Maximal function on generalized Lebesgue spaces $L^{p(\cdot)}$

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∞(Ω) in W1,p(·)(Ω) . Mathematics subject classification (2000): 42B25, 46E30.