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

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