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

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