Some notes on commutators of the fractional maximal function on variable Lebesgue spaces

Let $0<\alpha <n$ and $M_{\alpha }$ be the fractional maximal function. The nonlinear commutator of $M_{\alpha }$ and a locally integrable function $b$ is given by $[b,M_{\alpha }](f)=b{M}_{\alpha }(f)-M_{\alpha }(bf)$. In this paper, we mainly give some necessary and sufficient conditions for the boundedness of $[b,M_{\alpha }]$ on variable Lebesgue spaces when $b$ belongs to Lipschitz or $BMO(\text{rn})$ spaces, by which some new characterizations for certain subclasses of Lipschitz and $BMO(\text{rn})$ spaces are obtained.