Global hypoellipticity and global solvability for vector fields on compact Lie groups

TL;DR

This paper establishes necessary and sufficient conditions for global hypoellipticity and solvability of vector fields on compact Lie groups, extending understanding of regularity properties and normal form reductions.

Contribution

It provides a complete characterization of global hypoellipticity and solvability for vector fields on compact Lie groups, including perturbations and variable coefficient cases.

Findings

Characterization of global hypoellipticity conditions

Analysis of zero-order perturbations and weaker regularity notions

Reduction of variable coefficient operators to normal forms

Abstract

We present necessary and sufficient conditions to have global hypoellipticity and global solvability for a class of vector fields defined on a product of compact Lie groups. In view of Greenfield's and Wallach's conjecture, about the non-existence of globally hypoelliptic vector fields on compact manifolds different from tori, we also investigate different notions of regularity weaker than global hypoellipticity and describe completely the global hypoellipticity and global solvability of zero-order perturbations of our vector fields. We also present a class of vector fields with variable coefficients whose operators can be reduced to a normal form, and we prove that the study of the global properties of such operators is equivalent to the study of the respective properties for their normal forms.