Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces

We show that the dual of the variable exponent Hörmander space is isomorphic to the Hörmander space (when the exponent satisfies the conditions , the Hardy-Littlewood maximal operator is bounded on for some and is an open set in ) and that the Fréchet envelope of is the space . Our proofs rely heavily on the properties of the Banach envelopes of the -Banach local spaces of 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 , , are also given (e.g., all quasi-Banach subspace of is isomorphic to a subspace of , or is not isomorphic to a complemented subspace of the Shapiro space ). Finally, some questions are proposed.