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 $M_{\alpha}$ on Orlicz spaces. As an application of this results we consider the boundedness of fractional maximal commutator $M_{b,\alpha}$ and nonlinear commutator of fractional maximal operator $[b,M_{\alpha}]$ on Orlicz spaces, when $b$ belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces are given.