Analysis and Geometry in Metric Spaces: Sobolev Mappings, the Heisenberg Group, and the Whitney Extension Theorem

摘要

This thesis focuses on analysis in and the geometry of the Heisenberg group as well as geometric properties of Sobolev mappings. It begins with a detailed introduction to the Heisenberg group. After, we see a new and elementary proof for the structure of geodesics in the sub-Riemannian Heisenberg group. We also prove that the Carnot-Carath'{e}odory metric is real analytic away from the center of the group. ududNext, we prove a version of the classical Whitney Extension Theorem for curves in the Heisenberg group. Given a real valued function defined on a compact set in Euclidean space, the classical Whitney Extension Theorem from 1934 gives necessary and sufficient conditions for the existence of a $C^k$ extension defined on the entire space. We prove a version of the Whitney Extension Theorem for $C^1$, horizontal curves in the Heisenberg group.ududWe then turn our attention to Sobolev mappings.udIn particular, given a Lipschitz map from a compact subset $Z$ of Euclidean space into a Lipschitz connected metric space, we construct a Sobolev extension defined on any bounded domain containing $Z$.ududFinally, we generalize a classical result of Dubovitskiu{i} for smooth maps to the case of Sobolev mappings. In 1957, Duvovitskiu{i} generalized Sard's classical theorem by establishing a bound on the Hausdorff dimension of the intersection of the critical set of a smooth map and almost every one of its level sets. We show that Dubovitskiu{i}'s theorem udcan be generalized to $W_{m loc}^{k,p}(mathbb{R}^n,mathbb{R}^m)$ mappings for all positive integers $k$ and $p>n$.