Weighted Besov spaces with variable exponents

doi:10.1016/j.jmaa.2021.125478

Journal of Mathematical Analysis and Applications

Volume 505, Issue 1, 1 January 2022, 125478

https://doi.org/10.1016/j.jmaa.2021.125478 Get rights and content

Abstract

In this paper, we introduce Besov spaces with variable exponents and variable Muckenhoupt weights. Then we give a approximation characterization, the lifting property, embeddings, the duality and interpolation of these spaces.

Introduction

When Leopold studied pseudo-differential operators in [27], [28], [29], he introduced the related Besov space with variable smoothness $B_{p, p}^{s (\cdot)}$ . Besov then generalized Leopold's results by considering the Triebel-lizorkin space $F_{p, q}^{s (\cdot)}$ and the Besov space $B_{p, q}^{s (\cdot)}$ in $R^{n}$ with $p \neq q$ in [4], [5], [6], [7]. The second author studied the Besov space $B_{p (\cdot), q}^{s}$ and the Triebel-Lizorkin space $F_{p (\cdot), q}^{s}$ with variable integrable exponent $p (\cdot)$ for fixed exponents q and s in [42], [43]. In [17], Diening, Hästö and Roudenko introduced the variable exponent Triebel-Lizorkin space $F_{p (\cdot), q (\cdot)}^{s (\cdot)}$ , gave a discretization by the so called φ-transform, atomic and molecular decompositions of these function spaces. Furthermore, they obtained the trace results for $F_{p (\cdot), q (\cdot)}^{s (\cdot)}$ as an application. In [41], Vybíral proved the Sobolev and Jawerth embeddings of spaces $F_{p (\cdot), q (\cdot)}^{s (\cdot)}$ . In [1], Almeida and Hästö introduced the variable exponent Besov spaces $B_{p (\cdot), q (\cdot)}^{s (\cdot)}$ and obtained their characterization by approximations and embeddings. In [18], Drihem gave the atomic decomposition of Besov spaces $B_{p (\cdot), q (\cdot)}^{s (\cdot)}$ . In [23], Izuki and Noi proved the duality and the reflexivity of Besov spaces $B_{p (\cdot), q (\cdot)}^{s (\cdot)}$ and Triebel-Lizorkin spaces $F_{p (\cdot), q (\cdot)}^{s (\cdot)}$ . Almeida and Hästö studied both real and complex interpolations of the variable exponent Besov spaces $B_{p (\cdot), q (\cdot)}^{s (\cdot)}$ and Triebel-Lizorkin spaces $F_{p (\cdot), q (\cdot)}^{s (\cdot)}$ , 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 $B_{p (\cdot), q (\cdot)}^{s (\cdot)}$ and $F_{p (\cdot), q (\cdot)}^{s (\cdot)}$ by local means and ball means of differences in [25]. Drihem gave the duality of Triebel-Lizorkin spaces $F_{1, q (\cdot)}^{s (\cdot)}$ 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 $B_{p, q}^{s, τ}$ [30], [44] and the variable Besov space $B_{p (\cdot), q (\cdot)}^{s (\cdot)}$ , Yang, Zhuo and Yuan introduced a more generalized scale of function space with variable smoothness and variable integrability $B_{p (\cdot), q (\cdot)}^{s (\cdot), τ}$ 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 $F_{p (\cdot), q (\cdot)}^{s (\cdot), τ}$ in [46]. Drihem introduced variable Triebel-Lizorkin spaces $F_{p (\cdot), q (\cdot)}^{s (\cdot), τ (\cdot)}$ in [19].

For the weighted case, Bui obtained the interpolation properties of the weighted Besov space $B_{p, q}^{s, w} (R^{n})$ and the Triebel-Lizorkin space $F_{p, q}^{s, w} (R^{n})$ in [8]. Haroske and Skrzypczak studied the embedding of the Besov space $B_{p, q}^{s, w} (R^{n})$ and the Triebel-Lizorkin space $F_{p, q}^{s, w} (R^{n})$ with Muckenhoupts $A_{\infty}$ weights in [21], [22]. Meyries and Veraar [31] studied the validity of the continuous Sobolev embeddings for weighted ( $B_{p, q}^{s, w} (R^{n})$ and $F_{p, q}^{s, w} (R^{n})$ ) in the case of $p_{0} < p_{1}$ . 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 $B_{p (\cdot), q (\cdot)}^{s (\cdot), w} (R^{n})$ with weight w in variable Muckenhoupt class and variable exponents $p (\cdot)$ , $q (\cdot)$ and $s (\cdot)$ . 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 $p (\cdot) : R^{n} \to [1, \infty)$ , the modular $ρ_{p (\cdot)}$ is defined by $ρ_{p (\cdot)} (f) : = \int_{R^{n}} | f (x) |^{p (x)} d x, for measurable function f on R^{n} .$ Then the Lebesgue space with variable exponent $L^{p (\cdot)} (R^{n})$ is defined by $L^{p (\cdot)} (R^{n}) : = {f is measurable: ρ_{p (\cdot)} (f / λ) < \infty for some λ > 0} .$ The Lebesgue space $L^{p (\cdot)} (R^{n})$ becomes a Banach function space equipped with the norm ${‖ f ‖}_{L^{p (\cdot)}} : = \inf {λ > 0 : ρ_{p (\cdot)} (f / λ) \leq 1} .$

The space $L_{loc}^{p (\cdot)} (R^{n})$ is

Equivalent norms

Theorem 3

Let $s (\cdot) \in C_{loc}^{\log} (R^{n}) \cap L^{\infty} (R^{n})$ , $p (\cdot)$ , $q (\cdot) \in P (R^{n}) \cap C^{\log} (R^{n})$ , and $w \in A_{p (\cdot)}$ . Also let $f \in S^{'} (R^{n})$ . Then $f \in B_{p (\cdot), q (\cdot)}^{s (\cdot), w} (R^{n})$ if and only if, there exists a sequence of continuous functions ${u_{j}}_{j \in N_{0}}$ such that $f = \sum_{j = 0}^{\infty} u_{j}$ in $S^{'} (R^{n})$ , ${‖ {2^{j s (\cdot)} u_{j}}_{j = 0}^{\infty} ‖}_{ℓ^{q (\cdot)} (L_{w}^{p (\cdot)})} < \infty$ , $supp F (u_{0}) \subset {ξ : | ξ | \leq 1},$ $supp F (u_{j}) \subset {ξ : 2^{j - 2} \leq | ξ | \leq 2^{j}}, j \in N .$ In the case, ${‖ f ‖}_{B_{p (\cdot), q (\cdot)}^{s (\cdot), w}} \sim \inf {‖ {2^{j s (\cdot)} u_{j}}_{j = 0}^{\infty} ‖}_{ℓ^{q (\cdot)} (L_{w}^{p (\cdot)})},$ where the infimum is taken for all above decompositions of f.

Proof

Suppose that f satisfies the conditions of Theorem 3. Let ${ϕ_{j}}_{j = 0}^{\infty}$

Embeddings

Theorem 4

Let $s (\cdot)$ , $s_{1} (\cdot)$ , $s_{2} (\cdot) \in L^{\infty} (R^{n})$ and $p (\cdot)$ , $q (\cdot)$ , $q_{0} (\cdot)$ , $q_{1} (\cdot) \in P_{0} (R^{n})$ .

(i) If $w \in A_{p (\cdot)}$ and $q_{0} (\cdot) \leq q_{1} (\cdot)$ , then $B_{p (\cdot), q_{0} (\cdot)}^{s (\cdot), w} (R^{n}) ↪ B_{p (\cdot), q_{1} (\cdot)}^{s (\cdot), w} (R^{n}) .$ (ii) If $w \in A_{p (\cdot)}$ and ${(s_{0} - s_{1})}^{-} > 0$ , then $B_{p (\cdot), q_{0} (\cdot)}^{s_{0} (\cdot), w} (R^{n}) ↪ B_{p (\cdot), q_{1} (\cdot)}^{s_{1} (\cdot), w} (R^{n}) .$

Proof

(i) Without loss of generality, we consider that ${‖ f ‖}_{B_{p (\cdot), q_{0} (\cdot)}^{s (\cdot), w}} = 1$ . Then, by the definition of modular, we have $ρ_{B_{p (\cdot), q_{0} (\cdot)}^{s (\cdot), w}} (f / μ) = \sum_{j = 0}^{\infty} \inf {λ_{j} : \int_{R^{n}} {(\frac{| 2^{j s (x)} ϕ_{j} ⁎ f (x) w (x) |}{λ_{j}^{\frac{1}{q_{0} (x)}}})}^{p (x)} d x \leq 1} = 1 .$ If $λ_{j} \leq 1$ with $j \in N_{0}$ , then we have $λ_{j}^{\frac{1}{q_{0} (x)}} \leq λ_{j}^{\frac{1}{q_{1} (x)}} .$ Thus by (8), we obtain $1$

Lifting property

Definition 6

For $α \in R$ , the Bessel potential operator $I^{α}$ from $S (R^{n})$ to $S (R^{n})$ is defined by $I^{α} (f) : = F^{- 1} ({(1 + | \cdot |^{2})}^{\frac{α}{2}} F f) .$

As in [38], $I^{α}$ extends to an isomorphism in $S^{'} (R^{n})$ . Owing to Theorem 5, we have the following result.

Theorem 6

If $p (\cdot), q (\cdot) \in C^{\log} (R^{n}) \cap P (R^{n})$ , $w \in A_{p (\cdot)}$ and $s (\cdot) \in C_{loc}^{\log} (R^{n}) \cap L^{\infty} (R^{n})$ , then the operator $I^{α}$ is a linear continuous bijective operator from $B_{p (\cdot), q (\cdot)}^{s (\cdot), w} (R^{n})$ onto $B_{p (\cdot), q (\cdot)}^{s (\cdot) - α, w} (R^{n})$ .

Proof

By Theorem 5, we only consider $f \in S (R^{n})$ . Let ${ϕ_{j}}_{j = 0}^{\infty}$ as in Definition 3. Let $φ_{j}$ , $j \in N_{0}$ by $φ_{j} = ϕ_{j} ⁎ F^{- 1} (2^{j α} {(1 + | \cdot |^{2})}^{- \frac{α}{2}}),$

Duality

In this section we consider the dual spaces of $B_{p (\cdot), q (\cdot)}^{s (\cdot), w} (R^{n})$ by Triebel's method in [39].

Definition 7

Let $φ \in S (R^{n})$ satisfy $(F φ) (ξ) = 1$ for $1 / \sqrt{2} \leq | ξ | \leq \sqrt{2}$ $0 \leq F φ \in C_{0}^{\infty} ({ξ \in R^{n} : 1 / \sqrt{2} - ε \leq | ξ | \leq \sqrt{2} + ε}),$ and $ϱ \in S (R^{n}), 0 \leq F ϱ \in C_{0}^{\infty} ({ξ \in R^{n} : 1 / \sqrt{2} + δ \leq | ξ | \leq \sqrt{2} - δ}),$ where ε and δ are positive numbers. We construct ${φ_{l}}_{l = 0}^{\infty}$ and ${ϱ_{l}}_{l = 0}^{\infty}$ by $(F φ_{l}) (ξ) = (F φ) (2^{- l} ξ), (F ϱ_{l}) (ξ) = (F ϱ) (2^{- l} ξ) .$ We choose ε and δ sufficiently small so that $F φ_{l} \cdot F ϱ_{j} \equiv 0 for l \neq j .$

Lemma 13

Let $p (\cdot)$ , $q (\cdot) \in C^{\log} (R^{n}) \cap P (R^{n})$ , $w \in A_{p (\cdot)}$ , and $s \in C_{loc}^{\log} (R^{n}) \cap L^{\infty} (R^{n})$ . Moreover, let $g \in {(B_{p (\cdot), q (\cdot)}^{s (\cdot), w} (R^{n}))}^{'}$ (the

Interpolation

In this some section, we will use the K-method in [37], [38] for $B_{p (\cdot), q (\cdot)}^{s (\cdot), w} (R^{n})$ . To proceed, we recall notions. Let $A_{0}$ and $A_{1}$ be a couple of (real or complex) Banach spaces, continuously embedded into a linear Hausdorff space. Then $A_{0} + A_{1} : = {a : \exists a_{0} \in A_{0}, a_{1} \in A_{1}, a = a_{0} + a_{1}},$ with the norm $K (t, a) = K (t, a, A_{0}, A_{1}) : = \inf_{\begin{matrix} a = a_{0} + a_{1} \\ a_{0} \in A_{0}, a_{1} \in A_{1} \end{matrix}} ({‖ a_{0} ‖}_{A_{0}} + t {‖ a_{1} ‖}_{A_{1}}), t \in (0, \infty),$ and $A_{0} \cap A_{1}$ with ${‖ a ‖}_{A_{0} \cap A_{1}} : = {‖ a ‖}_{A_{0}} + {‖ a ‖}_{A_{1}},$ are Banach spaces, see [9]. Let $θ \in (0, 1)$ , we define the interpolation spaces ${(A_{0}, A_{1})}_{θ, p}$ , by ${(A_{0}, A_{1})}_{θ, p} = {a : a \in A_{0} + A_{1}$

With weights in $A_{\infty}$

Usually, one defined the weighted constant exponent Besov spaces with weights in $A_{\infty}$ . To define weighted variable exponent Besov spaces with weights in $A_{\infty}$ , we need use an alternative form of weighted Lebsgue spaces, which was used in [16]. For a wight w and $p (\cdot) \in P_{0} (R^{n})$ , we define ${\tilde{L}}_{w}^{p (\cdot)} = {f : {‖ f w^{1 / p (\cdot)} ‖}_{L^{p (\cdot)}} < \infty}$ and ${‖ f ‖}_{{\tilde{L}}_{w}^{p (\cdot)}} = {‖ f w^{1 / p (\cdot)} ‖}_{L^{p (\cdot)}}$ . To obtain the boundedness of the Hardy-Littlewood maximal operator on ${\tilde{L}}_{w}^{p (\cdot)}$ , Diening and Hästö introduced the following class ${\tilde{A}}_{p (\cdot)}$ in [16].

Definition 8

Let $p (\cdot) \in P (R^{n})$

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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View full text

Weighted Besov spaces with variable exponents

Abstract

Introduction

Section snippets

Preliminaries

Equivalent norms

Embeddings

Lifting property

Duality

Interpolation

With weights in A∞

Acknowledgments

J. Funct. Anal.

J. Funct. Anal.

J. Math. Anal. Appl.

J. Funct. Anal.

J. Math. Anal. Appl.

Nonlinear Anal.

J. Funct. Anal.

Interpolation in variable exponent spaces

Rev. Mat. Complut.

Interpolation Spaces: An Introduction

Embeddings of spaces of differentiable functions of variable smoothness

Proc. Steklov Inst. Math.

On spaces of functions of variable smoothness defined by pseudodifferential operators

Proc. Steklov Inst. Math.

Equivalent normings of spaces of functions of variable smoothness

Proc. Steklov Inst. Math.

Interpolation, embedding, and extension of spaces of functions of variable smoothness

Proc. Steklov Inst. Math.

Weighted Besov and Triebel spaces: interpolation by the real method

Hiroshima Math. J.

Intermediate spaces and interpolation, the complex method

Matematika

The boundedness of classical operators on variable Lp spaces

Ann. Acad. Sci. Fenn. Math.

Weighted norm inequalities for the maximal operator on variable Lebesgue spaces

J. Math. Anal. Appl.

Extrapolation and weighted norm inequalities in the variable Lebesgue spaces

Trans. Am. Math. Soc.

Maximal function on generalized Lebesgue spaces Lp(⋅)

Math. Inequal. Appl.

Lebesgue and Sobolev Spaces with Variable Exponents

Maximal functions in variable exponent spaces: limiting cases of the exponent

Ann. Acad. Sci. Fenn., Math.

Muckenhoupt weights in variable exponent spaces

Variable Triebel-Lizorkin-type spaces

Bull. Malays. Math. Sci. Soc.

With weights in $A_{\infty}$

The boundedness of classical operators on variable $L^{p}$ spaces

Maximal function on generalized Lebesgue spaces $L^{p (\cdot)}$