Dual of Modulation Spaces with Variable Smoothness and Integrability
In this article, we first give a proof for the denseness of the Schwartz class in the modulation spaces with variable smoothness and integrability. Then, we study the dual spaces of such modulation spaces.
The modulation spaces were introduced by Feichtinger  on a locally compact Abelian group in 1983 through short-time Fourier transform. His original motivation for modulation spaces was to introduce a new theory of function spaces and to offer an alternative to the class of Besov spaces. In recent years, it is gradually recognized that the modulation spaces are very useful for studying time-frequency behavior of functions. Therefore, the modulation spaces, -modulation spaces, and their applications have received a lot of attention and research, such as [2–10] and the references therein. Particularly, in [11–14], Wang and other authors showed that from PDE point of view, the combination of frequency-uniform decomposition operators and Banach function spaces is important in making nonlinear estimates, where is a Banach function space defined on .
On the other hand, function spaces with variable exponents have received extensive attention recently. Even though the study on variable Lebesgue spaces can be traced back to [15, 16] by Orlicz, the modern development started from the article  by Kováčik and Rákosník in 1991. In , Fan and Zhao obtained the results in  again through the method of Musielak-Orlicz spaces. Thereafter, variable Lebesgue and Sobolev spaces have been widely studied (see, for example, [19–23]). In addition, function spaces with variable exponents have a wealth of applications in many fields, such as in fluid dynamics , image processing , and partial differential equations .
The function spaces with variable smoothness and variable integrability were firstly introduced by Diening et al. in , where they studied Triebel-Lizorkin spaces with variable exponents . Then, Almeida and Hästö introduced the Besov space with variable smoothness and integrability in . Since then, many articles about these function spaces appeared, such as [29, 30]. In the past few years, many function spaces with variable exponents have appeared, such as Besov-type spaces with variable exponent, Bessel potential spaces with variable exponent, and Hardy spaces with variable exponent (see [31–35]). Recently, we studied the modulation spaces with variable smoothness and integrability and gave some properties about these spaces in . Since the modulation spaces and the function spaces with variable exponents have rich applications, we believe that the modulation spaces with variable exponents will also have many application areas, and we will continue to explore these application areas, especially in partial differential equations and time frequency analysis.
The dual is an important content when we study function spaces; for example, Triebel  has obtained duality of the usual Besov spaces and applied it to real interpolation and Sobolev embedding, Izuki  has given the duality of Herz spaces with variable exponent and applied it to characterize the above spaces by wavelet expansions. In [8, 12], the dual of modulation spaces was studied, respectively. In , Izuki and Noi were concerned with the dual of Triebel-Lizorkin spaces and Besov spaces with variable exponents. In this paper, we will study the dual of modulation spaces with variable smoothness and integrability.
The paper is organized as follows. In Section 2, we review some notions and notations about semimodular spaces and function spaces with variable exponents. In the theories of function spaces, the research on denseness of the Schwartz class has always been an important topic, by which we can obtain many conclusions such as duality of function spaces and boundedness of some operators. Therefore, in Section 3, we study the denseness of the Schwartz class in the modulation spaces with variable smoothness and integrability. In Section 4, we give the dual of modulation spaces with variable exponents.
In this section, we review some notions and conventions and state some basic results. Throughout this article, we let denote constants that are independent of the main parameters involved but whose value may differ from line to line. By , we mean that there exists a positive constant such that . The symbol means that . The symbol for denotes the maximal integer not more than . We also set and . We write and for . It is easy to see that . For any multi-index , we denote , and for , we denote . We also denote the sequence Lebesgue space by and Lebesgue space by for which the norm is written by .
Let be the Schwartz function space and be its strongly topological dual space which is also known as the space of all tempered distributions. For , we define the Fourier transform and the inverse Fourier transform , respectively, by
2.1. Modular Spaces
In what follows, let be a vector space over or . The function spaces studied in this paper fit into the framework semimodular spaces, and we refer to monograph  for a detailed exposition of these concepts.
Definition 1. A function is called a semimodular on if it satisfies (i) for all , with (ii) for all implies (iii) is left-continuous on for every
A semimodular is called a modular if implies , and it is called continuous if the mapping is continuous on for every . A semimodular can also be qualified by the term (quasi)convex; that is, for all and , there exists such that where in the convex case and in the quasiconvex case. By semimodular, we can obtain a normed space as follows:
Definition 2. If is a (semi)modular on , then is called a (semi)modular space.
In , the authors have proven that the is a (quasi)normed space with the Luxemburg (quasi)norm , where the infimum of the empty set is infinity by definition. The following conclusion can be found in , and we omit the proof here.
Theorem 3 (norm-modular unit ball property). Let be a semimodular on and . Then, if and only if . If is continuous, then and are equivalent, so are and ϱ.
2.2. Function Spaces with Variable Exponents
A measurable function is called a variable exponent function if it is bounded away from zero; namely, the range of the is for some . For a measurable function and a measurable set , let and For simplicity, we abbreviate and .
We denote by for the set of all measurable functions such that and denote by for the set of all measurable functions such that .
In order to make the Hardy-Littlewood maximal function bounded in the variable exponent Lebesgue spaces, one need to add some conditions to the variable exponent function, that is, so-called log-Hölder continuity, which was first introduced in .
Definition 4. Let . (i)If there exists such thatfor all , then is called locally log-Hölder continuous, abbreviated as . (ii)If is locally log-Hölder continuous and there exists such thatfor all , then is called globally log-Hölder continuous, abbreviated as .
If a variable exponent satisfies , we say that it belongs to the class . The class is defined similarly.
Remark 5. (i)One can notice that all functions always belong to (ii)Let , then if and only if . If satisfies (5), then (iii)We define the conjugate exponent function by the formula If is in , then is also in
We define and we adopt the convention in order that is left-continuous. The variable exponent modular of a measurable function on is defined by
According to Definition 2, one can define the corresponding semimodular space, namely, the variable exponent Lebesgue space which is denoted by , and the Luxemburg (quasi)norm of the is defined by
Now let us recall the mixed Lebesgue sequence space which was introduced by Almeida and Hästö in .
Definition 6. Let and be a measurable subset of . The mixed Lebesgue sequence space is the collection of all sequences of -functions such that where with the convention for all .
Remark 7. Let . (i)If , then , and we use the notation(ii)By Proposition 3.3 of , if is constant, then we have (iii)In , Almeida and Hästö proved that is a quasinorm for all , and is a norm when pointwise or is a constant. In , Kempka and Vybíral proved that is a norm if satisfy either for almost every or , and is a constant almost everywhere
To define modulation space with variable exponents, we need some general definitions from the constant exponent case. For , let be the unit cube with the center at ; then, constitutes a decomposition of . Let and be a smooth function satisfying for and for . Let be a translation of : Then, we see that in and for all . If we denote , for , then we have
We denote is nonempty, and for every sequence , one can construct an operator sequence as follows:
are said to be frequency-uniform decomposition operators. Let and ; the modulation space can be defined as
Further details about the frequency-uniform decomposition techniques and their applications to PDE can be found in the book  and articles [11, 12, 14].
Definition 8. Let and be the corresponding frequency-uniform decomposition operators. For and , the modulation space with variable smoothness and integrability is defined to be the set of all distributions such that
For above modulation space, we can define the following modular: which can be used to define the norm. In , we have shown that the space given by Definition 8 is independent of the choice of and the corresponding frequency-uniform decomposition operators. Thus, we can choose according to our requirements, and we will omit in the notation of the norm and modular.
In , the authors showed that the maximal function is not a good tool in the variable exponent space ; hence, they used so-called -functions which were also used in . Similarly, in our article, we define the so-called -functions on by with , , and . Note that when and that is independent of . These functions are different from the -functions since we use the uniform decomposition of rather than the dyadic decomposition.
Now let us review some useful results about -functions which have been proven in .
Lemma 9 (see ). Let and ; then, there exists a positive constant such that for all and , where is the constant from (4) for .
Remark 10. By Lemma 9, we can move the term inside the convolution as follows: which helps us to treat the variable smoothness in many cases.
Lemma 11 (see ). Let , for and every sequence of -functions, there exists a constant such that
Remark 12. In some cases, although we need to require that , we can weaken this condition by the following identity:
In , we have proven that , and in this section, we will prove that is also dense in . For this purpose, we need the following conclusions as in . The first one is the generalization of Lemma A.6 in  and Lemma 3.4 in .
Lemma 13. Let , , and . Then, for all and with , there exists a constant such that
Proof. As in the proof of Lemma 3.4 of , for , there exists such that , which implies . Then, for and a fixed dyadic cube , when , we have In addition, for and , we have when is large enough, which implies . Since we get Hence, for , we have where depends only on , , and . For any , there exists a such that . Then, we get the desired conclusion.
Definition 14. (i)Let be a compact subset of ; then, we denote the space of all elements with by (ii)Let and be a sequence of compact subsets of ; then, we denote by the space of all sequences in such that and for
Lemma 15. Let , , and be a sequence of compact subsets of such that . If and , then for all , there exists a constant such that
Proof. Let and ; then, for any by Lemmas 9 and 13, we have Therefore, for and , by Lemma 11, we obtain which completes the proof.
For a real number , we denote
Proposition 16. Let , , and be a sequence of compact subsets of such that . If , then for all and , there exists a constant such that
Proof. According to Lemma 15, by the similar argument in the proof of Theorem 4.15 of , we have Since then for and , by the same argument in 1.6.3 of , we have Thus, In addition, since then together with above inequality, (33) and Lemma 15, we can get the conclusion.
Remark 17. In fact, in the conclusion of the above lemma, the “” in can be replaced by .
Theorem 18 (density). Let and , then is dense in .
Proof. Let , for and , we put
where for . Then, we have . In fact, by Proposition 16, we obtain
when , in which the last limit can be deduced by Lemma 2.2 of . Hence, in when .
Next, we should approximate by some functions in for . Let satisfy and . Then, for any and , we have and Since , we have by Lemma 3.2.8 of . Now, we prove that is an approximation of in . By (38), we have Let , then and . For each with , let ; then, , where . By Remark 17 and the embedding properties of , we get Then, combining (39) and , we obtain Therefore, is dense in .
4. Dual Spaces of
For a quasi-Banach space , we denote the dual space of by . In this section, we show that for and .
Lemma 19. Let , , and be sequences of locally Lebesgue integrable functions satisfying and . Then, we have
The above lemma has been proven in ; hence, we omit the proof here.
Proposition 20. Let and . We denote by the collection of all satisfying that there exists such that and If we define