Lipschitz functions on submanifolds in Heisenberg groups

TL;DR

This paper investigates the differentiability of Lipschitz functions on submanifolds within Heisenberg groups, establishing almost everywhere Pansu differentiability and providing applications like approximation and a coarea formula.

Contribution

It proves the almost everywhere tangential Pansu differentiability of Lipschitz functions on intrinsic submanifolds in Heisenberg groups and introduces related approximation and coarea results.

Findings

Almost everywhere Pansu differentiability of Lipschitz functions

Lusin-type approximation on $ ext{Heisenberg}$-rectifiable sets

A coarea formula for $ ext{Heisenberg}$-rectifiable sets

Abstract

We study the behavior of Lipschitz functions on intrinsic $C^1$ submanifolds of Heisenberg groups: our main result is their almost everywhere tangential Pansu differentiability. We also provide two applications: a Lusin-type approximation of Lipschitz functions on $\HH$-rectifiable sets, and a coarea formula on $\HH$-rectifiable sets that completes the program started in~\cite{JNGV}.