Maximal function on generalized lebesgue spaces Lp(·)

Abstract

We prove the boundedness of the Hardy-Littlewood maximal function on the generalized Lebesgue space Lp(·) (ℝd) under a continuity assumption on p that is weaker than uniform Holder continuity. We deduce continuity of mollifying sequences and density of C∞ (Ω̄) in W1,p(·)(Ω).