Normed Köthe space structure and Köthe duals of variable-exponent Fofana spaces and their preduals
Fofana spaces, introduced on the basis of Wiener amalgam spaces, have attracted some interest recently and have found applications in the field of partial differential equations. In this paper, we study the normed Köthe structure of the predual spaces $\mathcal{H}(p^{\prime}(\cdot),q^{\prime},α^{\prime})$ of Fofana spaces with variable exponent $(L^{p(\cdot)}, \ell^{q})^α$, where $1\leq p(\cdot) \leq α\leq q\leq \infty$. As an application of this investigation, we identify the Köthe duals and Köthe biduals of these two spaces, and we obtain a predual of $\mathcal{H}(p^{\prime}(\cdot),q^{\prime},α^{\prime})$. Finally, we prove that neither $(L^{p(\cdot)}, \ell^{q})^α$ nor $\mathcal{H}(p^{\prime}(\cdot),q^{\prime},α^{\prime})$ is reflexive.
math.FA↗