Characterization of Lipschitz Space via the Commutators of Fractional Maximal Functions on Variable Lebesgue Spaces

TL;DR

This paper characterizes variable Lipschitz spaces through the boundedness of commutators involving fractional maximal functions in variable Lebesgue spaces, linking pointwise and integral definitions of the space.

Contribution

It provides new characterizations of variable Lipschitz spaces using commutators of fractional maximal functions in variable Lebesgue spaces.

Findings

Characterization of variable Lipschitz spaces via commutators.

Equivalence between pointwise and integral variable Lipschitz spaces.

Boundedness criteria for fractional maximal commutators.

Abstract

We obtain some new characterizations of a variable version of Lipschitz spaces in terms of the boundedness of commutators of sharp maximal functions, fractional maximal functions or fractional maximal commutators in the context of the variable Lebesgue spaces, where the symbols of the commutators belong to the variable Lipschitz space. A useful tool is that a symbol $b$ belongs a variable Lipschitz space of pointwise type if and only if $b$ belongs to a variable Lipschitz space of integral type.