# Malliavin Derivatives in Spaces with Variable Exponents

Authors: Bochi Xu; Yongqiang Fu; Boping Tian
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

Spaces with variable exponentsL
p
x
(
H
,
μ
) and L
p
x
(
H
,
μ
;
H
) are
introduced. After discussing some approximation results of L
p
x
(
H
,
μ
), Sobolev spaces on H with variable exponents are introduced. At last, we define Malliavin derivatives in L
p
x
(
H
,
μ
) and discuss some properties of Malliavin derivatives in L
p
x
(
H
,
μ
).


---

## Body

## 1. Introduction

Variable exponent spaces play an important role in the study of some nonlinear problems in natural science and engineering. In decades, there is a rapid development on the subject of variable exponent function spaces. Basic properties of the spaces have been discussed by Kovác
˘ik 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 function spaces with variable exponents and applications to partial differential equations. In [6], Aoyama discusses the properties of Lebesgue spaces with variable exponents on a probability space.Malliavin derivatives have many applications on mathematical finance (see Malliavin and Thalmaier [7] and di Nunno et al. [8]), and we are interested in the behavior of Malliavin derivatives in spaces with variable exponents. In this paper, in Section 2, motivated by [6, 9, 10], we will first introduce L
p
(
x
)
(
H
,
μ
) and L
p
(
x
)
(
H
,
μ
;
H
) and give some approximation results of L
p
(
x
)
(
H
,
μ
), which are useful in the definition on Malliavin derivatives in the following parts. In Section 3 we discuss gradients in L
p
(
x
)
(
H
,
μ
) and give properties of the linear operator D. At the end of this section, we define variable exponent Sobolev spaces on H. After the above preparation, we give Malliavin derivatives in L
p
(
x
)
(
H
,
μ
) and discuss some properties of the Malliavin derivative operator M in L
p
(
x
)
(
H
,
μ
) in the last section.


## 2. Preliminaries

LetH be a separable Hilbert space. A Borel probability measure N
m
,
Q in H with mean m and compact covariance operator Q is called Gaussian measure if the Fourier transform satisfies
(1)
∫
H
‍
e
i
〈
h
,
x
〉
N
m
,
Q
(
d
x
)
=
e
i
〈
m
,
h
〉
e
(
-
1
/
2
)
〈
Q
h
,
h
〉
,
∀
x
∈
H
.
N
m
,
Q is called nondegenerate if  Ker
⁡
Q
=
{
0
}. We are given a nondegenerate Gaussian measure μ
=
N
0
,
Q in the Hilbert space H. Since the operator Q is compact, there exists a complete orthonormal system {
e
k
} in H and a sequence {
λ
k
} of positive numbers such that
(2)
Q
e
k
=
λ
k
e
k
,
k
∈
ℕ
.
We denote by C
b
(
H
) the space of all mappings ϕ
:
H
→
ℝ, which are both continuous and bounded. C
b
(
H
) is a Banach space with the norm ∥
φ
∥
0
=
sup
⁡
x
∈
H
⁡
|
φ
(
x
)
|. And denote by C
b
k
(
H
), k
∈
ℕ, the space of all mappings ϕ
:
H
→
ℝ which are continuous and bounded together with their derivatives of order not bigger than k.Given a variable exponentp
:
H
→
[
1
,
∞
], it is assumed to be a Borel measurable function. On the set of all Borel measurable functions, the moduli ρ and ρ
~ are defined, respectively, by
(3)
ρ
(
φ
)
=
∫
H
∖
H
∞
‍
|
φ
(
x
)
|
p
(
x
)
μ
(
d
x
)
+
ess
sup
⁡
x
∈
H
∞
|
φ
(
x
)
|
,
ρ
~
(
F
)
=
∫
H
∖
H
∞
‍
|
F
(
x
)
|
H
p
(
x
)
μ
(
d
x
)
+
ess
sup
⁡
x
∈
H
∞
|
F
(
x
)
|
H
,
where H
∞
=
{
x
:
p
(
x
)
=
∞
},  φ
:
H
→
ℝ, and F
:
H
→
H.Definition 1.
The spaceℒ
p
(
x
)
(
H
,
μ
) is the set of Borel measurable functions φ
:
H
→
ℝ such that ρ
(
φ
)
<
∞, and it is endowed with the Luxemburg norm:
(4)
∥
φ
∥
L
p
(
x
)
(
H
,
μ
)
=
inf
⁡
{
λ
>
0
:
ρ
(
φ
λ
)
≤
1
}
.Define the equivalence relationr
1
:
φ
~
ψ if and only if ∥
φ
-
ψ
∥
L
p
(
x
)
(
H
,
μ
)
=
0. Denote by L
p
(
x
)
(
H
,
μ
) the quotient ℒ
p
(
x
)
(
H
,
μ
) with respect to the equivalence relation r
1.Definition 2.
The spaceℒ
p
(
x
)
(
H
,
μ
;
H
) is the set of Borel measurable functions F
:
H
→
H such that ρ
~
(
F
)
<
∞, and it is endowed with the Luxemburg norm:
(5)
∥
F
∥
L
p
(
x
)
(
H
,
μ
;
H
)
=
inf
⁡
{
λ
>
0
:
ρ
¯
(
F
λ
)
≤
1
}
.Define the equivalence relationr
2
:
F
~
G if and only if ∥
F
-
G
∥
L
p
(
x
)
(
H
,
μ
;
H
)
=
0. Denote by L
p
(
x
)
(
H
,
μ
;
H
) the quotient ℒ
p
(
x
)
(
H
,
μ
;
H
) with respect to the equivalence relation r
2.Proposition 3 (see Lemma3.2
.
20 in [5]).
Let variable exponentsp
(
x
)
,
q
(
x
)
,
r
(
x
)
∈
[
1
,
∞
] be such that 1
/
r
(
x
)
=
(
1
/
p
(
x
)
)
+
(
1
/
q
(
x
)
),  μ-a.e.; then the inequality
(6)
∥
φ
ψ
∥
L
r
(
x
)
(
H
,
μ
)
≤
2
∥
φ
∥
L
p
(
x
)
(
H
,
μ
)
∥
ψ
∥
L
q
(
x
)
(
H
,
μ
)
holds for every φ
∈
L
p
(
x
)
(
H
,
μ
) and φ
∈
L
q
(
x
)
(
H
,
μ
), where in the case r
=
p
=
q
=
∞ we use the convention r
/
p
=
r
/
q
=
1.Proposition 4.
The variable exponentp satisfies 1
≤
p
-
≤
p
(
x
)
≤
p
+
<
∞,  μ-a.e. Let F
∈
L
p
(
x
)
(
H
,
μ
;
H
) and set F
k
(
x
)
=
〈
F
(
x
)
,
e
k
〉; k
∈
ℕ. Then F
k
∈
L
p
(
x
)
(
H
,
μ
) for all k
∈
ℕ and
(7)
F
(
x
)
=
∑
k
=
1
∞
‍
F
k
(
x
)
e
k
,
μ
-
a
.
e
.
,
where the series is convergent in L
p
(
x
)
(
H
,
μ
;
H
).Proof.
ForF
k, k
∈
ℕ,
(8)
∫
H
‍
|
F
k
(
x
)
|
p
(
x
)
μ
(
d
x
)
≤
∫
H
‍
|
F
(
x
)
|
H
p
(
x
)
μ
(
d
x
)
<
∞
,
so F
k
∈
L
p
(
x
)
(
H
,
μ
). Since {
e
k
} is a complete orthonormal system in H and F
(
x
)
∈
H for x
∈
H, we have
(9)
F
(
x
)
=
∑
k
=
1
∞
‍
F
k
(
x
)
e
k
,
μ
-a
.
e
.
Set P
n
F
(
x
)
=
∑
k
=
1
n
F
k
(
x
)
e
k, and we have
(10)
lim
⁡
n
→
∞
P
n
F
(
x
)
=
F
(
x
)
,
μ
-a
.
e
.
,
|
F
(
x
)
|
H
p
(
x
)
=
(
∑
k
=
1
∞
‍
|
F
k
(
x
)
|
2
)
p
(
x
)
/
2
≥
(
∑
k
=
n
+
1
∞
‍
|
F
k
(
x
)
|
2
)
p
(
x
)
/
2
=
|
P
n
F
(
x
)
-
F
(
x
)
|
H
p
(
x
)
,
so by Lebesgue’s Dominated Convergence Theorem, we have
(11)
lim
⁡
n
→
∞
∫
H
‍
|
P
n
F
(
x
)
-
F
(
x
)
|
H
p
(
x
)
μ
(
d
x
)
=
0
.
Thus, the series are convergent in L
p
(
x
)
(
H
,
μ
;
H
). The proof is completed.Denote the linear span of all real and imaginary parts functionsφ
h
(
x
)
=
e
i
〈
h
,
x
〉, x
∈
H, and h
∈
H, by ℰ
(
H
).Proposition 5 (see Lemma 2.2 in [9]).
For allφ
∈
C
b
(
H
) there exists a two-index sequence {
φ
n
,
k
}
⊂
ℰ
(
H
) such that ∥
φ
n
,
k
∥
0
≤
∥
φ
∥
0, n
,
k
∈
ℕ, and
(12)
lim
⁡
n
→
∞
lim
⁡
k
→
∞
φ
n
,
k
(
x
)
=
φ
(
x
)
,
x
∈
H
.C
c
(
H
) is the space of all mappings φ
:
H
→
ℝ which are continuous and have compact support in H. We have the following proposition about C
c
(
H
).Proposition 6.
If the variable exponentp satisfies 1
≤
p
-
≤
p
(
x
)
≤
p
+
<
∞,  μ-a.e., then C
c
(
H
) is dense in L
p
(
x
)
(
H
,
μ
).Proof.
Note thatS
≔
S
(
H
,
μ
) is the set of all the simple functions. It is easy to check that S
⊂
L
p
(
x
)
(
H
,
μ
). Let f
∈
L
p
(
x
)
(
H
,
μ
) with f
≥
0. Since f is a Borel measurable function, there exist {
f
n
}
⊂
S with 0
≤
f
n
↗
f,  μ-a.e., and |
f
n
(
x
)
-
f
(
x
)
|
p
(
x
)
≤
|
f
(
x
)
|
p
(
x
), so by Lebesgue's Dominated Convergence Theorem, we have f
n
→
f in L
p
(
x
)
(
H
,
μ
). Thus, f is in the closure of S. If we drop the assumption f
≥
0, then we split f into positive and negative parts which belong to the closure of S. Thus S is dense in L
p
(
x
)
(
H
,
μ
). For any ɛ
>
0 and g
∈
L
p
(
x
)
(
H
,
μ
), there exists s
∈
S such that ∥
g
-
s
∥
L
p
(
x
)
(
H
,
μ
)
<
ɛ
/
2.
On the other hand, forp
+
<
∞, C
c
(
H
) is dense in L
p
+
+
1
(
H
,
μ
). So there exists g
~
∈
C
c
(
H
) such that ∥
s
-
g
~
∥
L
p
+
+
1
(
H
,
μ
)
<
ɛ
/
4. By Proposition 3, we have
(13)
∥
s
-
g
~
∥
L
p
(
x
)
(
H
,
μ
)
≤
2
∥
s
-
g
~
∥
L
p
+
+
1
(
H
,
μ
)
<
ɛ
2
.
Thus, ∥
g
-
g
~
∥
L
p
(
x
)
(
H
,
μ
)
≤
ɛ and C
c
(
H
) is dense in L
p
(
x
)
(
H
,
μ
). The proof is completed.Proposition 7.
If the variable exponentp satisfies 1
≤
p
-
≤
p
(
x
)
≤
p
+
<
∞,  μ-a.e., then ℰ
(
H
) is dense in L
p
(
x
)
(
H
,
μ
).Proof.
SinceC
c
(
H
)
⊂
C
b
(
H
), by Proposition 6  C
b
(
H
) is dense in L
p
(
x
)
(
H
,
μ
). So for any ɛ
>
0 and f
∈
L
p
(
x
)
(
H
,
μ
), there exists φ
∈
C
b
(
H
) such that
(14)
∥
f
-
φ
∥
L
p
(
x
)
(
H
,
μ
)
<
ɛ
2
.
By Proposition 5 and diagonal rule, there exists a sequence {
φ
n
}
⊂
ℰ
(
H
) such that |
φ
n
(
x
)
|
≤
∥
φ
∥
0, n
∈
ℕ, and
(15)
lim
⁡
n
→
∞
φ
n
(
x
)
=
φ
(
x
)
,
x
∈
H
.
By Lebesgue's Dominated Convergence Theorem, we have
(16)
lim
⁡
n
→
∞
∫
H
‍
|
φ
n
(
x
)
-
φ
(
x
)
|
μ
(
d
x
)
=
0
.
And as
(17)
∫
H
‍
|
φ
n
(
x
)
-
φ
(
x
)
|
p
(
x
)
μ
(
d
x
)
≤
(
2
∥
φ
∥
0
)
p
+
∫
H
‍
|
φ
n
(
x
)
-
φ
(
x
)
|
μ
(
d
x
)
,
we have φ
n
→
φ in L
p
(
x
)
(
H
,
μ
). Suppose that N
∈
ℕ, ∥
φ
n
-
φ
∥
L
p
(
x
)
(
H
,
μ
)
<
ɛ
/
2 for n
>
N, and we have ∥
φ
n
-
f
∥
L
p
(
x
)
(
H
,
μ
)
<
ɛ for n
>
N. Thus, ℰ
(
H
) is dense in L
p
(
x
)
(
H
,
μ
).Proposition 8.
Suppose thatW
z, z
∈
H, is a white noise function on H. If the variable exponent p satisfies 1
≤
p
-
≤
p
(
x
)
≤
p
+
<
∞,  μ-a.e., then
(18)
∥
W
z
∥
L
p
(
x
)
(
H
,
μ
)
≤
2
(
(
2
m
)
!
2
m
m
!
|
z
|
H
m
)
1
/
2
m
,
where p
+
<
2
m and m
∈
ℕ.Proof.
SinceW
z is a Gaussian random variable with mean 0 and covariance |
z
|
H, by Proposition 3, we have
(19)
∥
W
z
∥
L
p
(
x
)
(
H
,
μ
)
≤
2
∥
1
∥
L
(
2
m
p
(
x
)
)
/
(
2
m
-
p
(
x
)
)
(
H
,
μ
)
∥
W
z
∥
L
2
m
(
H
,
μ
)
=
2
(
(
2
m
)
!
2
m
m
!
|
z
|
H
m
)
1
/
2
m
.
The proof is completed.


## 3. Variable Exponent Sobolev Spaces on Separable Hilbert SpaceH

For anyφ
∈
ℰ
(
H
) and k
∈
ℕ, we denote by D
k
φ its derivative in the direction of e
k; that is,
(20)
D
k
φ
(
x
)
=
lim
⁡
ɛ
→
0
1
ɛ
(
φ
(
x
+
ɛ
e
k
)
-
φ
(
x
)
)
,
x
∈
H
.
Denote the gradient of φ by D
φ.We will consider the following linear mapping:(21)
D
:
ℰ
(
H
)
⊂
L
p
(
x
)
(
H
,
μ
)
⟶
L
p
(
x
)
(
H
,
μ
;
H
)
,
φ
⟼
D
φ
,
where 1
≤
p
-
≤
p
(
x
)
≤
p
+
<
∞,  μ-a.e.Lemma 9 (see Lemma 2.6 in [9]).
Supposeφ, ψ
∈
ℰ
(
H
); then
(22)
∫
H
‍
D
k
φ
ψ
d
μ
=
-
∫
H
‍
φ
D
k
ψ
d
μ
+
1
λ
k
∫
H
‍
x
k
φ
ψ
d
μ
,
where x
∈
H and x
k
=
〈
x
,
e
k
〉.Lemma 10 (see Corollary 2.7 in [9]).
Supposeφ
,
ψ
∈
ℰ
(
H
) and z
∈
Q
1
/
2
(
H
); then
(23)
∫
H
‍
〈
D
φ
,
z
〉
ψ
d
μ
=
-
∫
H
‍
〈
D
ψ
,
z
〉
φ
d
μ
+
∫
H
‍
W
Q
-
1
/
2
z
φ
ψ
d
μ
.Given a linear operatorA, not necessarily closed, if the closure of its graph happens to be the graph of some operator, that operator is called the closure of A, and we say that A is closable. Denote the closure of A by A
¯. And A is closable if and only if for any sequence {
x
n
}
⊂
D
(
A
) such that
(24)
lim
⁡
n
→
∞
x
n
=
0
,
lim
⁡
n
→
∞
A
x
n
=
y
,
one has y
=
0.Theorem 11.
The mappingD is a closable linear operator.Proof.
Let{
φ
n
}
⊂
ℰ
(
H
) such that
(25)
φ
n
⟶
0
in
L
p
(
x
)
(
H
,
μ
)
,
D
φ
n
⟶
F
in
L
p
(
x
)
(
H
,
μ
;
H
)
.
By the definition of closable operators, we only need to prove that F
=
0.
For anyψ
∈
ℰ
(
H
) and z
∈
Q
1
/
2
(
H
), by Lemma 10, we have
(26)
∫
H
‍
〈
D
φ
n
,
z
〉
ψ
d
μ
=
-
∫
H
‍
〈
D
ψ
,
z
〉
φ
n
d
μ
+
∫
H
‍
W
Q
-
1
/
2
z
φ
n
ψ
d
μ
.
By Proposition 3, we have
(27)
|
∫
H
‍
〈
D
φ
n
-
F
,
z
〉
ψ
d
μ
|
≤
∫
H
‍
|
D
φ
n
-
F
|
H
|
z
|
H
|
ψ
|
H
d
μ
≤
2
|
z
|
H
∥
D
φ
n
-
F
∥
L
p
(
x
)
(
H
,
μ
;
H
)
×
∥
ψ
∥
L
q
(
x
)
(
H
,
μ
)
,
|
∫
H
‍
〈
D
ψ
,
z
〉
φ
n
d
μ
|
≤
2
|
z
|
H
∥
φ
n
∥
L
p
(
x
)
(
H
,
μ
)
∥
D
ψ
∥
L
q
(
x
)
(
H
,
μ
;
H
)
,
where (
1
/
p
(
x
)
)
+
(
1
/
q
(
x
)
)
=
1; x
∈
H. Since ψ is bounded, by Propositions 3 and 8, we have
(28)
|
∫
H
‍
W
Q
-
1
/
2
z
φ
n
ψ
d
μ
|
≤
2
C
∥
φ
n
∥
L
p
(
x
)
(
H
,
μ
)
∥
W
Q
-
1
/
2
z
∥
L
q
(
x
)
(
H
,
μ
)
≤
C
p
+
|
Q
-
1
/
2
z
|
H
1
/
2
∥
φ
n
∥
L
p
(
x
)
(
H
,
μ
)
.
Thus, ∫
H
‍
〈
F
,
z
〉
ψ
d
μ
=
0, as n
→
∞. Since F
k
∈
L
p
(
x
)
(
H
,
μ
), for a fixed k
∈
ℕ, suppose {
ψ
k
n
}
⊂
ℰ
(
H
) such that ψ
k
n
→
|
F
k
|
p
(
x
)
-
1
sgn
⁡
(
F
k
) in L
q
(
x
)
(
H
,
μ
). We have
(29)
lim
⁡
n
→
∞
∫
H
‍
F
k
ψ
k
n
d
μ
=
0
;
that is, ∫
H
‍
|
F
k
|
p
(
x
)
d
μ
=
0. Thus, we have F
k
(
x
)
=
0,  μ-a.e., and F
(
x
)
=
0 , μ-a.e. The proof is completed.We denote byD
¯ the closable operator of D and by W
1
,
p
(
x
)
(
H
,
μ
) the domain of D
¯.Proposition 12.
For anyk
∈
ℕ, the linear operator D
k is closable and its closure D
¯
k satisfies D
¯
k
φ
=
〈
D
¯
φ
,
e
k
〉 for φ
∈
W
1
,
p
(
x
)
(
H
,
μ
).Proof.
For anyk
∈
ℕ, let {
φ
n
}
⊂
ℰ
(
H
) such that
(30)
φ
n
⟶
0
in
L
p
(
x
)
(
H
,
μ
)
,
D
k
φ
n
⟶
F
k
in
L
p
(
x
)
(
H
,
μ
)
.
By the definition of closable operators, we only need to prove that F
k
=
0. For any ψ
∈
ℰ
(
H
) and z
∈
H, by Lemma 9, we have
(31)
∫
H
‍
D
k
φ
n
ψ
d
μ
=
-
∫
H
‍
φ
n
D
k
ψ
d
μ
+
1
λ
k
∫
H
‍
x
k
φ
n
ψ
d
μ
.
Similar to the proof of Theorem 11, we have ∫
H
‍
F
k
ψ
d
μ
=
0 as n
→
∞ and F
k
=
0,  μ-a.e. Thus D
k is closable. And
(32)
D
¯
k
φ
=
lim
⁡
n
→
∞
D
k
φ
n
=
lim
⁡
n
→
∞
〈
D
φ
n
,
e
k
〉
=
〈
D
¯
φ
,
e
k
〉
.
The proof is completed.


## 4. Malliavin Derivatives inD
1
,
p
(
x
)
(
H
,
μ
)

For anyφ
∈
ℰ
(
H
), define the linear operator:
(33)
M
0
:
ℰ
(
H
)
⊂
L
p
(
x
)
(
H
,
μ
)
⟶
L
p
(
x
)
(
H
,
μ
;
H
)
,
M
0
φ
(
x
)
=
Q
1
/
2
D
φ
(
x
)
,
φ
∈
ℰ
(
H
)
,
x
∈
H
,
where 1
≤
p
-
≤
p
(
x
)
≤
p
+
<
∞,  μ-a.e. And
(34)
M
k
φ
(
x
)
=
〈
M
0
φ
(
x
)
,
e
k
〉
=
λ
k
1
/
2
D
k
φ
(
x
)
,
where φ
∈
ℰ
(
H
) and x
∈
H.Proposition 13 (see Corollary 2.10 in [9]).
Supposeφ
,
ψ
∈
ℰ
(
H
) and z
∈
H; then the following identity holds:
(35)
∫
H
‍
〈
M
0
φ
,
z
〉
ψ
d
μ
=
-
∫
H
‍
〈
M
0
ψ
,
z
〉
φ
d
μ
+
∫
H
‍
W
z
φ
ψ
d
μ
.
In a similar way to Theorem 11, one has Theorem 14.Theorem 14.
The mappingM
0 is a closable linear operator.Proof.
Let{
φ
n
}
⊂
ℰ
(
H
) such that
(36)
φ
n
⟶
0
in
L
p
(
x
)
(
H
,
μ
)
,
M
0
φ
n
⟶
F
in
L
p
(
x
)
(
H
,
μ
;
H
)
.
We only need to prove that F
=
0.
For anyψ
∈
ℰ
(
H
) and z
∈
H, by Proposition 13, we have
(37)
∫
H
‍
〈
M
0
φ
n
,
z
〉
ψ
d
μ
=
-
∫
H
‍
〈
M
0
ψ
,
z
〉
φ
n
d
μ
+
∫
H
‍
W
z
φ
n
ψ
d
μ
.
And we have ∫
H
‍
〈
F
,
z
〉
ψ
d
μ
=
0, as n
→
∞. Similar to Theorem 11, we have F
(
x
)
=
0,  μ-a.e. The proof is completed.We denote byM the closable operator of M
0 and by D
1
,
p
(
x
)
(
H
,
μ
) the domain of M. For any φ
∈
D
1
,
p
(
x
)
(
H
,
μ
), we call M
φ the Malliavin derivative of φ.Proposition 15.
Ifp satisfies 1
≤
p
(
x
)
≤
p
+
<
∞,  μ-a.e., then W
1
,
p
(
x
)
(
H
,
μ
)
⊂
D
1
,
p
(
x
)
(
H
,
μ
) and M
φ
=
Q
1
/
2
D
¯
φ for any φ
∈
W
1
,
p
(
x
)
(
H
,
μ
).Proof.
For anyφ
∈
W
1
,
p
(
x
)
(
H
,
μ
), there exists {
φ
n
}
⊂
ℰ
(
H
) such that φ
n
→
φ in L
p
(
x
)
(
H
,
μ
). As
(38)
∫
H
‍
|
Q
1
/
2
D
φ
n
(
x
)
-
Q
1
/
2
D
¯
φ
(
x
)
|
H
p
(
x
)
μ
(
d
x
)
≤
∫
H
‍
∥
Q
1
/
2
∥
p
(
x
)
|
D
φ
n
(
x
)
-
D
¯
φ
(
x
)
|
H
p
(
x
)
μ
(
d
x
)
≤
max
⁡
{
∥
Q
1
/
2
∥
,
∥
Q
1
/
2
∥
p
+
}
∫
H
‍
|
D
φ
n
(
x
)
-
D
¯
φ
(
x
)
|
H
p
(
x
)
μ
(
d
x
)
we have M
φ
=
Q
1
/
2
D
¯
φ and φ
∈
D
1
,
p
(
x
)
(
H
,
μ
).To prove Proposition17, we need the following lemma.Lemma 16 (see Lemma 2.3 in [9]).
For allφ
∈
C
b
1
(
H
) there exists a two-index sequence {
φ
n
,
k
}
⊂
ℰ
(
H
) such that ∥
φ
n
,
k
∥
0
+
∥
D
φ
n
,
k
∥
0
≤
∥
φ
∥
0
+
∥
D
φ
∥
0, n
,
k
∈
ℕ, and
(39)
lim
⁡
n
→
∞
lim
⁡
k
→
∞
φ
n
,
k
(
x
)
=
φ
(
x
)
,
x
∈
H
,
lim
⁡
n
→
∞
lim
⁡
k
→
∞
D
φ
n
,
k
(
x
)
=
D
φ
(
x
)
,
x
∈
H
.Proposition 17.
Ifp satisfies 1
≤
p
(
x
)
≤
p
+
<
∞,  μ-a.e., then one has g
(
φ
)
∈
D
1
,
p
(
x
)
(
H
,
μ
) and M
g
(
φ
)
=
g
′
(
φ
)
M
φ for any φ
∈
D
1
,
p
(
x
)
(
H
,
μ
) and g
∈
C
b
1
(
H
).Proof.
First, we prove thatg
(
ψ
)
∈
D
1
,
p
(
x
)
(
H
,
μ
) for any ψ
∈
ℰ
(
H
). By Lemma 16, for g
(
ψ
), there exists a two-index sequence {
ϕ
n
,
k
}
⊂
ℰ
(
H
) such that ∥
ϕ
n
,
k
∥
0
+
∥
D
ϕ
n
,
k
∥
0
≤
∥
g
(
ψ
)
∥
0
+
∥
D
g
(
ψ
)
∥
0, n, k
∈
ℕ, and
(40)
lim
⁡
n
→
∞
lim
⁡
k
→
∞
ϕ
n
,
k
(
x
)
=
g
(
ψ
)
(
x
)
,
x
∈
H
,
lim
⁡
n
→
∞
lim
⁡
k
→
∞
D
ϕ
n
,
k
(
x
)
=
D
g
(
ψ
)
(
x
)
,
x
∈
H
.
By diagonal rule and Lebesgue’s Dominated Convergence Theorem, ϕ
n
→
g
(
ψ
) in L
p
(
x
)
(
H
,
μ
). Since M is closed, we have M
g
(
ψ
)
=
lim
⁡
n
→
∞
⁡
M
0
ϕ
n and g
(
ψ
)
∈
D
1
,
p
(
x
)
(
H
,
μ
).
Now we will prove the proposition. Forφ
∈
D
1
,
p
(
x
)
(
H
,
μ
), there exists {
φ
n
}
⊂
ℰ
(
H
) such that
(41)
lim
⁡
n
→
∞
φ
n
=
φ
in
L
p
(
x
)
(
H
,
μ
)
.
By the first part of this proof, g
(
φ
n
)
∈
D
1
,
p
(
x
)
(
H
,
μ
). And
(42)
M
g
(
φ
n
)
=
Q
1
/
2
D
g
(
φ
n
)
=
g
′
(
φ
n
)
Q
1
/
2
D
φ
n
=
g
′
(
φ
n
)
M
φ
n
.
Since M
φ
n
∈
L
p
(
x
)
(
H
,
μ
;
H
) and g
′
(
φ
n
)
∈
C
b
(
H
), we have
(43)
lim
⁡
n
→
∞
M
g
(
φ
n
)
=
lim
⁡
n
→
∞
g
′
(
φ
n
)
M
φ
n
=
g
′
(
φ
)
M
φ
.
As M is closed, M
g
(
φ
)
=
g
′
(
φ
)
M
φ. The proof is completed.Proposition 18.
Assume thatp satisfies 1
≤
p
(
x
)
≤
p
+
<
∞,  μ-a.e. For φ, ψ
∈
D
1
,
p
(
x
)
(
H
,
μ
); suppose ψ and M
ψ are bounded; then φ
ψ
∈
D
1
,
p
(
x
)
(
H
,
μ
) and M
(
φ
ψ
)
=
ψ
M
φ
+
φ
M
ψ.Proof.
First, forφ
∈
ℰ
(
H
), as ψ
∈
D
1
,
p
(
x
)
(
H
,
μ
), there exists {
ψ
n
}
⊂
ℰ
(
H
) such that ψ
n
→
ψ in L
p
(
x
)
(
H
,
μ
). And since
(44)
M
0
(
φ
ψ
n
)
=
ψ
n
M
0
φ
+
φ
M
0
ψ
n
,
we have
(45)
lim
⁡
n
→
∞
M
0
(
φ
ψ
n
)
=
lim
⁡
n
→
∞
(
ψ
n
M
0
φ
+
φ
M
0
ψ
n
)
=
ψ
M
0
φ
+
φ
M
ψ
.
As φ
ψ
n
∈
ℰ
(
H
), φ
ψ
n
→
φ
ψ in L
p
(
x
)
(
H
,
μ
), and M
(
φ
ψ
)
=
lim
⁡
n
→
∞
⁡
M
0
(
φ
ψ
n
) in L
p
(
x
)
(
H
,
μ
;
H
), we have φ
ψ
∈
D
1
,
p
(
x
)
(
H
,
μ
).
Secondly, forφ
∈
D
1
,
p
(
x
)
(
H
,
μ
), there exists {
φ
n
}
⊂
ℰ
(
H
) such that φ
n
→
φ in L
p
(
x
)
(
H
,
μ
). By the first part of the proof,
(46)
M
(
φ
n
ψ
)
=
ψ
M
0
φ
n
+
φ
n
M
ψ
.
Since ψ and M
ψ are bounded, we have
(47)
M
(
φ
ψ
)
=
lim
⁡
n
→
∞
M
0
(
φ
n
ψ
)
=
ψ
M
φ
+
φ
M
ψ
,
and φ
ψ
∈
D
1
,
p
(
x
)
(
H
,
μ
). The proof is completed.


## 5. Conclusion

In this work some new results on Malliavin derivatives are given. Malliavin derivatives are extended fromD
1,2
(
H
,
μ
) to variable exponent spaces and some properties of Malliavin derivatives in variable exponent spaces are obtained. We introduce L
p
(
x
)
(
H
,
μ
) and L
p
(
x
)
(
H
,
μ
;
H
), and this work generalizes classical variable exponents spaces based on Kovác
˘ik and Rákosník [1]. Then, we prove that ℰ
(
H
) is dense in L
p
(
x
)
(
H
,
μ
) and by this approximation result we give extension of Malliavin derivatives which are given by P. Malliavin [11]. Some properties of Malliavin derivatives in D
1,2
(
H
,
μ
) are also discussed.



---

*Source: 198060-2014-04-03.xml*