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 considers 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.