Isometric representation of Lipschitz-free spaces over convex domains in finite-dimensional spaces

TL;DR

This paper establishes a canonical isometric isomorphism between Lipschitz-free spaces over convex domains in finite-dimensional spaces and a quotient of $L^1$ vector fields with divergence constraints, providing a new geometric insight.

Contribution

It introduces a novel isometric representation of Lipschitz-free spaces over convex domains as a quotient of divergence-free vector fields in $L^1$, linking geometric and functional analytic structures.

Findings

Lipschitz-free space over convex domain is isometric to a quotient of $L^1$ vector fields

Characterizes the structure of Lipschitz-free spaces in finite-dimensional convex settings

Provides a new perspective on the geometry of Lipschitz-free spaces

Abstract

Let $E$ be a finite-dimensional normed space and $\Omega$ a nonempty convex open set in $E$. We show that the Lipschitz-free space of $\Omega$ is canonically isometric to the quotient of $L^1(\Omega,E)$ by the subspace consisting of vector fields with zero divergence in the sense of distributions on $E$.