# Local Characterizations of Besov and Triebel-Lizorkin Spaces with Variable Exponent

Authors: Baohua Dong; Jingshi Xu
Journal: Journal of Function Spaces
 (2014)

---

## Abstract

We introduce new Besov and Triebel-Lizorkin spaces with variable
integrable exponent, which are different from those introduced by the second author early.
Then we characterize these spaces by the boundedness of the local Hardy-Littlewood
maximal operator on variable exponent Lebesgue space. Finally the completeness and
the lifting property of these spaces are also given.


---

## Body

## 1. Introduction

Variable exponent function spaces have attracted many attentions because of their applications in some aspects, such as partial differential equations with nonstandard growth [1], electrorheological fluids [2], and image restoration [3–5]. In fact, since the variable Lebesgue and Sobolev spaces were systemically studied by Kováčik and Rákosník in [6], there are many spaces introduced, such as, Bessel potential spaces with variable exponent, Besov and Triebel-Lizorkin spaces with variable exponents, Morrey spaces with variable exponents, and Hardy spaces with variable exponent; see [7–20] and references therein. When the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue spaces, many results in classical harmonic analysis and function theory hold for the variable exponent case; see [21–23].Letp
(
·
)
:
ℝ
n
→
[
1
,
∞
) be a measurable function. Denote by L
p
(
·
)
(
ℝ
n
) the space of all measurable functions f on ℝ
n such that for some λ
>
0
(1)
∫
ℝ
n
‍
|
f
(
x
)
λ
|
p
(
x
)
d
x
<
∞
,
with the norm
(2)
∥
f
∥
L
p
(
·
)
(
ℝ
n
)
:
=
{
∫
ℝ
n
‍
|
f
(
x
)
λ
|
p
(
x
)
d
x
⩽
1
}
.
Then L
p
(
·
)
(
ℝ
n
) is a Banach space with the norm ∥
·
∥
L
p
(
·
)
(
ℝ
n
).We will use the following notations:p
-
:
=
ess
inf
⁡
{
p
(
x
)
:
x
∈
ℝ
n
} and p
+
:
=
ess
sup
⁡
{
p
(
x
)
:
x
∈
ℝ
n
}. The set 𝒫
(
ℝ
n
) consists of all p
(
·
) satisfying p
-
>
1 and p
+
<
∞. Moreover, we define 𝒫
0
(
ℝ
n
) to be the set of measurable functions p
(
·
) on ℝ
n with the range in (
0
,
∞
) such that 0
<
p
-
≤
p
+
<
∞. Given p
(
·
)
∈
𝒫
0
(
ℝ
n
), one can define the space L
p
(
·
)
(
ℝ
n
) as above. This is equivalent to defining it to be the set of all functions f such that |
f
|
p
0
∈
L
p
(
·
)
/
p
0
(
ℝ
n
), where 0
<
p
0
<
p
- and p
(
·
)
/
p
0
∈
𝒫
(
ℝ
n
). We also define a quasinorm on this space by ∥
f
∥
L
p
(
·
)
(
ℝ
n
)
:
=
∥
|
f
|
p
0
∥
L
p
(
·
)
/
p
0
(
ℝ
n
)
1
/
p
0.Letf be a locally integrable function on ℝ
n; the local variant of the Hardy-Littlewood maximal operator is given by
(3)
ℳ
loc
⁡
f
(
x
)
:
=
sup
⁡
Q
∋
x
,
|
Q
|
⩽
C
1
|
Q
|
∫
Q
‍
|
f
(
y
)
|
d
y
,
∀
x
∈
ℝ
n
,
for some constant C. We denote ℬ
loc
⁡
(
ℝ
n
) the set of p
(
·
)
∈
𝒫
(
ℝ
n
) such that ℳ
loc
⁡ is bounded on L
p
(
·
)
(
ℝ
n
). In 2013 Danelia et al. gave characterizations of ℬ
loc
⁡
(
ℝ
n
), a vector-estimate for the local Hardy-Littlewood maximal operator if p
(
·
)
∈
ℬ
loc
⁡
(
ℝ
n
), and a Littlewood-Paley square-function characterization of the variable exponent Lebesgue spaces L
p
(
·
)
(
ℝ
n
) when p
(
·
) belongs to ℬ
loc
⁡
(
ℝ
n
) in [24].In 2001 Rychkov used the boundedness of the local Hardy-Littlewood maximal operator to prove a stronger result of the Peetre type for spacesF
p
,
q
s
,
ω
(
ℝ
n
) and B
p
,
q
s
,
ω
(
ℝ
n
) and gave the lifting property for these spaces in [25].Motivated by the previous papers, the goal of this paper is to introduce new Besov and Triebel-Lizorkin spaces with variable exponent. To state our result, we need some notations.Throughout this paper|
S
| denotes the Lebesgue measure for a measurable set S
⊂
ℝ
n
.
ℕ
0 denotes the set of all nonnegative integers. Let 𝒟
:
=
𝒞
0
∞
(
ℝ
n
) and 𝒟
′ be the dual space of 𝒟. For s
∈
ℝ,s
+
:
=
max
⁡
{
s
,
0
} and [
s
] is the largest integer less than or equal to s.Given a functionφ on ℝ
n, let L
φ
∈
ℕ
0 denote the maximal number such that φ has vanishing moments up to order L
φ. In other words, ∫
ℝ
n
x
α
φ
(
x
)
d
x
=
0 for all multiindices α with |
α
|
⩽
L
φ. If no moments of φ vanish, then put L
φ
=
-
1. A pair of functions (
φ
0
,
φ
) is called satisfying the M
s condition, if ∫
ℝ
n
φ
0
(
x
)
d
x
≠
0 and L
φ
⩾
[
s
].Take a functionφ
0
∈
𝒟 satisfying M
s condition, which is possible for any s. (Indeed, the assumption L
φ
⩾
[
s
] is void for s
<
0 and is satisfied automatically for 0
⩽
s
<
1. For s
⩾
1, any φ
0
∈
𝒟 with Fourier transform φ
^
0
(
ξ
)
=
1
+
O
(
|
ξ
|
[
s
]
+
1
) near the origin will do the job.) The notation 𝒮
e
′
(
ℝ
n
) was introduced by Schott in [26]. More precisely, let it be the set of all f
∈
𝒟
′ for which the estimate
(4)
|
〈
f
,
γ
〉
|
⩽
A
sup
⁡
{
|
D
α
γ
(
x
)
|
exp
⁡
(
N
|
x
|
)
:
x
∈
ℝ
n
,
|
α
|
⩽
N
}
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
all
γ
∈
𝒟
,
is valid with some constants A
=
A
f, N
=
N
f. It is evidently that 𝒮
e
′
(
ℝ
n
) includes temperate distributions S
′
(
ℝ
n
).Now, we give the definition of Besov spaces and Triebel-Lizorkin spaces with variable exponent.Definition 1.
Letp
(
·
)
∈
𝒫
0
(
ℝ
n
),s
∈
ℝ, 0
<
q
≤
∞, φ
0 as above, φ
(
x
)
:
=
φ
0
(
x
)
-
2
-
n
φ
0
(
x
/
2
), and φ
j
(
x
)
:
=
2
j
n
φ
(
2
j
x
) for j
∈
ℕ.(i)
The Besov space with variable exponentB
p
(
·
)
s
,
q
(
ℝ
n
) is the set of f with
(5)
{
f
∈
𝒮
e
′
(
ℝ
n
)
:
∥
f
∥
B
p
(
·
)
s
,
q
(
ℝ
n
)
<
∞
}
,
where
(6)
∥
f
∥
B
p
(
·
)
s
,
q
(
ℝ
n
)
:
=
(
∑
j
=
0
∞
‍
2
j
s
q
∥
φ
j
*
f
∥
L
p
(
·
)
(
ℝ
n
)
q
)
1
/
q
.
(ii)
Forp
+
<
∞, the Triebel-Lizorkin space with variable exponent F
p
(
·
)
s
,
q
(
ℝ
n
) is the set of f with
(7)
{
f
∈
𝒮
e
′
(
ℝ
n
)
:
∥
f
∥
F
p
(
·
)
s
,
q
(
ℝ
n
)
<
∞
}
,
where
(8)
∥
f
∥
F
p
(
·
)
s
,
q
(
ℝ
n
)
:
=
∥
(
∑
j
=
0
∞
‍
2
j
s
q
|
φ
j
*
f
(
·
)
|
q
)
1
/
q
∥
L
p
(
·
)
(
ℝ
n
)
.The key point is to prove that different choices ofφ
0 in Definition 1 do not really change the spaces, leading to equivalent quasinorms. For f
∈
S
′
(
ℝ
n
) that has been proved by the second author in 2008, see [19]. To go on, we recall variant Peetre-type maximal functions which was introduced by Rychkov in [25]. Let
(9)
φ
j
,
A
,
B
*
f
(
x
)
:
=
sup
⁡
y
∈
ℝ
n
|
φ
j
*
f
(
x
-
y
)
|
m
j
,
A
,
B
(
y
)
,
x
∈
ℝ
n
,
j
∈
ℕ
0
,
where
(10)
m
j
,
A
,
B
(
y
)
:
=
(
1
+
2
j
|
y
|
)
A
2
|
y
|
B
,
A
,
B
⩾
0
.Now it is the position to state our main result.Theorem 2.
Lets
∈
ℝ, 0
<
q
<
∞, and p
(
·
)
∈
𝒫
0
(
ℝ
n
) with 0
<
p
0
<
min
⁡
{
p
-
,
q
} such that p
(
·
)
/
p
0
∈
ℬ
loc
⁡
(
ℝ
n
). Suppose that φ
0, ζ
0
∈
𝒟 and the pairs (
φ
0
,
φ
:
=
φ
0
-
2
-
n
φ
0
(
·
/
2
)
), and (
ζ
0
,
ζ
:
=
ζ
0
-
2
-
n
ζ
0
(
·
/
2
)
) satisfy the M
s condition. Then there are positive constants A
0
:
=
A
0
(
s
,
q
), B
0
:
=
B
0
(
n
), and C such that for each A
⩾
A
0, B
>
B
0
/
p
-, and all f
∈
𝒮
e
′
(
ℝ
n
) one has
(11)
∥
(
∑
j
=
0
∞
‍
2
j
s
q
|
ζ
j
,
A
,
B
*
f
(
·
)
|
q
)
1
/
q
∥
L
p
(
·
)
(
ℝ
n
)
⩽
C
∥
f
∥
F
p
(
·
)
s
,
q
(
ℝ
n
)
,
(12)
(
∑
j
=
0
∞
‍
2
j
s
q
∥
ζ
j
,
A
,
B
*
f
∥
L
p
(
·
)
(
ℝ
n
)
q
)
1
/
q
⩽
C
∥
f
∥
B
p
(
·
)
s
,
q
(
ℝ
n
)
.
Since φ
j
,
A
,
B
*
⩾
|
φ
j
*
f
| for any φ
0, one immediately gets a consequence of Theorem 2.Corollary 3.
The spacesF
p
(
·
)
s
,
q
(
ℝ
n
) and F
p
(
·
)
s
,
q
(
ℝ
n
) with s
∈
ℝ, 0
<
q
<
∞, and p
(
·
)
∈
𝒫
0
(
ℝ
n
) with 0
<
p
0
<
min
⁡
{
p
-
,
q
} such that p
(
·
)
/
p
0
∈
ℬ
loc
⁡
(
ℝ
n
) are independent of the particular choice of the function φ
0 in Definition 1. The quasinorms arising for different φ
0 are equivalent.The proof of Theorem2 will be given in Section 2. In Section 3 we study the completeness and the lifting property of these spaces by using Theorem 2. We will use the notation a
≲
b if there exists a constant C
>
0 such that a
⩽
C
b. If a
≲
b and b
≲
a we will write a
≈
b. Finally we claim that C is always a positive constant but it may change from line to line. Other notations will be explained when we meet them.


## 2. Proof of Theorem2

We will use the idea of [25] by Rychkov to prove Theorem 2. First we need some lemmas.Lemma 4 (see [25, Theorem 1.6]).
Let a functionφ
0
∈
𝒟 have nonzero integral, and let φ
(
x
)
=
φ
0
(
x
)
-
2
-
n
φ
0
(
x
/
2
). Then for any N
⩾
0 there exist two functions ψ
0, ψ
∈
𝒟, such that ψ has vanishing moments up to order N and
(13)
f
=
∑
j
=
0
∞
‍
ψ
j
*
φ
j
*
f
,
∀
f
∈
𝒟
′
,
where ψ
j
(
x
)
:
=
2
j
n
ψ
j
(
2
j
x
) and φ
j
(
x
)
:
=
2
j
n
φ
(
2
j
x
) for j
∈
ℕ.Before the next lemma we denote a special convolution operator which is given by(14)
K
B
f
(
x
)
:
=
∫
ℝ
n
‍
|
f
(
y
)
|
2
-
B
|
x
-
y
|
d
y
(
B
⩾
0
)
.Lemma 5 (see [25, Lemma 2.10]).
Let0
<
r
<
∞, φ
0
∈
𝒟, ∫
ℝ
n
‍
φ
0
d
x
≠
0, and φ
=
φ
0
-
2
-
n
φ
0
(
·
/
2
) and A
>
n
/
r, B
⩾
0. Then there is a constant C depending only on n, r, φ
0, A, B such that for all f
∈
𝒮
e
′
(
ℝ
n
) and each x
∈
ℝ
n, j
∈
ℕ
0, one has
(15)
φ
j
,
A
,
B
*
f
(
x
)
r
⩽
C
∑
k
=
j
∞
‍
2
(
j
-
k
)
(
A
r
-
n
)
d
d
d
d
d
×
{
ℳ
loc
⁡
(
|
φ
k
*
f
|
r
)
(
x
)
+
K
B
r
(
|
φ
k
*
f
|
r
)
(
x
)
}
.Lemma 6 (see [24, Corollary 3.2]).
Letp
(
·
)
∈
ℬ
loc
⁡
(
ℝ
n
) and 1
<
q
<
∞; then there exists a positive constant C such that for all sequences {
f
j
}
j
=
0
∞ of locally integrable functions on ℝ
n
(16)
∥
(
∑
j
=
0
∞
‍
(
ℳ
loc
⁡
f
j
)
q
)
1
/
q
∥
L
p
(
·
)
(
ℝ
n
)
⩽
C
∥
(
∑
j
=
0
∞
‍
|
f
j
|
q
)
1
/
q
∥
L
p
(
·
)
(
ℝ
n
)
.Lemma 7 (see [23, Lemma 2.1
.
14]).
LetX be a real or complex vector space and ϱ be a semimodular on X. Then ∥
x
∥
ϱ
⩽
1 and ϱ
(
x
)
⩽
1 are equivalent. If ϱ is continuous, then also ∥
x
∥
ϱ
<
1 and ϱ
(
x
)
<
1 are equivalent, as are ∥
x
∥
ϱ
=
1 and ϱ
(
x
)
=
1.Lemma 8.
Letp
(
·
)
∈
ℬ
loc
⁡
(
ℝ
n
) and 1
<
q
<
∞. Then there exists a positive constant B
0
=
B
0
(
n
)
>
0 such that for B
⩾
B
0
/
p
-
(17)
∥
(
∑
i
=
0
∞
‍
(
K
B
f
i
)
q
)
1
/
q
∥
L
p
(
·
)
(
ℝ
n
)
⩽
C
∥
(
∑
i
=
0
∞
‍
|
f
i
|
q
)
1
/
q
∥
L
p
(
·
)
(
ℝ
n
)
,
where C is a positive constant and {
f
i
}
i
=
0
∞ are locally integrable functions on ℝ
n.Proof.
By homogeneity, it suffices to consider the case(18)
∥
(
∑
i
=
0
∞
‍
|
f
i
|
q
)
1
/
q
∥
L
p
(
·
)
(
ℝ
n
)
⩽
1
.
Let ℝ
n
=
⋃
Q
∈
ℐ
1
‍
Q, where ℐ
1 is the set of all unit dyadic cubes in ℝ
n. Then it is easy to get
(19)
K
B
f
(
x
)
=
∫
ℝ
n
‍
|
f
(
y
)
|
2
-
B
|
x
-
y
|
d
y
≲
∑
ℐ
1
∋
Q
‍
2
-
B
dist
⁡
(
x
,
Q
)
∫
Q
‍
|
f
(
y
)
|
d
y
.
Since p
-
>
1, by Minkowski’s inequality and Hölder’s inequality,
(20)
(
∑
i
=
0
∞
K
B
f
i
(
x
)
q
‍
)
1
/
q
≲
[
∑
i
=
0
∞
(
∑
ℐ
1
∋
Q
2
-
B
dist
⁡
(
x
,
Q
)
‍
∫
Q
|
f
i
(
y
)
|
d
y
‍
)
q
]
1
/
q
≲
∑
ℐ
1
∋
Q
2
-
B
dist
⁡
(
x
,
Q
)
‍
[
∑
i
=
0
∞
(
∫
Q
|
f
i
(
y
)
|
d
y
‍
)
q
]
1
/
q
≲
{
∑
ℐ
1
∋
Q
2
-
B
dist
⁡
(
x
,
Q
)
p
(
x
)
/
2
ddddddddd
×
[
∑
i
=
0
∞
(
∫
Q
|
f
i
(
y
)
|
d
y
‍
)
q
]
p
(
x
)
/
q
}
1
/
p
(
x
)
×
(
∑
ℐ
1
∋
Q
2
-
B
dist
⁡
(
x
,
Q
)
p
′
(
x
)
/
2
)
1
/
p
′
(
x
)
.
Since B
>
0 and p
-
>
1, the latter factor is uniformly bounded in x. We take the p
(
x
)th power of the above inequality and integrate it. We get
(21)
∫
ℝ
n
(
∑
i
=
0
∞
K
B
f
i
(
x
)
q
)
p
(
x
)
/
q
d
x
≲
∑
ℐ
1
∋
Q
∫
ℝ
n
2
-
B
dist
⁡
(
x
,
Q
)
p
(
x
)
/
2
d
x
‍
×
[
∑
i
=
0
∞
(
∫
Q
|
f
i
(
y
)
|
d
y
‍
)
q
‍
]
p
(
x
)
/
q
.
It is easy to know that
(22)
∫
ℝ
n
2
-
B
dist
⁡
(
x
,
Q
)
p
(
x
)
/
2
d
x
≲
∫
Q
d
x
for p
-
>
1 and also that
(23)
∫
Q
‍
|
f
i
(
y
)
|
d
y
⩽
ℳ
loc
⁡
f
i
(
x
)
,
∀
x
∈
Q
.
By these two observations and (21), we have
(24)
∫
ℝ
n
(
∑
i
=
0
∞
‍
K
B
f
i
(
x
)
q
)
p
(
x
)
/
q
d
x
≲
∫
ℝ
n
[
∑
i
=
0
∞
‍
(
ℳ
loc
⁡
f
i
(
x
)
)
q
]
p
(
x
)
/
q
d
x
.
Applying Lemmas 6 and 7 we obtain
(25)
∫
ℝ
n
(
∑
i
=
0
∞
‍
K
B
f
i
(
x
)
q
)
p
(
x
)
/
q
d
x
≲
1
.
Using Lemma 7 again we obtain
(26)
∥
(
∑
i
=
0
∞
‍
(
K
B
f
i
)
q
)
1
/
q
∥
L
p
(
·
)
(
ℝ
n
)
≲
1
.
Thus we have
(27)
∥
(
∑
i
=
0
∞
‍
(
K
B
f
i
)
q
)
1
/
q
∥
L
p
(
·
)
(
ℝ
n
)
≲
∥
(
∑
i
=
0
∞
‍
|
f
i
|
q
)
1
/
q
∥
L
p
(
·
)
(
ℝ
n
)
.
This finishes the proof.We give a notation of norm inL
p
(
·
)
(
ℓ
q
) which will be used in the following context:
(28)
∥
{
g
j
}
j
=
0
∞
∥
L
p
(
·
)
(
ℓ
q
)
:
=
∥
(
∑
j
=
0
∞
‍
|
g
j
|
q
)
1
/
q
∥
L
p
(
·
)
(
ℝ
n
)
.Lemma 9 (see [27, Lemma 2]).
Let0
<
q
≤
∞, δ
>
0. For any sequence {
g
j
}
0
∞ of nonnegative numbers denote
(29)
G
j
=
∑
k
=
0
∞
‍
2
-
|
k
-
j
|
δ
g
k
.
Then
(30)
∥
{
G
j
}
0
∞
∥
ℓ
q
≤
C
∥
{
g
j
}
0
∞
∥
ℓ
q
holds, where C is constant and only depends on q, δ.Lemma 10 (see [19, Lemma 3]).
Let0
<
q
≤
∞, δ
>
0, and p
(
·
)
∈
P
0
(
ℝ
n
). For any sequence {
g
j
}
0
∞ of nonnegative measurable functions on ℝ
n, denote
(31)
G
j
(
x
)
=
∑
k
=
0
∞
‍
2
-
|
k
-
j
|
δ
g
k
(
x
)
,
x
∈
ℝ
n
.
Then
(32)
∥
{
G
j
}
0
∞
∥
L
p
(
·
)
(
ℓ
q
)
≤
C
1
∥
{
g
j
}
0
∞
∥
L
p
(
·
)
(
ℓ
q
)
,
∥
{
G
j
}
0
∞
∥
ℓ
q
(
L
p
(
·
)
)
≤
C
2
∥
{
g
j
}
0
∞
∥
ℓ
q
(
L
p
(
·
)
)
hold with some constants C
1
=
C
1
(
q
,
δ
) and C
2
=
C
2
(
p
(
·
)
,
q
,
δ
).Proof of Theorem2.
By Lemma4, take ψ
0, ψ
∈
𝒟, with large enough L
ψ so that (13) is true. It follows that
(33)
|
ζ
j
*
f
(
x
-
y
)
|
⩽
∑
k
=
0
∞
∫
ℝ
n
|
ζ
j
*
ψ
k
(
z
)
|
·
|
φ
k
*
f
(
x
-
y
-
z
)
|
d
z
‍
‍
⩽
∑
k
=
0
∞
I
j
k
sup
⁡
z
∈
ℝ
n
⁡
‍
|
φ
k
*
f
(
x
-
y
-
z
)
|
m
k
,
A
,
B
(
z
)
,
where
(34)
I
j
k
:
=
∫
ℝ
n
|
ζ
j
*
ψ
k
(
z
)
|
m
k
,
A
,
B
(
z
)
d
z
.
Because of the elementary inequality
(35)
m
j
,
A
,
B
(
z
)
⩽
m
j
,
A
,
B
(
y
)
m
j
,
A
,
B
(
z
-
y
)
,
m
j
,
A
,
B
(
y
)
⩾
{
m
k
,
A
,
B
(
y
)
,
j
⩾
k
,
2
(
j
-
k
)
A
m
k
,
A
,
B
(
y
)
,
j
<
k
we have the following fact:
(36)
ζ
j
,
A
,
B
*
f
(
x
)
⩽
{
∑
k
=
0
∞
‍
I
j
k
φ
k
,
A
,
B
*
f
(
x
)
,
if
j
⩾
k
∑
k
=
0
∞
‍
I
j
k
2
(
k
-
j
)
A
φ
k
,
A
,
B
*
f
(
x
)
,
if
j
<
k
.
To estimate I
j
k, note that
(37)
∥
ζ
j
*
ψ
k
∥
∞
≲
{
2
(
k
-
j
)
(
L
ζ
+
1
)
2
k
n
,
if
j
⩾
k
,
2
(
j
-
k
)
(
L
ψ
+
1
)
2
j
n
,
if
j
<
k
,
which follows easily from the moment conditions on ζ and ψ. Furthermore, ζ
j
*
ψ
k
(
z
) is supported in the ball {
|
z
|
≲
max
⁡
(
2
-
j
,
2
-
k
)
}, in which
(38)
m
k
,
A
,
B
(
z
)
≲
{
1
,
if
j
⩾
k
2
(
k
-
j
)
A
,
if
j
<
k
.
By the last two estimates,
(39)
I
j
k
≲
{
2
(
k
-
j
)
(
L
ζ
+
1
)
,
if
j
⩾
k
2
(
j
-
k
)
(
L
ψ
+
1
-
A
)
,
if
j
<
k
.
We put this estimate in (36) and see that if we choose L
ψ
>
2
A
-
[
s
] and take into account L
ζ
⩾
[
s
], then we arrive at
(40)
2
j
s
ζ
j
,
A
,
B
*
f
(
x
)
≲
∑
k
=
0
∞
‍
2
-
ɛ
|
k
-
j
|
2
k
s
φ
k
,
A
,
B
*
f
(
x
)
with some ɛ
>
0. It is easy to see that, in the right side of (40), we have essentially the convolution with the sequence {
2
-
ɛ
k
}, which is of course a bounded operator on any ℓ
q, 0
<
q
<
∞. Now by Lemmas 9 and 10 for ℓ
q and L
p
(
·
)
(
ℝ
n
) we easily obtain
(41)
∥
(
∑
j
=
0
∞
‍
2
j
s
q
|
ζ
j
,
A
,
B
*
f
(
·
)
|
q
)
1
/
q
∥
L
p
(
·
)
(
ℝ
n
)
≲
∥
(
∑
k
=
0
∞
‍
2
k
s
φ
k
,
A
,
B
*
f
(
·
)
q
)
1
/
q
∥
L
p
(
·
)
(
ℝ
n
)
,
(
∑
j
=
0
∞
‍
2
j
s
q
∥
ζ
j
,
A
,
B
*
f
∥
L
p
(
·
)
(
ℝ
n
)
q
)
1
/
q
≲
(
∑
k
=
0
∞
‍
2
k
s
q
∥
φ
j
,
A
,
B
*
f
∥
L
p
(
·
)
(
ℝ
n
)
q
)
1
/
q
.
In other words, we reduce matters to prove (11) and (12) with ζ
0
=
φ
0, ζ
=
φ. Below we do it only for (11); the argument for (12) is similar.
Let0
<
r
<
q and A
>
A
0
:
=
n
/
r
+
max
⁡
{
-
s
,
0
}. By Lemma 5 and a discrete version of the Hardy inequality
(42)
∑
j
=
0
∞
‍
2
j
θ
(
∑
k
=
j
∞
‍
|
b
k
|
)
τ
≲
∑
j
=
0
∞
‍
2
j
θ
|
b
k
|
τ
(
θ
,
τ
>
0
)
,
which we apply with θ
:
=
q
(
A
+
s
-
n
/
r
) and τ
=
q
/
r, we have
(43)
(
∑
j
=
0
∞
‍
2
j
s
q
(
φ
j
,
A
,
B
*
f
)
q
)
1
/
q
≲
(
∑
j
=
0
∞
‍
2
j
s
q
{
ℳ
loc
⁡
(
|
φ
j
*
f
|
r
)
(
x
)
ddddddddddd
+
K
B
r
(
|
φ
j
*
f
|
r
)
(
x
)
(
|
φ
j
*
f
|
r
)
}
q
/
r
∑
j
=
0
∞
‍
)
1
/
q
.
Note that B
r
·
p
-
/
r
⩾
B
0 and 1
<
q
/
r
<
∞. Let r
:
=
p
0; by Lemmas 6 and 8, the operators f
↦
{
ℳ
loc
⁡
(
|
f
|
r
)
}
1
/
r and f
↦
{
K
B
r
(
|
f
|
r
)
}
1
/
r are all bounded on L
p
(
·
)
(
ℓ
q
). Hence the desired estimate (11) with ζ
0
=
φ
0, ζ
=
φ, follows.
This finishes the proof.


## 3. Some Applications

In this section, we will consider the completeness, the lifting property, and the related quasinorms of these spaces introduced in previous section.Theorem 11.
Lets
∈
ℝ, 0
<
q
<
∞, and p
(
·
)
∈
𝒫
0
(
ℝ
n
) with 0
<
p
0
<
min
⁡
{
p
-
,
q
} such that p
(
·
)
/
p
0
∈
ℬ
loc
⁡
(
ℝ
n
). Then the quasinormed spaces B
p
(
·
)
s
,
q
(
ℝ
n
) and F
p
(
·
)
s
,
q
(
ℝ
n
) are quasi-Banach spaces.Proof.
We only give the proof forF
p
(
·
)
s
,
q
(
ℝ
n
) and for B
p
(
·
)
s
,
q
(
ℝ
n
); it can be proved by the similar way. We use the similar argument in [25]. Let f
∈
𝒮
e
′
(
ℝ
n
) and γ
∈
𝒟 with supp γ
⊂
B
(
0,1
). We set ζ
0
=
γ and j
=
0 in the left side of (11). Analyzing the proof of Theorem 2 shows that only finite numbers of derivatives of the kernels are involved in the estimates, and therefore we know
(44)
∥
sup
⁡
y
∈
ℝ
n
|
f
*
γ
(
·
-
y
)
|
(
1
+
|
y
|
)
A
2
|
y
|
B
∥
L
p
(
·
)
(
ℝ
n
)
≲
∥
f
∥
F
p
(
·
)
s
,
q
(
ℝ
n
)
sup
⁡
|
α
|
⩽
L
|
D
α
γ
|
,
where L is a constant and depends on p
-, q, s, n, but not on f and γ.
It is easy to know(45)
sup
⁡
y
∈
ℝ
n
⁡
|
f
*
γ
(
x
-
y
)
|
(
1
+
|
y
|
)
A
2
|
y
|
B
=
sup
⁡
u
∈
ℝ
n
⁡
|
f
*
γ
(
u
)
|
(
1
+
|
x
-
u
|
)
A
2
|
x
-
u
|
B
⩾
sup
⁡
|
u
-
x
|
⩽
1
⁡
|
f
*
γ
(
u
)
|
(
1
+
|
u
-
x
|
)
A
2
|
x
-
u
|
B
.
We take∥
·
∥
L
p
(
·
)
(
ℝ
n
) on both sides of the last inequality and get
(46)
∥
sup
⁡
y
∈
ℝ
n
⁡
|
f
*
γ
(
·
-
y
)
|
(
1
+
|
y
|
)
A
2
|
y
|
B
∥
L
p
(
·
)
(
ℝ
n
)
=
∥
sup
⁡
u
∈
ℝ
n
⁡
|
f
*
γ
(
u
)
|
(
1
+
|
·
-
u
|
)
A
2
|
·
-
u
|
B
∥
L
p
(
·
)
(
ℝ
n
)
⩾
∥
sup
⁡
|
·
-
u
|
⩽
1
⁡
|
f
*
γ
(
u
)
|
(
1
+
|
·
-
u
|
)
A
2
|
·
-
u
|
B
∥
L
p
(
·
)
(
ℝ
n
)
⩾
|
f
*
γ
(
u
)
|
∥
sup
⁡
|
v
|
⩽
1
⁡
1
(
1
+
|
v
|
)
A
2
|
v
|
B
·
χ
B
(
u
,
1
)
∥
L
p
(
·
)
(
ℝ
n
)
≳
|
f
*
γ
(
u
)
|
.
By (44) and (46) we have
(47)
|
f
*
γ
(
x
)
|
≲
∥
f
∥
F
p
(
·
)
s
,
q
(
ℝ
n
)
(
sup
⁡
|
α
|
⩽
L
|
D
α
γ
|
)
exp
⁡
(
N
|
x
|
)
,
where N, L are constants and depend on p
-, q, s, n, but not on f and γ.
Then we know that the following estimate(48)
|
f
*
γ
(
x
)
|
≲
∥
f
∥
F
p
(
·
)
s
,
q
(
ℝ
n
)
sup
⁡
⁡
{
|
D
α
γ
(
x
)
|
exp
⁡
⁡
(
N
|
x
|
)
:
x
∈
ℝ
n
,
|
α
|
⩽
L
}
is valid for all f
∈
𝒮
e
′
(
ℝ
n
) and γ
∈
𝒟 with some constants N, L which may depend on p
-, q, s, n, but not on f and γ. Thus we obtain that F
p
(
·
)
s
,
q
(
ℝ
n
) is continuously embedded in 𝒮
e
′
(
ℝ
n
).
Now we conclude the proof of the theorem in a normal way. If a sequence of distributions{
f
j
} is Cauchy sequence in F
p
(
·
)
s
,
q
(
ℝ
n
), then by (48) it converges “pointwise.” By the completeness of 𝒟
′, the sequence has a limit f in 𝒟
′. Again by (48), we have f
∈
𝒮
e
′
(
ℝ
n
), since Cauchy sequences are bounded. Finally, by Lebesgue’s theorem on dominated convergence it is easily seen that f
j
→
f in F
p
(
·
)
s
,
q
(
ℝ
n
).
This finishes the proof.In next context, we study the action of the Bessel potential operators in our Besov and Triebel-Lizorkin spaces with variable exponent. More precisely, we consider the followingt-dilated version:
(49)
𝒯
t
a
=
(
id
-
t
2
Δ
)
-
a
/
2
,
a
∈
ℝ
,
t
>
0
,
where id denotes the identity operator.Forf
∈
S
(
ℝ
n
) this operator acts by the rule 𝒯
t
a
f
=
(
K
a
)
t
*
f, where
(50)
K
a
=
ℱ
-
1
{
(
1
+
4
π
2
|
x
|
2
)
-
a
/
2
}
≔
formally
∫
e
2
π
i
〈
x
,
·
〉
(
1
+
4
π
2
|
x
|
2
)
-
a
/
2
d
x
∈
S
′
(
ℝ
n
)
.
It is well known that if a
>
0, then K
a
∈
L
1
(
ℝ
n
) and has the representation
(51)
K
a
(
x
)
≈
∫
0
∞
‍
u
a
-
n
e
-
π
|
x
|
2
/
u
2
e
-
u
2
/
4
π
d
u
u
(see Stein’s book [28] for these matters), from which it follows rather easily that K
a with a
>
0 is 𝒞
∞ away from the origin and
(52)
|
D
α
K
a
(
x
)
|
≲
e
-
E
|
x
|
,
|
x
|
⩾
1
with an absolute constant E
>
0. By the identity
(53)
K
a
=
(
id
-
Δ
)
N
K
a
+
2
N
,
N
∈
ℕ
,
we see that for a
⩽
0 the distribution K
a agrees in ℝ
n
∖
{
0
} with a 𝒞
∞ function, which again satisfies (52).By the same argument in page 170 of [25] we know that the convolution K
t
a
*
f can be defined as an element of 𝒮
e
′
(
ℝ
n
) for any f
∈
B
p
(
·
)
s
,
q
(
ℝ
n
)
∪
F
p
(
·
)
s
,
q
(
ℝ
n
), provided that t
⩽
t
0
(
n
)
·
p
-. The next theorem states explicitly where it acts.Theorem 12 (the lifting property).
Letp
(
·
)
∈
𝒫
0
(
ℝ
n
) with 0
<
p
0
<
min
⁡
{
p
-
,
q
} such that p
(
·
)
/
p
0
∈
ℬ
loc
⁡
(
ℝ
n
). Then there is a constant t
0
=
t
0
(
n
)
>
0 so that for all 0
<
q
<
∞, s
∈
ℝ, and every positive t
<
t
0
·
p
- one has
(54)
𝒯
t
a
:
B
p
(
·
)
s
,
q
(
ℝ
n
)
→
B
p
(
·
)
s
+
a
,
q
(
ℝ
n
)
isomorphically
,
𝒯
t
a
:
F
p
(
·
)
s
,
q
(
ℝ
n
)
→
F
p
(
·
)
s
+
a
,
q
(
ℝ
n
)
isomorphically
.Proof.
The idea of the proof comes from [25]. We use again (13) with φ and ψ having vanishing moments up to large order L. By an argument similar to that one used above to define 𝒯
t
a on Besov and Triebel-Lizorkin spaces with variable exponent, one can establish the identity
(55)
φ
l
*
(
K
t
a
*
f
)
=
∑
j
=
0
∞
‍
K
t
a
*
φ
l
*
ψ
j
*
(
φ
j
*
f
)
,
l
∈
ℕ
0
.
From [25] by choosing L sufficiently large, we have
(56)
2
l
(
s
+
a
)
|
φ
l
*
(
K
t
a
*
f
)
(
x
)
|
⩽
∑
j
=
0
∞
‍
2
-
ɛ
|
l
-
j
|
2
j
s
φ
j
,
A
,
B
*
f
(
x
)
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
ɛ
>
0
,
t
<
E
2
B
ln
⁡
2
.
Now by using Theorem 2 with A
=
A
0 and B
=
B
0
/
p
-, it follows easily that if f belongs to B
p
(
·
)
s
,
q
(
ℝ
n
) or F
p
(
·
)
s
,
q
(
ℝ
n
), then K
t
a
*
f is in B
p
(
·
)
s
+
a
,
q
(
ℝ
n
) or F
p
(
·
)
s
+
a
,
q
(
ℝ
n
), respectively. Then the condition on t becomes t
<
t
0
·
p
- with t
0
=
E
/
2
B
0
ln
⁡
2.
The fact that the maps in (54) are actually onto follows from the identity 𝒯
t
-
a
𝒯
t
a
=
id.
This finishes the proof.It follows from Theorem12 that ∥
𝒯
t
-
s
f
∥
F
p
(
·
)
0
,
q
(
ℝ
n
) is an equivalent quasinorm on F
p
(
·
)
s
,
q
(
ℝ
n
) for small t
>
0 and analogously for B
p
(
·
)
s
,
q
(
ℝ
n
). The next theorem gives a version of this result for s
∈
ℕ involving “pure” derivatives.Theorem 13.
Lets
∈
ℕ, 0
<
q
<
∞, and p
(
·
)
∈
𝒫
0
(
ℝ
n
) with 0
<
p
0
<
min
⁡
{
p
-
,
q
} such that p
(
·
)
/
p
0
∈
ℬ
loc
⁡
(
ℝ
n
). Then for any |
α
|
⩽
s
(57)
∥
f
∥
F
p
(
·
)
s
,
q
≈
∑
|
α
|
⩽
s
‍
∥
D
α
f
∥
F
p
(
·
)
0
,
q
,
∀
f
∈
𝒮
e
′
,
∥
f
∥
B
p
(
·
)
s
,
q
≈
∑
|
α
|
⩽
s
‍
∥
D
α
f
∥
B
p
(
·
)
0
,
q
,
∀
f
∈
𝒮
e
′
.Proof.
For brevity, we only give the outline of the proof for Triebel-Lizorkin space with variable exponent; for Besov space with variable exponent it can be proved by similar way. The “⩾” inequality follows immediately from Definition 1 by partial integration and invoking Theorem 2.
The “⩽” inequality for s even follows from Theorem 12. To obtain it for s odd, it suffices to consider the case s
=
1. In view of Theorem 12, it is sufficient to prove the estimate
(58)
∥
K
t
1
*
f
∥
F
p
(
·
)
0
,
q
≲
∑
|
α
|
⩽
1
‍
∥
D
α
f
∥
F
p
(
·
)
0
,
q
for
small
t
.
From [25] again, we have
(59)
φ
j
*
f
=
2
-
j
∑
ν
=
1
n
‍
(
φ
ν
)
j
*
∂
f
∂
x
ν
.
All this leads to the following counterpart of (56):
(60)
|
φ
l
*
(
K
t
a
*
f
)
(
x
)
|
≲
2
-
ɛ
l
φ
0
,
A
,
B
*
f
(
x
)
+
∑
j
=
1
∞
2
-
ɛ
|
l
-
j
|
∑
ν
=
1
n
‍
(
φ
ν
)
j
,
A
,
B
*
(
∂
f
∂
x
ν
)
(
x
)
,
ddddddddddddddddddddddd
ɛ
>
0
,
where, for ν
=
1
,
…
,
n, φ
ν
∈
𝒟 has vanishing moments up to order L
-
1 and satisfy
(61)
φ
=
∑
ν
=
1
n
‍
(
∂
∂
x
ν
)
φ
ν
.
From (58), it is easily deduced by virtue of Theorem 12.
This finishes the proof.



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*Source: 417341-2014-01-22.xml*