Sobolev extensions of Lipschitz mappings into metric spaces

TL;DR

This paper develops methods to extend Lipschitz mappings into the Heisenberg group and similar metric spaces to Sobolev mappings without dimension restrictions, broadening the scope of extension theory.

Contribution

It introduces a general approach for Sobolev extensions of Lipschitz maps into the Heisenberg group and Lipschitz $(n-1)$-connected metric spaces, removing previous dimension constraints.

Findings

Sobolev extensions are possible for Lipschitz maps into the Heisenberg group.

Extension results hold for mappings into Lipschitz $(n-1)$-connected metric spaces.

No restriction on the dimension of the domain for Sobolev extension.

Abstract

Wenger and Young proved that the pair $(\mathbb{R}^m,\mathbb{H}^n)$ has the Lipschitz extension property for $m \leq n$ 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.