On the Converse of Pansu’s Theorem

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 $$\mu $$ μ , then $$\mu $$ μ must be absolutely continuous with respect to the Haar measure of the group.