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

Let 0 ≤ α<n, Mα be the fractional maximal operator, M^ be the sharp maximal operator and b be the locally integrable function. Denote by [b, Mα] and [b, M^] be the commutators of the fractional maximal operator Mα and the sharp maximal operator M^. In this paper, we show some necessary and sufficient conditions for the boundedness of the commutators [b, Mα] 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