Lipschitz spaces over non-porous sets

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-Carathéodory metric.

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