Porosity, Differentiability and Pansu's Theorem

TL;DR

This paper investigates the differentiability of Lipschitz functions on Carnot groups using porosity, establishing that irregular points form a small set and providing a new proof of Pansu's theorem.

Contribution

It introduces porosity as a tool to analyze Lipschitz differentiability on Carnot groups and offers a novel proof of Pansu's theorem.

Findings

Directional derivatives act linearly outside a $\sigma$-porous set

Irregular points form a $\sigma$-porous set

Provides a new proof of Pansu's theorem

Abstract

We use porosity to study differentiability of Lipschitz maps on Carnot groups. Our first result states that directional derivatives of a Lipschitz function act linearly outside a $\sigma$-porous set. The second result states that irregular points of a Lipschitz function form a $\sigma$-porous set. We use these observations to give a new proof of Pansu's theorem for Lipschitz maps from a general Carnot group to a Euclidean space.