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

TL;DR

This work provides a unified isometric framework for Lipschitz-free spaces of domains $(Ω,d)$ embedded in a finite-dimensional host $E$. It identifies Lip_0(M) with a subspace $Y$ of $L^{\infty}(Ω;E^{*})$ via the gradient, and shows 𝔽(M) is canonically isometric to $L^{1}(Ω;E)/X$ where $X = { h ∈ L^{1}(Ω;E) : \mathrm{div}(h) = 0 \text{ in } \mathcal{D}'(E) }$, with a distributional characterization of Dirac functionals $δ(x)$ through $-\mathrm{div}(h) = δ_x - δ_{x_0}$. The core contributions are the mollifier-based, path-integral inverse for the gradient and the duality-based identification of 𝔽(M) as a quotient, which generalize and reconcile prior results for bounded Lipschitz domains and convex domains while avoiding boundary regularity assumptions. The results yield a boundary-agnostic, intrinsic-distance framework that subsumes the convex and Lipschitz-boundary cases as corollaries and opens avenues for extensions to Lipschitz manifolds and other length-space settings.

Abstract

For a domain $Ω$ in a finite-dimensional space $E$, we consider the space $M=(Ω,d)$ where $d$ is the intrinsic distance in $Ω$. We obtain an isometric representation of the space $\mathrm{Lip}_{0}(M)$ as a subspace of $L^{\infty}(Ω;E^{*})$ 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 $L^{1}(Ω;E)$. 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.

Figures (1)

• Figure 1: $\Omega$ and its closure in $\mathbb{R}^{2}$ vs $M$ and its completion embedded in $\mathbb{R}^{3}$

Theorems & Definitions (30)

• Theorem
• Theorem 2.1
• proof
• Remark 2.2
• Lemma 2.3
• proof
• Proposition 2.4
• proof
• Proposition 2.5
• Lemma 2.6: Poincaré for $1$-forms