Subelliptic Bourgain-Brezis Estimates on Groups

TL;DR

This paper extends Bourgain and Brezis' Euclidean results to stratified, nilpotent groups, demonstrating that divergence-free L^1 vector fields are dual to functions with L^Q sub-gradients, where Q is the group's homogeneous dimension.

Contribution

It generalizes Bourgain-Brezis estimates from Euclidean spaces to stratified, nilpotent groups, establishing a new duality result in this setting.

Findings

Divergence-free L^1 vector fields are in the dual space of functions with L^Q sub-gradients.

Extension of Bourgain-Brezis estimates to non-Euclidean, group-theoretic contexts.

Abstract

We show that divergence free vector fields which belong to L^1 on stratified, nilpotent groups are in the dual space of functions whose sub-gradient are L^Q integrable where Q is the homogeneous dimension of the group. This was first obtained on Euclidean space by Bourgain and Brezis.