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 $\dbarb$ on the Heisenberg group $\mathbb{H}^n$.