Lipschitz and bilipschitz maps on Carnot groups

William Meyerson

Vol. 263 (2013), No. 1, 143–170

DOI: 10.2140/pjm.2013.263.143

Abstract

Suppose A is an open subset of a Carnot group G 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 also 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.

Keywords

analysis on Carnot groups, Heisenberg groups, Grushin plane, subriemannian, wavelets

Mathematical Subject Classification 2010

Primary: 43A80

Milestones

Received: 27 March 2012

Revised: 14 August 2012

Accepted: 15 August 2012

Published: 14 May 2013