On the reflexivity of the spaces of variable integrability and summability

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 Hästö 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 $B_{p(\cdot )q(\cdot )}^{s(\cdot )}$ is reflexive.