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
Categories: math.FA math.CA