Sobolev Extensions of Lipschitz Mappings into Metric Spaces

Wenger and Young proved that the pair (R-m, H-n) has the Lipschitz extension property for m <= n where H-n is the sub-Riemannian Heisenberg group. That is, for some C > 0, any L-Lipschitz map from a subset of R-m into H-n can be extended to a CL-Lipschitz mapping on R-m. In this article, we construct Sobolev extensions of such Lipschitz mappings with no restriction on the dimensionm. We prove that any Lipschitz mapping from a compact subset of R-m into 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.