Abstract: The problem of the boundedness of fractional maximal operator and its commutator on generalized homogeneous Morrey spaces was solved by using function decomposition method, and the sufficient and necessary conditions for the boundedness of fractional maximal operator and its commutator on generalized homogeneous Morrey spaces were obtained. So some characteristics of fractional maximal operator on generalized Morrey spaces were generalized to generalized homogeneous Morrey spaces.

Key words: generalized homogeneous Morrey spaces, fractional maximal operator, commutator