Fractional maximal function and its commutators on Orlicz spaces

In this paper, we find necessary and sufficient conditions for the boundedness of fractional maximal operator M α on Orlicz spaces. As an application of this results we consider the boundedness of fractional maximal commutator M b , α and nonlinear commutator of fractional maximal operator [ b, M α ] on Orlicz spaces, when b belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces are given. © 2017, Springer International Publishing AG.