Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces

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 .