
n is quasi-continuous, so there exists a closed set F
n such that c
F
n
c
<
1
/
n
p
+
+
1 and Y
n is continuous on F
n. By Tietze’s extension theorem there exists Z
n
∈
C
b
Ω such that |
Z
n
|
≤
n and Z
n
=
Y
n on F
n. Then, we have  (32)
E
Y
n
−
Z
n
p
≤
E
Y
n
−
Z
n
p
1
F
n
+
E
Y
n
−
Z
n
p
1
F
n
c
≤
E
2
n
p
1
F
n
c
≤
2
n
p
+
c
F
n
c
≤
2
p
+
n
.Thus,(33)
E
X
−
Z
n
p
≤
2
p
+
−
1
E
X
−
Y
n
p
+
E
Y
n
−
Z
n
p
≤
2
p
+
−
1
E
X
−
Y
n
p
+
2
2
p
+
−
1
n
.As a consequence,X
∈
L
c
p
·.Proposition 15.
Suppose thatX
:
Ω
→
R has a quasi-continuous version and that there exists a function f
:
R
+
→
R
+ satisfying l
i
m
n
→
∞
f
t
/
t
p
=
∞ uniformly on Ω and E
f
X
<
∞. Then X
∈
L
c
p
·, where 1
<
p
−
≤
p
ω
≤
p
+
<
∞.Proof.
For eachε
>
0, there exists N
>
0 which is independent of ω such that for all t
≥
N,(34)
f
t
t
p
≥
1
ε
.So(35)
E
X
p
1
X
>
N
≤
ε
E
f
X
1
X
>
N
≤
ε
E
f
X
.Thus,l
i
m
N
→
∞
E
X
p
1
{
X
>
N
}
=
0.


## 3. Kolmogorov’s Criterion on Variable Exponent Function Spaces

Definition 16.
LetI be a set of indices. {
X
t
}
t
∈
I and {
Y
t
}
t
∈
I be two processes indexed by I. We say that Y is a quasi-modification of X if for all t
∈
I
X
t
=
Y
t q.s.To prove a Kolmogorov’s criterion for a process indexed byR
d with d
∈
N on variable exponent function spaces, we give the following lemma first.Lemma 17.
Let1
<
p
−
≤
p
ω
≤
p
+
<
∞ and {
X
t
}
t
∈
0
,
1
d be a process such that for all t
∈
0
,
1
d, X
t
∈
L
p
·. Assume that for a fixed P
∈
P there exist positive constants c and ε such that(36)
E
P
X
t
−
X
s
p
≤
c
t
−
s
d
+
ε
.ThenX admit a modification X
˜ such that	(37)
E
P
sup
s
≠
t
⁡
X
∼
t
-
X
∼
s
t
-
s
α
p
<
∞
,for everyα
∈
0
,
ε
/
p
+Proof.
FixP
∈
P. For m
∈
N, define the set of dyadic points in 0
,
1
d:(38)
D
m
≔
i
1
2
m
,
⋯
,
i
d
2
m
:
i
1
⋯
,
i
d
∈
0
,
1
,
⋯
,
2
m
,andD
=
∪
m
D
m. Set Δ
m
=
{
s
,
t
:
|
s
−
t
|
=
2
−
m
,
s
,
t
∈
D
m
}, and there are fewer than 2
m
+
l
d such pairs in Δ
m. For s
,
t
∈
D, we say that s
≤
t if each component of s is less than or equal to the corresponding component of t.SetK
i
=
s
u
p
s
,
t
∈
Δ
i
|
X
s
−
X
t
|. And by the hypothesis in the lemma,(39)
E
P
K
i
p
≤
∑
s
,
t
∈
Δ
i
E
P
X
s
,
−
,
X
t
p
≤
2
i
+
1
d
c
2
−
i
d
+
ε
=
c
2
d
2
−
i
ε
.It is easy to seec
2
d
2
−
i
ε
<
1 from some i
0 on.Fors
,
t
∈
D, there exists increasing sequences {
s
n
}
⊂
D and {
t
n
}
⊂
D such that s
n
,
t
n
∈
D
n, s
n
≤
s, t
n
≤
t and s
n
=
s, t
n
=
t from some n on.Let nows
,
t
∈
D and |
s
−
t
|
≤
2
−
m, and either s
m
=
t
m or s
m
,
t
m
∈
Δ
m and in any case,(40)
X
s
−
X
t
=
∑
i
=
m
∞
X
s
i
+
1
−
X
s
i
+
X
s
m
−
X
t
m
+
∑
i
=
m
∞
X
t
i
−
X
t
i
+
1
,where the series are actually finite sums. It follows that(41)
X
s
−
X
t
≤
K
m
+
2
∑
i
=
m
+
1
∞
K
i
≤
2
∑
i
=
m
∞
K
i
.Thus, settingM
α
=
s
u
p
s
,
t
∈
D
,
s
≠
t
X
s
−
X
t
/
s
−
t
α, we have(42)
M
α
≤
sup
m
∈
ℕ
2
m
+
1
α
sup
s
,
t
∈
D
,
s
≠
t
,
s
−
t
≤
2
−
m
X
s
−
X
t
≤
sup
m
∈
ℕ
2
m
+
1
α
+
1
∑
i
=
m
∞
K
i
≤
2
α
+
1
∑
i
=
0
∞
2
i
α
K
i
.As(43)
K
i
p
⋅
≤
max
E
P
K
i
p
1
/
p
−
,
E
P
K
i
p
1
/
p
+
≤
max
c
2
d
2
−
i
ε
1
/
p
−
,
c
2
d
2
−
i
ε
1
/
p
+
,for everyα
∈
0
,
ε
/
p
+, we have(44)
M
α
p
·
≤
2
α
+
1
∑
i
=
0
∞
2
i
α
K
i
p
·
≤
2
α
+
1
∑
i
=
0
i
0
−
1
2
i
α
c
2
d
2
−
i
ε
1
/
p
−
+
∑
i
=
i
0
∞
2
i
α
c
2
d
2
−
i
ε
1
/
p
+
≤
C
α
,
p
−
,
ε
+
C
d
,
α
,
p
+
,
ε
∑
i
=
i
0
∞
2
i
α
−
ε
/
p
+
<
∞
,whereα
∈
0
,
ε
/
p
+. It follows in particular that for almost every ω, X is uniformly continuous on D and it makes sense to set(45)
X
˜
t
ω
=
lim
s
→
t
,
s
∈
D
X
s
ω
.By Fatou’s lemma and the hypothesis,(46)
E
P
X
˜
t
−
X
t
p
≤
lim
inf
s
→
t
E
P
X
s
−
X
t
p
≤
lim
inf
s
→
t
c
t
−
s
d
+
ε
,so,X
˜
t
=
X
t a.s. and X
˜
t is a modification.Theorem 18.
Let1
<
p
−
≤
p
ω
≤
p
+
<
∞ and {
X
t
}
t
∈
[
0
,
1
]
d be a process such that for all t
∈
[
0
,
1
]
d, X
t
∈
L
p
·. Assume that there exists a positive constants c and ε such that(47)
E
X
t
−
X
s
p
≤
c
t
−
s
d
+
ε
.
ThenX admit a modification X
˜ such that(48)
E
sup
s
≠
t
X
˜
t
−
X
˜
s
t
−
s
α
p
<
∞
,
for everyα
∈
0
,
ε
/
p
+
.Proof.
Let setD the same as the proof of Lemma 17. We set.(49)
M
α
=
sup
s
,
t
∈
D
,
s
≠
t
X
s
−
X
t
s
−
t
α
,whereα
∈
0
,
ε
/
p
+. By Lemma 17, we know that for any P
∈
P, E
P
M
p is finite and uniformly bounded with respect to P so that(50)
E
M
p
=
s
u
p
P
∈
P
E
P
M
p
<
∞
.Thus,X is uniformly continuous on D q.s. and set(51)
X
˜
t
ω
=
lim
s
→
t
,
s
∈
D
X
s
ω
,
t
∈
0
,
1
d
.In the similar way in Lemma17, X
˜
t
=
X
t q.s. and X
˜
t is a modification.



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*Source: 1734174-2020-01-13.xml*