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 as long as \(\ell ^{q(\cdot )} (L^{p(\cdot )})\) is a Banach space. In this way we give an answer to the 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 \(B^{s(\cdot )}_{p(\cdot )q(\cdot )}\) is reflexive.