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 $$ on the Heisenberg group ${ℍ}^n$.

In questa nota dimostriamo stime asintotiche ottimali per le soluzioni deboli non negative del problema al contorno $$-{\mathrm{\Delta }}_{{\mathbb{H}}^n}u=u^{\left(Q+2\right)/\left(Q-2\right)}in\mathrm{\Omega },u=0\text{ in }\partial \mathrm{\Omega }.$$$-{\mathrm{\Delta }}_{{\mathbb{H}}^n}$