Regularity near the characteristic set in the non-linear Dirichlet problem and conformal geometry of sub-Laplacians on Carnot groups

Abstract

This paper constitutes the first part of a project devoted to the study of a class of nonlinear sub-elliptic problems which arise in function theory on CR manifolds. The infinitesimal groups naturally associated with these problems are non-commutative Lie groups whose Lie algebra admits a stratification. The fundamental role of such groups in analysis was envisaged by E. M. Stein [72] in his address at the Nice International Congress of Mathematicians in 1970, see also the recent monograph [73]. There has been since a tremendous development in the analysis of the so-called stratified nilpotent Lie groups, nowadays also known as Carnot groups, and in the study of the sub-elliptic partial differential equations, both linear and non-linear, which arise in this connection. Despite all the progress, our understanding of a large number of basic questions is not to present day as substantial as one may desire. Such situation is due primarily to the complexity of the underlying sub-Riemannian geometry, on the one hand, and to the considerable obstacles which are imposed by non-commutativity and by the presence of characteristic points on the other. To introduce the problems studied in this paper we recall that a Carnot group G is a simply connected nilpotent Lie group such that its Lie algebra g admits a stratification g = r ⊕ j=1 Vj , with [V1, Vj ] = Vj+1 for 1 ≤ j < r , [V1, Vr ] = {0}.