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 $\hat{I}¼$, then $\hat{I}¼$ must be absolutely continuous with respect to the Haar measure of the group.