Maximal function on generalized Lebesgue spaces L^p(⋅)

We prove the boundedness of the Hardy-Littlewood maximal function on the generalized Lebesgue space L p(•) (R d ) under a continuity assumption on p that is weaker than uniform Hölder continuity.We deduce continuity of mollifying sequences and density of C ∞ (Ω) in W 1,p(•) (Ω) .