Lipschitz and bilipschitz maps on Carnot groups (Q2391403)

In [Rev. Mat. Iberoam. 4, No. 1, 73--114 (1988; Zbl 0696.42011)] \textit{G.~David} proved that if \(f\) is a Lipschitz function from the unit cube in \(\mathbb R^n\) to a subset of some Euclidean space with positive \(n\)-dimensional Hausdorff measure, then there exists a subset \(K\) of the domain of \(f\) with positive \(n\)-dimensional Hausdorff measure such that \(f\) is bilipschitz on \(K\). In this paper, the author considers Lipschitz and bilipschitz maps on Heisenberg and Carnot groups. It is shown that if \(A\) is an appropriate subset of the \(k\)-th Heisenberg group \(H_k\) corresponding to the unit cube in \(\mathbb R^n\), and \(F\) is a Lipschitz function from \(A\) to another Heisenberg group whose image has positive Hausdorff \((2k+2)\)-dimensional measure, then there exists \(B\subset A\) with positive Hausdorff \((2k+2)\)-dimensional measure such that \(F\) is bilipschitz on \(B\). It is shown that this result is also valid for Carnot groups. Next, the author constructs Lipschitz maps from open sets in Carnot groups to Euclidean space that do not decrease dimension, and finally two counterexamples are discussed to explain why Carnot group structure is necessary for these results.