Duals of variable exponent Hörmander spaces ($0< p^- \le p^+ \le 1$) and some applications

Abstract

In this paper we characterize the dual $\bigl(\B^c_{p(\cdot)} (\Omega) \bigr)'$ of the variable exponent H\"or\-man\-der space $\B^c_{p(\cdot)} (\Omega)$ when the exponent $p(\cdot)$ satisfies the conditions $0 < p^- \le p^+ \le 1$, 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 $\R^n$. It is shown that the dual $\bigl(\B^c_{p(\cdot)} (\Omega) \bigr)'$ is isomorphic to the H\"ormander space $\B^{\mathrm{loc}}_\infty (\Omega)$ (this is the $p^+ \le 1$ counterpart of the isomorphism $\bigl(\B^c_{p(\cdot)} (\Omega) \bigr)' \simeq \B^{\mathrm{loc}}_{\widetilde{p'(\cdot)}} (\Omega)$, $1 < p^- \le p^+ < \infty$, recently proved by the authors) and hence the representation theorem $\bigl( \B^c_{p(\cdot)} (\Omega) \bigr)' \simeq l^{\N}_\infty$ is obtained. Our proof relies heavily on the properties of the Banach envelopes of the steps of $\B^c_{p(\cdot)} (\Omega)$ and on the extrapolation theorems in the variable Lebesgue spaces of entire analytic functions obtained in a precedent paper. Other results for $p(\cdot) \equiv p$, $0 < p < 1$, are also given (e.g. $\B^c_p (\Omega)$ does not contain any infinite-dimensional $q$-Banach subspace with $p < q \le 1$ or the quasi-Banach space $\B_p \cap \E'(Q)$ contains a copy of $l_p$ when $Q$ is a cube in $\R^n$). Finally, a question on complex interpolation (in the sense of Kalton) of variable exponent H\"ormander spaces is proposed.