Extension of Lipschitz-type operators on Banach function spaces

TL;DR

This paper extends Lipschitz-type operators on Banach function spaces by developing measure-theoretic inequalities and using interpolation methods to broaden their applicability.

Contribution

It introduces a measure-theoretic framework for Lipschitz operators on integrable function spaces and extends these results to interpolated Banach spaces.

Findings

Established measure-theoretic inequalities for Lipschitz operators.

Extended Lipschitz operator theory to interpolated Banach function spaces.

Provided a new framework for analyzing Lipschitz-type inequalities in Banach spaces.

Abstract

We study extension theorems for Lipschitz-type operators acting on metric spaces and with values on spaces of integrable functions. Pointwise domination is not a natural feature of such spaces, and so almost everywhere inequalities and other measure-theoretic notions are introduced.%more appropriate. Thus, we adapt the classical definition of Lipschitz map to the context of spaces of integrable functions by introducing such elements. We analyze Lipschitz type inequalities in two fundamental cases. The first concerns a.e. pointwise inequalities, while the second considers dominations involving integrals. These Lipschitz type inequalities provide the suitable frame to work with operators that take values on Banach function spaces. In the last part of the paper we use some interpolation procedures to extend our study to interpolated Banach function spaces.