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

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