Lipschitz and biLipschitz Maps on Carnot Groups

TL;DR

This paper investigates Lipschitz and biLipschitz maps between Carnot groups, establishing conditions under which Lipschitz functions are biLipschitz on large subsets, and explores the necessity of Carnot group structure through counterexamples.

Contribution

It proves that Lipschitz maps with positive measure images are biLipschitz on large subsets and constructs dimension-preserving Lipschitz maps to Euclidean space, highlighting Carnot group structure importance.

Findings

Lipschitz maps with positive measure images are biLipschitz on large subsets.

Constructed Lipschitz maps from Carnot groups to Euclidean space that preserve dimension.

Provided counterexamples showing the necessity of Carnot group structure.

Abstract

Suppose A is an open subset of a Carnot group G, where G has a discrete analogue, and H is another Carnot group. We show that a Lipschitz function from A to H whose image has positive Hausdorff measure in the appropriate dimension is biLipschitz on a subset of A of positive Hausdorff measure. We then construct Lipschitz maps from open sets in Carnot groups to Euclidean space that do not decrease dimension. Finally, we discuss two counterexamples to explain why Carnot group structure is necessary for these results.