Characterization of Lipschitz functions via commutators of maximal operators on slice spaces

Let 0 <n, M_ be the fractional maximal operator, M^ be the sharp maximal operator and b be the locally integrable function. Denote by b, M_ and b, M^ be the commutators of the fractional maximal operator M_ and the sharp maximal operator M^. In this paper, we show some necessary and sufficient conditions for the boundedness of the commutators b, M_ and b, M^ on slice spaces when the function b is the Lipschitz function, by which some new characterizations of the non-negative Lipschitz function are obtained