An extension to non-nilpotent groups of Rothschild-Stein lifting method

In their celebrated paper of 1976, Rothschild and Stein prove a lifting procedure that locally reduces to a free nilpotent Lie algebra any family of smooth vector fields $X_1,\dots ,X_q$, over a manifold $M$. Then, a large class of differential operators can be lifted, and fundamental solutions on the lifted space can be re-projected to fundamental solutions of the given operators on $M$. In case that the Lie algebra $\mathfrak{g}=Lie(X_1,\dots ,X_q)$ is finite dimensional but not nilpotent, this procedure could introduce a strong tilting of the space. In this paper we represent a global construction of a Lie group $G$ associated to $\mathfrak{g}$ that avoid this tilting problem. In particular $Lie(G)\cong \mathfrak{g}$ and a right $G$-action exists over $M$, faithful and transitive, inducing a natural projection $E\text{\hspace{0.17em}⁣}:G\to M$. We represent the group $G$ as a direct product $M\times G^z$ where the model fiber $G^z$ has a group structure. We prove that for any simply connected manifold $M$ – and a vast class of non-simply connected manifolds – a fundamental solution for a differential operator $L={\sum }_{\alpha \in {\mathbb{N}}^q}{r}_{\alpha }\cdot X^{\alpha }$ of finite degree over $M$ can be obtained, via a saturation method, from a fundamental solution for the associated lifted operator over the group $G$. This is a generalization of Biagi and Bonfiglioli analogous result for homogeneous vector fields over $M={\mathbb{R}}^n$.