Critical Sobolev-Type Identities and Inequalities on Stratified Lie Groups

In this note, we discuss the critical case of the extended Hardy inequality obtained by Badiale and Tarantello (Arch. Ration. Mech. Anal. 163:259–293, 2002). Moreover, weighted versions of such inequalities, the so-called weighted critical Sobolev-type inequalities, with sharp constants are established. Actually, we obtain weighted critical Sobolev-type identities. Furthermore, anisotropic versions of these identities with any homogeneous quasi-norm are presented. Finally, we discuss hypoelliptic versions of these results in the setting of stratified Lie groups.