Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces

Abstract

We show that the dual ${\left(B_{p\left(·\right)}^{\mathrm{l}\mathrm{o}\mathrm{c}}\left(\mathrm{\Omega }\right)\right)}^{\mathrm{\prime }}$ of the variable exponent Hörmander space $B_{p(·)}^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega })$ is isomorphic to the Hörmander space $B_{\mathrm{\infty }}^c(\mathrm{\Omega })$ (when the exponent $p(·)$ satisfies the conditions , the Hardy-Littlewood maximal operator $M$ is bounded on $L_{p(·)/p_{\mathrm{0}}}$ for some and $\mathrm{\Omega }$ is an open set in ${\mathbb{R}}^n$) and that the Fréchet envelope of $B_{p(·)}^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega })$ is the space $B_{\mathrm{1}}^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega })$. Our proofs rely heavily on the properties of the Banach envelopes of the $p_{\mathrm{0}}$-Banach local spaces of $B_{p(·)}^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega })$ and on the inequalities established in the extrapolation theorems in variable Lebesgue spaces of entire analytic functions obtained in a previous article. Other results for $p(·)\equiv p$, , are also given (e.g., all quasi-Banach subspace of $B_p^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega })$ is isomorphic to a subspace of $l_p$, or $l_{\mathrm{\infty }}$ is not isomorphic to a complemented subspace of the Shapiro space $h_{p^{-}}$). Finally, some questions are proposed.