Cylindrical and Horizontal Extensions of Critical Sobolev Type Inequalities and Identities

In this note, we discuss the critical case of the extended Hardy inequality as presented in Badiale and Tarantello (Arch Ration Mech Anal 163:259–293, 2002). Moreover, we obtain critical Sobolev type inequalities with sharp constants and sharp remainder formulae. Furthermore, we show anisotropic versions of these results with any homogeneous quasi-norm. Finally, we also discuss horizontal versions of such results in the setting of stratified Lie groups.