Characterizations of Sobolev and BV functions on Carnot groups

TL;DR

This paper provides new characterizations of Sobolev and BV functions on Carnot groups using nonlocal gradients and $L^p$ Taylor approximations, extending classical Euclidean results.

Contribution

It introduces two novel characterizations of Sobolev and BV functions on Carnot groups, adapting Euclidean techniques to sub-Riemannian settings.

Findings

Established a nonlocal approximation characterization of horizontal gradients.

Developed an $L^p$ Taylor approximation approach for Sobolev and BV functions.

Extended Bourgain, Brezis, and Mironescu's results to Carnot groups.

Abstract

We establish two characterizations of real-valued Sobolev and BV functions on Carnot groups. The first is obtained via a nonlocal approximation of the distributional horizontal gradient, while the second is based on an $L^p$ Taylor approximation, in the spirit of the results by Bourgain, Brezis and Mironescu.