Fréchet envelopes of nonlocally convex variable exponent Hörmander spaces
Summary: We show that the dual \((B_{p \left(\cdot\right)}^{\mathrm{loc}} \left(\Omega\right))'\) of the variable exponent Hörmander space \(B_{p(\cdot)}^{\mathrm{loc}}(\Omega)\) is isomorphic to the Hörmander space \(B_{\infty}^c(\Omega)\) (when the exponent \(p(\cdot)\) satisfies the conditions \(0 < p^- \leq p^+ \leq 1\), the Hardy-Littlewood maximal operator \(M\) is bounded on \(L_{p(\cdot) / p_0}\) for some \(0 < p_0 < p^-\) and \(\Omega\) is an open set in \(\mathbb{R}^n\)) and that the Fréchet envelope of \(B_{p(\cdot)}^{\mathrm{loc}}(\Omega)\) is the space \(B_1^{\mathrm{loc}}(\Omega)\). Our proofs rely heavily on the properties of the Banach envelopes of the \(p_0\)-Banach local spaces of \(B_{p(\cdot)}^{\mathrm{loc}}(\Omega)\) 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 \(p(\cdot) \equiv p\), \(0 < p < 1\), are also given (e.g., all quasi-Banach subspace of \(B_p^{\mathrm{loc}}(\Omega)\) is isomorphic to a subspace of \(l_p\), or \(l_{\infty}\) is not isomorphic to a complemented subspace of the Shapiro space \(h_{p^-}\)). Finally, some questions are proposed.
scientific article; zbMATH DE number 6929342
scientific article