Existence of solutions and regularity near the characteristic boundary for sub-Laplacian equations on Carnot groups

Abstract

We prove that the best constant in the Folland?Stein embedding theorem on Carnot groups is achieved. This implies the existence of a positive solution of the Yamabe-type equation on Carnot groups. The second goal of the paper is to show a certain regularity of the Green?s function and solutions of the Yamabe equation involving the sub-Laplacian near the characteristic boundary of a domain in the considered groups.