Abstract

In an article of 1977, Folland built a global solution for the sum of squares operator L generated by homogeneous vector fields respecting the H ̈ormander’s condition. The main idea by Folland, was to consider the Carnot group G generated by the same vector fields, and look at L as the projection of the sub-Laplacian over G. Biagi and Bonfiglioli recently generalized this approach to any differential operator of finite degree, in the case that the Lie algebra g generated by the vector fields is nilpotent.
I present a similar result that applies to any Riemannian manifold M and vector fields respecting the H ̈ormander’s condition and such that g is finitely generated. Therefore, I’m dropping the nilpotency hypothesis, maintaining the representation as a direct product of the Lie group associated to g. This allows to expand the saturation approach of Biagi-Bonfiglioli and therefore to project on M a vast class of fundamental solutions, also transposing some estimate tools that behave well with respect to saturation.