Integral representation and supports of functionals on Lipschitz spaces

TL;DR

This paper explores the relationship between measures and continuous linear functionals on Lipschitz spaces, establishing a unified framework using compactifications and defining supports for all functionals.

Contribution

It introduces a comprehensive approach to characterize functionals on Lipschitz spaces via measures and compactifications, extending support concepts beyond existing frameworks.

Findings

Characterizes continuous functionals via measures on the Samuel compactification.

Defines support for all elements of the dual space with properties similar to those in Lipschitz-free spaces.

Establishes a Jordan-like decomposition for functionals as differences of positive functionals.

Abstract

We analyze the relationship between Borel measures and continuous linear functionals on the space $\mathrm{Lip}_0(M)$ of Lipschitz functions on a complete metric space $M$. In particular, we describe continuous functionals arising from measures and vice versa. In the case of weak$^\ast$ continuous functionals, i.e. members of the Lipschitz-free space $\mathcal{F}(M)$, measures on $M$ are considered. For the general case, we show that the appropriate setting is rather the uniform (or Samuel) compactification of $M$ and that it is consistent with the treatment of $\mathcal{F}(M)$. This setting also allows us to give a definition of support for all elements of $\mathrm{Lip}_0(M)^\ast$ with similar properties to those in $\mathcal{F}(M)$, and we show that it coincides with the support of the representing measure when such a measure exists. We deduce that the members of $\mathrm{Lip}_0(M)^\ast$ that can be expressed as the difference of two positive functionals admit a Jordan-like decomposition into a positive and a negative part.