A subelliptic Bourgain-Brezis inequality

TL;DR

This paper generalizes Bourgain-Brezis approximation results to stratified homogeneous groups and applies it to establish a Gagliardo-Nirenberg inequality for the $ar{ullstop}_b$ operator on the Heisenberg group, advancing analysis on non-isotropic spaces.

Contribution

It introduces an approximation lemma for non-isotropic Sobolev spaces on stratified groups and derives a new Gagliardo-Nirenberg inequality for the Heisenberg group.

Findings

Approximation of functions in $ ext{dot} NL^{1,Q}$ by $L^{\infty}$ functions.

Extension of Bourgain-Brezis results to stratified homogeneous groups.

Establishment of a Gagliardo-Nirenberg inequality for $ar{ullstop}_b$ on $ ext{Heisenberg}$ group.

Abstract

We prove an approximation lemma on (stratified) homogeneous groups that allows one to approximate a function in the non-isotropic Sobolev space $\dot{NL}^{1,Q}$ by $L^{\infty}$ functions, generalizing a result of Bourgain-Brezis \cite{MR2293957}. We then use this to obtain a Gagliardo-Nirenberg inequality for $\bar{\partial}_b$ on the Heisenberg group $\mathbb{H}^n$.