Characterization of Lipschitz spaces via commutators of the Hardy-Littlewood maximal function

TL;DR

This paper explores the boundedness of maximal commutators involving the Hardy-Littlewood maximal function on Lebesgue and Morrey spaces, providing new characterizations of Lipschitz spaces through these operators.

Contribution

It introduces novel characterizations of Lipschitz spaces based on the boundedness of commutators of the Hardy-Littlewood maximal function.

Findings

Boundedness of $M_b$ and $[b,M]$ on Lebesgue spaces established.

Boundedness of $M_b$ and $[b,M]$ on Morrey spaces established.

New characterizations of Lipschitz spaces derived from these boundedness results.

Abstract

Let $M$ be the Hardy-Littlewood maximal function and $b$ be a locally integrable function. Denote by $M_b$ and $[b,M]$ the maximal commutator and the (nonlinear) commutator of $M$ with $b$. In this paper, the author consider the boundedness of $M_b$ and $[b,M]$ on Lebesgue spaces and Morrey spaces when $b$ belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces are given.