On some intrinsic differentiability properties for absolutely continuous functions between Carnot groups and the area formula

Abstract

We discuss Q-absolutely continuous functions between Carnot groups, following Malý's definition for maps of several variables ([43]). Such maps enjoy nice regularity properties, like continuity, Pansu differentiability a.e., weak differentiability and an area formula. Furthermore, we extend Stein's result concerning the sharp condition for continuity and differentiability a.e. of a Sobolev map in terms of the integrability of the weak gradient: more precisely, we prove that a Sobolev map between Carnot groups with horizontal gradient of its sections uniformly bounded in $L^{Q,1}$ admits a representative which is Q-absolutely continuous.