Wenger and Young proved that the pair $(\mathbb{R}^m,\mathbb{H}^n)$ has theLipschitz extension property for $m \leq n$ where $\mathbb{H}^n$ is thesub-Riemannian Heisenberg group. That is, for some $C>0$, any $L$-Lipschitz mapfrom 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 Sobolevextensions 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 domaincontaining the set. More generally, we prove this result in the case ofmappings into any Lipschitz $(n-1)$-connected metric space.
展开▼
机译：Wenger和Young证明，对$（\ mathbb {R} ^ m，\ mathbb {H} ^ n）$具有$ m \ leq n $的Lipschitz扩展属性，其中$ \ mathbb {H} ^ n $是sub-黎曼海森堡小组。也就是说，对于某些$ C> 0 $，从$ \ mathbb {R} ^ m $的子集到$ \ mathbb {H} ^ n $的任何$ L $ -Lipschitz映射都可以扩展为a $ CL $ -Lipschitz映射到$ \ mathbb {R} ^ m $。在本文中，我们构建了此类Lipschitz映射的Sobolevextensions，而对维度$ m $没有限制。我们证明了任何Lipschitz映射都从$ \ mathbb {R} ^ m $的紧凑子集到$ \ mathbb {H} ^ n $可以扩展到包含该集合的任何有界域上的Sobolev映射。更笼统地说，我们证明了在映射到任何Lipschitz $（n-1）$连接的度量空间的情况下的结果。
展开▼
