#### Abstract

We will prove the duality and reflexivity of variable exponent Triebel-Lizorkin and Besov spaces. It was shown by many authors that variable exponent Triebel-Lizorkin spaces coincide with variable exponent Bessel potential spaces, Sobolev spaces, and Lebesgue spaces when appropriate indices are chosen. In consequence of the results, these variable exponent function spaces are shown to be reflexive.

#### 1. Introduction

Recently, variable exponent function spaces have been studied by many authors [1โ13], and in particular, the papers about the variable exponent Triebel-Lizorkin and Besov spaces have been published in [14โ18]. Diening et al. [15] and Almeida and Hรคstรถ [14] studied the spaces and and Xu studied [16โ18] the spaces and .

In this paper, we will show the duality of variable exponent Triebel-Lizorkin spaces and Besov spaces under suitable conditions by using the same arguments in Triebel [19]. The duality follows from the fact that and are reflexive under same conditions.

Xu [16] showed that variable exponent Bessel potential space coincides with if and . Diening et al. [15] showed that the variable exponent Lebesgue space coincides with under suitable assumptions on . Gurka et al. [7] showed that coincides with variable exponent Sobolev spaces if and . In consequences of these results, we have the duality and reflexivity of , although the duality and reflexivity of have been obtained in Kovรกฤik and Rรกkosnรญk [9] under the assumptions on which are weaker than ours.

#### 2. Definitions of Variable Exponent Function Spaces

We first introduce variable exponent Lebesgue spaces. Let be a measurable function on with range in . Let denote the set of all complex-valued functions on such that, for some , The set becomes a Banach function space when it is equipped with the Luxemburg-Nakano norm: If is a constant function, then the above norm coincides with the usual -norm and so the notation is not confusional. In Kovรกฤik and Rรกkosnรญk [9], variable exponent Lebesgue spaces are defined on arbitrary measurable subset of . Denote by the set of measurable functions on with range in such that For a complex-valued locally Lebesgue-integrable function on , let denote the Hardy-Littlewood maximal function, where the supremum is taken over all balls centered at . There exists such that the Hardy-Littlewood maximal operator is not bounded on [11], although the operator is bounded on for . Some sufficient conditions on for maximal operator to be bounded on are known. Let be the set of such that the Hardy-Littlewood maximal operator is bounded on .

Let be the Schwartz space of all complex-valued rapidly decreasing and infinitely differentiable functions on . Let be the set of all the tempered distribution on . For , let denote the Fourier transform of and the inverse Fourier transform of . We write for the sake of simplicity.

Let and . The variable exponent Bessel potential space is the collection of such that the norm:

Let and . The variable exponent Sobolev space is the collection of such that the derivatives (in the sense of distribution) up to the order belong to and the norm: where is a multi-index and .

Let be a function such that and on , where is an open ball centered at 0 with radius . Set for and . We also set It follows that where .

*Definition 2.1. *Let , , and . Let , as above. The variable exponent Triebel-Lizorkin space is the collection of such that
Let , , and . The variable exponent Besov space is the collection of such that

Here, and are the spaces of all sequences of measurable functions on such that quasi-norms: are finite, respectively.

*Definition 2.2. *Let and . For a sequence of compact subsets of , denotes the space of all sequences of such that
and . For a compact subset of , denotes the space of all elements such that
and .

#### 3. Preliminaries

We need the following fundamental properties of .

Theorem 3.1 (see [5, Theorem 8.1]). *Let . Then, the following conditions are equivalent:**
(a),**
(b) for some ,**
(c), where
*

In [1, 5], some other conditions equivalent to the above are given. Let and . Then we write If , then, we write .

*Remark 3.2. *Let and as in Theorem 3.1. Then, for any , we have . Furthermore, for any , we have also by Jensen inequality. The proofs are found in [5]. Consequently, if , then, for any . For any , we have also .

The next theorem gives a generalized Hรถlder inequality, which is shown in [9, 13].

Theorem 3.3 (see [9, 13]). * Let . Then,
**
for every and .*

The next theorem is shown in [6].

Theorem 3.4. *Let and . Then,
*

The next theorem is shown in [9].

Theorem 3.5. *
(i) Let . Then, is a Banach space.**
(ii) If , then is dense in .*

Let such that is not a constant function. Then, according to [9, Example 2.9, Theorem 2.10], for every , there exists a function such that its translation if . However, we can prove that all elements of belong to for every . Let . Then, we have where .

Hence, by the same arguments of Triebel [20], we have the following two theorems.

Theorem 3.6. *Let be a compact subset of and an arbitrary multi-index. Let , such that . Then, there exists a positive constant such that
**
for all .*

Let and . Then, makes sense for any by the classical Hรถlder inequality and (3.7).

Let be a real number and the spaces of such that

Theorem 3.7. *Let and . Let be a sequence of compact subsets of . Let be the diameter of . If , then there exists a number such that
**
for and .*

#### 4. Basic Properties of Variable Exponent Triebel-Lizorkin and Besov Spaces

We will first consider the quasi-norms on and .

*Definition 4.1. *
(i) The set is the collection of all systems with for , such that
for every multi-index , there exists a positive number such that
for , and and there exists a positive number such that
for .

(ii) The set is the collection of all systems such that
for every multi-index , there exists a positive number such that
for , and , and
for .

Theorem 4.2. *
(i) Let , , and . Let be as in Definition 2.1 and . Then, the norm coincides with and the norm with .**
(ii) Let , , and . Let , . Then, and are equivalent quasi-norms on . Similarly, and are equivalent quasi-norms on .**
(iii) Let , , and . Let and . Then, and are equivalent quasi-norms on . Similarly, and are equivalent quasi norms on .*

*Proof. *Let be as in Definition 2.1. It is obvious that for some positive number , which depends only on the definition of Fourier transform . Let . Then . Hence, the norm coincides with . Similarly, the variable exponent Besov norm coincides with . This proves (i). The proof of (ii) is done as in the proof of [20, page 46, Proposition 1] with using Theorem 3.7, the scalar case of Theorem 3.7, and Definition 4.1.

We will prove (iii). Let , , and . Let for . Then,
because
for and . Here, means converges to in . It follows that
where we use Theorem 3.7 with , , and . Hence, we have
We will prove the opposite inequality. Let be a real function with for and
We set , for . Furthermore, let be a real function such that for and
We define by
for ( for ). Then, and . Let . It is obvious that and by Theorem 4.8 and (4.10), where is some sequence of compact subsets of . Then,
for any . This means that is independent of and in . Hence, we have . This implies that
This proves the case of (iii). The proof of the case of (iii) is essentially the same as one of the case.

Corollary 4.3. *Let , , and . **
(i) if and only if there exist continuous functions such that , ,
**
In particular, coincides with , where the infimum is taken over all representation .**
(ii) if and only if, there exist continuous functions such that , , , and for . In particular, the norm coincides with .*

Proposition 4.4. *
(i) Let , , and . Then,
**
(ii) Let , , , , and . Then
**
(iii) Let and . If , then
*

The proof is almost same as in [20]. Furthermore, Almeida and Hรคstรถ [14] proved the above inclusion for and .

Theorem 4.5. *Let , , and . Let be either or . Then, . Furthermore, is dense in and is a quasi Banach space (a Banach space if ).*

*Proof. *To prove the inclusion, we may restrict ourselves to with by Proposition 4.4. Let and . We recall that the topology of the complete locally convex space is generated by seminorms:
yields a one-to-one mapping from onto itself, and, in particular, with generates the topology of . If is a sufficiently large natural number, then
where is a positive number. This proves the left-hand side of the inclusion with . We prove the right-hand side of the inclusion with . Let be the above system. It is well known that
in for all and . By the Paley-Wiener-Schwartz Theorem, the definition of implies that are analytic functions, and that these functions are regular distributions. We put if , with . If and , then denotes the value of the functional of of the test function . We obtain
by the classical Hรถlder inequality. Because both and are analytic functions, the last estimate makes sense. By applying (3.7) with to , we have
If is a sufficiently large natural number, then, for any , we have
by the left-hand side of the inclusion for . Hence, we have the inclusion for .

The proof of the completeness and the density of is given by the same arguments of step 4 and step 5 of the proof of [20, page 48, Theorem] with replaced by .

As we mentioned in Section 1, the next theorem is found in [15, 16].

Theorem 4.6. *Let and . Then, coincides with . Let and . Then, coincides with .*

*Definition 4.7. *Let such that for and
and let such that
where and are positive numbers. We construct and by
We choose and sufficiently small so that
for .

Diening [3] shows that Young inequality holds if and only if is a constant function. However, the next theorem, which is a part of Corollary 3.6 of [3], holds.

Theorem 4.8. *Let . Let be an integrable function on with range in and set for all . Assume that the least decreasing radial majorant of is integrable, that is, . Then, for any , one has
**
Hence, there exist a positive number such that
**
where only depends on and .*

The duality pair of and is denoted by , although, for , we also write as . In order to prove the duality of and (Theorem 5.1), we will first prove the following two lemmas which correspond to [19, Lemmasโโ7.1.3, andโ7.1.5].

Lemma 4.9. *Let , as in Definition 4.7 and
**
in . Then, and
**
where does not depend on .*

*Proof. *We use the same argument of the proof of [19, Lemma 7.1.3]. Let . Then, it holds that
where . Using and Theorem 3.7 with and , we find that
where is a system of Definition 4.7. Thus, we get
from Definition 2.1 and Theorem 4.2.

Lemma 4.10. *Let and be ones as in Definition 4.7, , and the functional in Lemma 4.9. Let be a system of functions such that . Set
**
for , and
**
Then,
**
where depends only on the definition of Fourier transform.*

*Proof. *We use the same argument of the proof of [19, Lemma 7.1.5]. We first prove that , where is defined by (3.1). The consideration of the proof of Lemma 4.9 implies that for ,
where and if . This implies and the right-hand side of (4.40) has a sense. We assume . Then, and . By (4.32), we have
in and
Since , (4.29), (4.37), and (4.38) imply that
From this, (4.40) follows for . Let , such that
in as and
Thus, (4.40) holds for and . For any , the least decreasing radial majorant of is integrable. By the definition of ,
Hence, converges to in as by Theorem 4.8 and converges to in as . Equation (4.40) follows from the last relation.

#### 5. Duality and Reflexivity of Variable Exponent Function Spaces

Theorem 5.1. *Let , , and . Let be either or . Then,
**
where
**
and . The spaces and are reflexive.*

*Proof. *We will only prove (5.1) because the reflexivity of follows from Theorem 3.1.*Step 1. *We will first prove the case. We use the arguments of the proof of [19, Theoremsโโ7.1.7,โandโ7.2.2]. Let , , and replaced by as in Definition 2.1. Then, in , , and in . It follows from Definition 2.1 that
in . Here, for . Using Hรถlder inequality, we get
where does not depend on , and . Hence, we see and since is dense in .*Step 2. * Let . We assume that and the functions and have the same sense as in Definition 4.7, Lemmas 4.9, and 4.10. We show that . For this purpose, we construct functions as (4.37) and set
for . If , we set . We have
where we used . We set
and . Then,
where we used
Hence, we have
In particular, . Now, we construct the same function by (4.39). Using Lemma 4.10, (4.37) and Theorem 4.8, we have
We set
Then, it follows from Theorem 3.4, (5.11), and (5.10) that
If is not a constant function, then . So, we normalize the . We put and
Then, we have
Hence we have
The last estimate, (4.32), (4.37), and Corollary 4.3 lead us to
We specialize the function in Definition 4.7 by setting for
where is a sufficiently small positive number. Then, we find a function such that
for . As in (4.28), we construct , and, as in (4.32), we put
in . The function has properties similar to , notably, (5.17) holds. Changing the function , we obtain . Then, (5.17) implies
This completes Theorem 5.1 for the . *Step 3. *Finally, we will prove the case. Let and . Then, we have and see that there exists a positive number such that
by the same manner as in the Step 1. *Step 4. *Let and as in (4.32). Then, we have and
by the same argument in the proof of Lemma 4.9. Furthermore, Lemma 4.10 also holds for by the same argument in the proof of Lemma 4.10. Let as in (4.37). Then, . We set
for . Then, we have by the following calculating: