On some intrinsic differentiability properties for Absolutely continuous functions between Carnot groups and the Area formula

TL;DR

This paper investigates the regularity and differentiability properties of absolutely continuous functions between Carnot groups, establishing new results on the relationship between Sobolev regularity and intrinsic differentiability, including an area formula extension.

Contribution

It extends Stein's classical result by proving that Sobolev maps with bounded horizontal gradients are Q-absolutely continuous, and explores intrinsic differentiability properties of such functions between Carnot groups.

Findings

Q-absolutely continuous functions are continuous and Pansu differentiable a.e.

Sobolev maps with bounded horizontal gradient admit a representative that is Q-absolutely continuous.

Extension of the Area formula to these classes of functions.

Abstract

We discuss Q-absolutely continuous functions between Carnot groups, following Maly's definition for maps of several variables. 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.