An Isometric Representation for the Lipschitz-Free Space of Length Spaces Embedded in Finite-Dimensional Spaces

TL;DR

This paper provides an isometric representation of Lipschitz-free spaces over length spaces embedded in finite-dimensional spaces, extending previous results and offering new insights into their structure.

Contribution

It introduces a novel isometric representation of Lipschitz-free spaces for length spaces in finite-dimensional spaces, generalizing existing results for specific domain types.

Findings

Isometric representation of (M) as a subspace of L^\u221e(;E^*)

Representation of (M) as a quotient of L^1(;E)

Comparison with prior results for Lipschitz and convex domains

Abstract

For a domain $\Omega$ in a finite-dimensional space $E$, we consider the space $M=(\Omega,d)$ where $d$ is the intrinsic distance in $\Omega$. We obtain an isometric representation of the space $\mathrm{Lip}_{0}(M)$ as a subspace of $L^{\infty}(\Omega;E^{*})$ and we use this representation in order to obtain the corresponding isometric representation for the Lipschitz-free space $\mathcal{F}(M)$ as a quotient of the space $L^{1}(\Omega;E)$. We compare our result with those existent in the literature for bounded domains with Lipschitz boundary, and for convex domains, which can be then deduced as a corollaries of our result.