Sobolev extensions of Lipschitz mappings into metric spaces

Abstract

Wenger and Young proved that the pair $({\mathbb{R}}^m,{\mathbb{H}}^n)$ has the Lipschitz extension property for $m\le n$ where ${\mathbb{H}}^n$ is the sub-Riemannian Heisenberg group. That is, for some $C>0$, any $L$-Lipschitz map from a subset of ${\mathbb{R}}^m$ into ${\mathbb{H}}^n$ can be extended to a $CL$-Lipschitz mapping on ${\mathbb{R}}^m$. In this paper, we construct Sobolev extensions of such Lipschitz mappings with no restriction on the dimension $m$. We prove that any Lipschitz mapping from a compact subset of ${\mathbb{R}}^m$ into ${\mathbb{H}}^n$ may be extended to a Sobolev mapping on any bounded domain containing the set. More generally, we prove this result in the case of mappings into any Lipschitz $(n-1)$-connected metric space.

arXiv comment20 pages. In this version, more are details included and proofs explained at referee request. The statements of results are expanded upon