Sobolev-type Embedding on Cauchy-Riemann Manifolds and Nonlinear Sub-elliptic PDE on Heisenberg Group

Abstract

In this thesis, we study the optimal constants in the Sobolev-type inequality (AB inequality) on compact Cauchy-Riemann manifolds. We apply the results to give new proof to special case of CR Yamabe problem. We also consider the AB inequality for functions with constraints on CR manifolds. We show for those functions the optimal value can be smaller and use the analytic results to touch Nirenberg problem on CR sphere. In the last chapter, we discuss existence of positive solutions to some nonlinear sub-elliptic equations involving critical Sobolev exponents on Heisenberg group.