Maximal function on Musielak–Orlicz spaces and generalized Lebesgue spaces

Abstract We consider the Hardy–Littlewood maximal operator M on Musielak–Orlicz Spaces L φ ( R d ) . We give a necessary condition for the continuity of M on L φ ( R d ) 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 ( | f | q ) ) 1 q .