A subelliptic Bourgain–Brezis inequality

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. We then use this to obtain a Gagliardo–Nirenberg inequality for ${\bar{\partial }}_b$ on the Heisenberg group ${\mathbb{H}}^n$.