## Abstract

We show that the dual ${\left({B}_{p\left(\xb7\right)}^{\mathrm{l}\mathrm{o}\mathrm{c}}\left(\mathrm{\Omega}\right)\right)}^{\mathrm{\prime}}$ of the variable exponent Hörmander space ${B}_{p(\xb7)}^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega})$ is isomorphic to the Hörmander space ${B}_{\mathrm{\infty}}^{c}(\mathrm{\Omega})$ (when the exponent $p(\xb7)$ satisfies the conditions $$, the Hardy-Littlewood maximal operator $M$ is bounded on ${L}_{p(\xb7)/{p}_{\mathrm{0}}}$ for some $$ and $\mathrm{\Omega}$ is an open set in ${\mathbb{R}}^{n}$) and that the Fréchet envelope of ${B}_{p(\xb7)}^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega})$ 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 ${B}_{p(\xb7)}^{\mathrm{l}\mathrm{o}\mathrm{c}}(\mathrm{\Omega})$ 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 $p(\xb7)\equiv p$, $$, 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.

## Citation

Joaquín Motos. María Jesús Planells. César F. Talavera. "Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces." Abstr. Appl. Anal. 2016 1 - 9, 2016. https://doi.org/10.1155/2016/1393496

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