Commutators of the maximal and sharp functions with weighted Lipschitz functions

TL;DR

This paper characterizes the boundedness of commutators of the Hardy-Littlewood maximal and sharp functions with weighted Lipschitz functions on weighted Lebesgue spaces, providing new characterizations of these spaces.

Contribution

It introduces new characterizations for weighted Lipschitz spaces and establishes boundedness criteria for maximal and sharp function commutators in weighted Lebesgue spaces.

Findings

Boundedness of $M_b$ and $[b,M]$ characterized for weighted Lipschitz symbols.

New characterizations of weighted Lipschitz spaces are provided.

Results extend to nonlinear commutators of the sharp function.

Abstract

Let $M$ be the Hardy-Littlewood maximal function. Denote by $M_b$ and $[b,M]$ the maximal and the nonlinear commutators of $M$ with a function $b$. The boundedness of $M_b$ and $[b,M]$ on weighted Lebesgue spaces are characterized when the symbols $b$ belong to weighted Lipschitz (weighted Morrey-Campanato) spaces. Some new characterizations for weighted Lipschitz spaces are obtained. Similar results are also established for the nonlinear commutator of the sharp function.