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

TL;DR

This paper characterizes Lipschitz functions through the boundedness of certain commutators of fractional and sharp maximal operators on slice spaces, providing new insights into Lipschitz function properties.

Contribution

It offers necessary and sufficient conditions for the boundedness of these commutators when the function is Lipschitz, leading to new characterizations of Lipschitz functions.

Findings

Boundedness conditions for commutators on slice spaces

New characterizations of non-negative Lipschitz functions

Connections between Lipschitz continuity and maximal operator commutators

Abstract

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