Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces

We consider the Hardy-Littlewood maximal operator M on Musielak-Orlicz Spaces L-phi(R-d). We give a necessary condition for the continuity of M on L-phi(Rd) which generalizes the concept of Muckenhoupt classes. In the special case of generalized Lebesgue spaces L-P(.)(R-d) 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 M-q is continuous for some q > 1, where M-q f = (M(vertical bar f vertical bar(q))) (l/q). (c) 2005 Elsevier SAS. All rights reserved.