Existence of a global fundamental solution for H\"ormander operators

We prove that for a simply connected manifold M - and a vast class of non-simply connected manifolds - the existence of a fundamental solution for a differential operator L=∑α∈ Nq r_α· X^α of finite degree over M, follows via a saturation method from the existence of a fundamental solution for the associated "lifted" operator over a group G. Given some smooth complete vector fields X₁,, Xq on M, suppose that they generate a finite dimensional Lie algebra g satisfying the H\"ormander's condition. The simply connected Lie group G is the unique such that Lie(G) g. It has the property that a right G-action exists over M, faithful and transitive, inducing a natural projection E G→ M. We generalize an approach developed by Biagi and Bonfiglioli. In particular we represent the group G as a direct product M× Gᶻ where the model fiber Gᶻ has a group structure. Our approach doesn't need any nilpotency hypothesis on the group G, therefore it broadens the spectrum of cases where techniques of the kind "lifting and approximation", as those introduce by the works of Rothschild, Stein, Goodman, can be applied.