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

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 $L^{\infty }(\Omega ;E^{\ast })$ 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.

Revised version