Weighted Besov spaces with variable exponents
Introduction
When Leopold studied pseudo-differential operators in [27], [28], [29], he introduced the related Besov space with variable smoothness . Besov then generalized Leopold's results by considering the Triebel-lizorkin space and the Besov space in with in [4], [5], [6], [7]. The second author studied the Besov space and the Triebel-Lizorkin space with variable integrable exponent for fixed exponents q and s in [42], [43]. In [17], Diening, Hästö and Roudenko introduced the variable exponent Triebel-Lizorkin space , gave a discretization by the so called φ-transform, atomic and molecular decompositions of these function spaces. Furthermore, they obtained the trace results for as an application. In [41], Vybíral proved the Sobolev and Jawerth embeddings of spaces . In [1], Almeida and Hästö introduced the variable exponent Besov spaces and obtained their characterization by approximations and embeddings. In [18], Drihem gave the atomic decomposition of Besov spaces . In [23], Izuki and Noi proved the duality and the reflexivity of Besov spaces and Triebel-Lizorkin spaces . Almeida and Hästö studied both real and complex interpolations of the variable exponent Besov spaces and Triebel-Lizorkin spaces , and proved the embedding property of Besov spaces in [2]. Noi and Sawano considered complex interpolation of Besov spaces and Triebel-Lizorkin spaces with variable exponents in [36]. Kempka and Vybíral gave characterizations of and by local means and ball means of differences in [25]. Drihem gave the duality of Triebel-Lizorkin spaces in [20]. Kempka introduced 2-microlocal Besov and Triebel-Lizorkin spaces of variable integrability in [24]. Nakai and Sawano gave the real variable theory of Hardy spaces with variable exponents in [33].
On the basis of the Besov-type space [30], [44] and the variable Besov space , Yang, Zhuo and Yuan introduced a more generalized scale of function space with variable smoothness and variable integrability which covers both Besov-type spaces and Besov spaces with variable smoothness and integrability in [45]. In the spirit of [45], Yang, Zhuo and Yuan introduced space in [46]. Drihem introduced variable Triebel-Lizorkin spaces in [19].
For the weighted case, Bui obtained the interpolation properties of the weighted Besov space and the Triebel-Lizorkin space in [8]. Haroske and Skrzypczak studied the embedding of the Besov space and the Triebel-Lizorkin space with Muckenhoupts weights in [21], [22]. Meyries and Veraar [31] studied the validity of the continuous Sobolev embeddings for weighted ( and ) in the case of . For the classical theory of Besov spaces, we refer the readers to [38], [40].
Inspirited by the aforementioned works, in this paper, we will introduce the weighted Besov space with weight w in variable Muckenhoupt class and variable exponents , and . Then we will obtain the approximation characterization, the lifting property, embeddings, the duality and interpolation of these new spaces.
Section snippets
Preliminaries
In this section, we first recall some definitions and notation, and then we state our results. Given a measurable function , the modular is defined by Then the Lebesgue space with variable exponent is defined by The Lebesgue space becomes a Banach function space equipped with the norm
The space is
Equivalent norms
Theorem 3 Let , , , and . Also let . Then if and only if, there exists a sequence of continuous functions such that in , , In the case, where the infimum is taken for all above decompositions of f. Proof Suppose that f satisfies the conditions of Theorem 3. Let
Embeddings
Theorem 4 Let , , and , , , . (i) If and , then (ii) If and , then
Proof (i) Without loss of generality, we consider that . Then, by the definition of modular, we have If with , then we have Thus by (8), we obtain
Lifting property
Definition 6 For , the Bessel potential operator from to is defined by
As in [38], extends to an isomorphism in . Owing to Theorem 5, we have the following result.
Theorem 6 If , and , then the operator is a linear continuous bijective operator from onto . Proof By Theorem 5, we only consider . Let as in Definition 3. Let , by
Duality
In this section we consider the dual spaces of by Triebel's method in [39]. Definition 7 Let satisfy for and where ε and δ are positive numbers. We construct and by We choose ε and δ sufficiently small so that
Lemma 13 Let , , , and . Moreover, let (the
Interpolation
In this some section, we will use the K-method in [37], [38] for . To proceed, we recall notions. Let and be a couple of (real or complex) Banach spaces, continuously embedded into a linear Hausdorff space. Then with the norm and with are Banach spaces, see [9]. Let , we define the interpolation spaces , by
With weights in
Usually, one defined the weighted constant exponent Besov spaces with weights in . To define weighted variable exponent Besov spaces with weights in , we need use an alternative form of weighted Lebsgue spaces, which was used in [16]. For a wight w and , we define and . To obtain the boundedness of the Hardy-Littlewood maximal operator on , Diening and Hästö introduced the following class in [16].
Definition 8 Let
Acknowledgments
The authors would like to thank the referee for his careful reading and suggestions, particularly, Lemma 18. The second author is partially supported by the National Natural Science Foundation of China (Grant No. 11761026) and Guangxi Natural Science Foundation (Grant No. 2020GXNSFAA159085).
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