Primitives of volume forms in Carnot groups

TL;DR

This paper extends a classical Euclidean result to Carnot groups, showing that functions with zero average can be represented as divergence of vector fields in these groups, with bounds depending on the group's homogeneous dimension.

Contribution

It proves a divergence representation theorem for functions in Carnot groups, generalizing Euclidean results to a broader geometric setting.

Findings

Established divergence representation in Carnot groups for functions with zero average.

Extended Euclidean divergence results to non-commutative, stratified Lie groups.

Provided bounds depending on the homogeneous dimension Q.

Abstract

In the Euclidean space it is known that a function $f\in L^2$ of a ball, with vanishing average,is the divergence of a vector field $F\in L^2$ with$$\| F\|\_{ L^2(B)} \le C \|f\|\_{L^2(B)}.$$In this Note we prove a similar result in any Carnot group $\mathbb{G}$ for a vanishing average $f\in L^p$, $1\le p < Q$, where $Q$ is the so-called homogeneous dimension of $\mathbb{G}$.