Characterization of Lipschitz spaces via commutators of the Hardy-Littlewood maximal function

Let $M$ be the Hardy-Littlewood maximal function and $b$ be a locally integrable function. Denote by $M_b$ and $[b,M]$ the maximal commutator and the (nonlinear) commutator of $M$ with $b$. In this paper, the author consider the boundedness of $M_b$ and $[b,M]$ on Lebesgue spaces and Morrey spaces when $b$ belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces are given.