From the introduction: 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 in his address at the Nice International Congress of Mathematicians in 1970 [Actes Congr. Internat. Math. 1970, 1, 173–189 (1971; Zbl 0252.43022)], see also the recent monograph [Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series. 43. Princeton, NJ: Princeton University Press (1993; Zbl 0821.42001)].
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 nonlinear, 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.