Sobolev's theorem and duality for Herz-Morrey spaces of variable exponent

Abstract

In this paper, we consider the Herz-Morrey space H p(·),q,! {x0} (G) of variable exponent consisting of all measurable functions f on a bounded open set G ⊂ R n satisfying kfk H p(·),q,! {x0} (G) = ˆ 2dG H p(·),q,! (G),H p(·),q,! {x0} (G) and ˜ H p(·),q,! (G). Following Fiorenza-Rakotoson (18), Di Fratta-Fiorenza (17) and Gogatishvili-Mustafayev (19), we next discuss the duality properties among these Herz- Morrey spaces.