A Note on Variable Exponent Hörmander Spaces

Abstract

In this paper we introduce the variable exponent Hormander spaces and we study some of their properties. In particular, it is shown that \({{(\mathcal{B}_{p_{(\cdot)}}^{c}(\Omega))^{\prime}}}\) is isomorphic to \({{\mathcal{B}^{loc}_{\widetilde{p^\prime(\cdot)}(\Omega)}}}\) (Ω open set in \({{\mathbb{R}^n, p? > 1}}\) and the Hardy–Littlewood maximal operator M is bounded in \({L_p(\cdot))}\) extending a Hormander’s result to our context. As a consequence, a number of results on sequence space representations of variable exponent Hormander spaces are given.