Carnot rectifiability and Alberti representations

TL;DR

This paper characterizes Carnot-rectifiable metric measure spaces through Alberti representations and differentiability of Lipschitz maps into Carnot groups, introducing Pansu differentiability spaces as a new framework.

Contribution

It provides multiple characterizations of Carnot-rectifiability using Alberti representations and differentiability, and develops the concept of Pansu differentiability spaces.

Findings

Characterization of Carnot-rectifiable spaces via Alberti representations.

Development of Pansu differentiability spaces as an analogue of Lipschitz differentiability spaces.

Establishment of differentiability criteria using Carnot groups and Pansu derivatives.

Abstract

A metric measure space is said to be Carnot-rectifiable if it can be covered up to a null set by countably many biLipschitz images of compact sets of a fixed Carnot group. In this paper, we give several characterisations of such notion of rectifiability both in terms of Alberti representations of the measure and in terms of differentiability of Lipschitz maps with values in Carnot groups. In order to obtain this characterisation, we develop and study the analogue of the notion of Lipschitz differentiability space by Cheeger, using Carnot groups and Pansu derivatives as models. We call such metric measure spaces Pansu differentiability spaces (PDS).