Commutators of some maximal functions with Lipschitz functions on mixed Morrey spaces

Abstract

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