Characterization of Lipschitz functions in terms of variable exponent Lebesgue spaces

TL;DR

This paper characterizes Lipschitz functions using variable exponent Lebesgue spaces, providing new insights into their boundedness and relationships with maximal functions, and establishing equivalences between Lipschitz and Lebesgue norms.

Contribution

It introduces novel characterizations of Lipschitz spaces via variable exponent Lebesgue spaces and explores boundedness of maximal and nonlinear commutators in this context.

Findings

New characterizations of Lipschitz spaces obtained.

Boundedness criteria for maximal and nonlinear commutators established.

Equivalence relations between Lipschitz and Lebesgue norms derived.

Abstract

Our aim is to characterize the Lipschitz functions by variable exponent Lebesgue spaces. We give some characterizations of the boundedness of the maximal or nonlinear commutators of the Hardy-Littlewood maximal function and sharp maximal function in variable exponent Lebesgue spaces when the symbols $b$ belong to the Lipschitz spaces, by which some new characterizations of Lipschitz spaces and nonnegative Lipschitz functions are obtained. Some equivalent relations between the Lipschitz norm and the variable exponent Lebesgue norm are also given.