On the reflexivity of the spaces of variable integrability and summability

TL;DR

This paper proves the reflexivity of variable mixed Lebesgue-sequence spaces and related Besov spaces under certain conditions, resolving an open problem and characterizing their dual spaces.

Contribution

It establishes the reflexivity of $ ext{ell}^{q(ullet)}(L^{p(ullet)})$ spaces and identifies their duals, answering a question posed by H"asto in 2017.

Findings

Reflexivity of $ ext{ell}^{q(ullet)}(L^{p(ullet)})$ spaces under specified conditions.

Explicit dual space characterization for these variable spaces.

Application of results to the reflexivity of variable Besov spaces.

Abstract

In this paper, we show that under the condition $ 1<p_-, q_-, p_+, q_+<\infty$, the space $\ell^{q(\cdot)} (L^{p(\cdot)})$ is reflexive. In this way we give an answer to open problem posed by H\"ast\"o in 2017 about the reflexivity of the variable mixed Lebesgue-sequence spaces $\ell^{q(\cdot)} (L^{p(\cdot)})$. What is important here is that the dual space of $\ell^{q(\cdot)} (L^{p(\cdot)})$ is specified. As its direct corollary, we show that the corresponding Besov space $B^{s(\cdot)}_{p(\cdot)q(\cdot)}$ is reflexive.