Global solvability and propagation of regularity of sums of squares on compact manifolds

TL;DR

This paper studies the global solvability and regularity propagation of sums of squares of vector fields on compact manifolds, using invariant vector fields on Lie groups to analyze the operators.

Contribution

It introduces a method to analyze solvability via left-invariant vector fields on Lie groups, simplifying the study of sums of squares on compact manifolds.

Findings

Established conditions for global solvability of sums of squares operators.

Proved a general result on propagation of regularity for these operators.

Developed tools linking operator analysis to invariant vector fields on Lie groups.

Abstract

We investigate global solvability, in the framework of smooth functions and Schwartz distributions, of certain sums of squares of vector fields defined on a product of compact Riemannian manifolds $T \times G$, where $G$ is further assumed to be a Lie group. As in a recent article due to the authors, our analysis is carried out in terms of a system of left-invariant vector fields on $G$ naturally associated with the operator under study, a simpler object which nevertheless conveys enough information about the original operator so as to fully encode its solvability. As a welcome side effect of the tools developed for our main purpose, we easily prove a general result on propagation of regularity for such operators.