On the converse of Pansu's Theorem

TL;DR

This paper generalizes Pansu's differentiability theorem to Radon measures on Carnot groups, showing that Lipschitz maps are Pansu-differentiable almost everywhere only if the measure is absolutely continuous with respect to Haar measure.

Contribution

It extends Pansu's theorem to a broader class of measures, establishing a new link between differentiability and measure absolute continuity in Carnot groups.

Findings

Lipschitz maps are Pansu-differentiable almost everywhere under certain measures.

Measures for differentiability must be absolutely continuous w.r.t. Haar measure.

Generalization applies to Radon measures on Carnot groups.

Abstract

We provide a suitable generalisation of Pansu's differentiability theorem to general Radon measures on Carnot groups and we show that if Lipschitz maps between Carnot groups are Pansu-differentiable almost everywhere for some Radon measures $\mu$, then $\mu$ must be absolutely continuous with respect to the Haar measure of the group.