Lipschitz spaces over non-porous sets

TL;DR

This paper proves that for non-porous subsets of Euclidean space, the Lipschitz space over the set is linearly isomorphic to the Lipschitz space over the entire space, extending to Carnot groups.

Contribution

It establishes a linear isomorphism between Lipschitz spaces over non-porous sets and the whole space, generalizing to Carnot groups with Carnot-Carathéodory metrics.

Findings

Lipschitz spaces over non-porous sets are linearly isomorphic to those over the ambient space.

The result applies to subsets with positive Lebesgue measure and Carnot groups.

The isomorphism holds under the condition of non-porosity of the set.

Abstract

Let $M$ be a subset of $\mathbb{R}^n$. If $M$ is not porous, in particular if it has positive $n$-dimensional Lebesgue measure, we prove that the Lipschitz spaces $\mathrm{Lip}_0(M)$ and $\mathrm{Lip}_0(\mathbb{R}^n)$ are linearly isomorphic. The result also holds more generally if $\mathbb{R}^n$ is replaced with a Carnot group equipped with its Carnot-Carath\'eodory metric.