Porosity, Differentiability and 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.