Characterization of BMO spaces via commutators of fractional maximal function on Lorentz spaces

Let 0 ≤ α<n and b be the locally integrable function. In this paper, we consider the maximal commutator of fractional maximal function Mb,α and the nonlinear commutator of fractional maximal function [b, Mα] on Lorentz spaces. We give some necessary and sufficient conditions for the boundedness of the commutators Mb,α and [b, Mα] on Lorentz spaces when the function b belongs to BMO spaces, by which some new characterizations of certain subclasses of BMO spaces are obtained.