A detour on Sobolev inequalities in the Heisenberg group and applications to critical subelliptic problems

Abstract: Thanks to important applications to geometric control theory and PDEs, great attention has been recently devoted to the study of geometric inequalities in the context of stratified Lie groups. In the first part of the course we present some classical results and open problems related to the Sobolev inequalities in the Heisenberg group. In the second part, we apply these inequalities to obtain existence results for several nonlinear critical problems involving subelliptic operators in the Heisenberg group, even with non–standard growth conditions, also known as subelliptic (p, q) operators.