Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces

Abstract

We consider the Hardy-Littlewood maximal operator M on Musielak-Orlicz Spaces Lφ(ℝd). We give a necessary condition for the continuity of M on Lφ(ℝd) which generalizes the concept of Muckenhoupt classes. In the special case of generalized Lebesgue spaces Lp(·)(ℝd) we show that this condition is also sufficient. Moreover, we show that the condition is "left-open" in the sense that not only M but also Mq is continuous for some q > 1, where Mqf = (M( f q))1/q. © 2005 Elsevier SAS. All rights reserved.