The distributional divergence of horizontal vector fields vanishing at infinity on Carnot groups

TL;DR

This paper introduces a BV-type space in Carnot groups to characterize distributions that are divergence of continuous horizontal vector fields vanishing at infinity, extending Euclidean results to a broader geometric setting.

Contribution

It generalizes Euclidean divergence characterization results to Carnot groups, providing a new framework for analyzing divergence equations in this non-commutative setting.

Findings

Defined a BV-type space in Carnot groups.

Characterized distributions as divergences of horizontal vector fields.

Extended Euclidean divergence results to Carnot group setting.

Abstract

We define a BV -type space in the setting of Carnot groups (i.e., simply connected Lie groups with stratified nilpotent Lie algebra) that allows one to characterize all distributions F for which there exists a continuous horizontal vector field {\Phi}, vanishing at infinity, that solves the equation divH{\Phi} = F. This generalize to the setting of Carnot groups some results by De Pauw and Pfeffer, [12], and by De Pauw and Torres, [13], for the Euclidean setting.