Global Solvability for Involutive Systems on Non-Compact Manifolds

TL;DR

This paper provides necessary and sufficient conditions for the closedness of the range of certain first-order differential operators on non-compact manifolds with involutive structures, extending previous compact manifold results to a broader setting.

Contribution

It establishes a comprehensive criterion for the closedness of operator ranges on non-compact manifolds and links global hypoellipticity to this property, generalizing prior work.

Findings

Criteria for closedness of operator ranges on non-compact manifolds

Extension of hypoellipticity results to non-compact settings

Connection between global hypoellipticity and range closedness

Abstract

We establish necessary and sufficient conditions for the closedness of the range of a class of first-order differential operators associated with an involutive structure on $M\times\mathbb{T}^m$, where $M$ is a non-compact manifold satisfying suitable geometric assumptions and $\mathbb{T}^m$ is the $m$-dimensional torus. In addition, we prove that a weaker notion of global hypoellipticity ensures the closedness of the range for differential operators on smooth paracompact manifolds, thereby extending to the non-compact setting a result previously obtained by G.~Ara\'ujo, I.~Ferra, and L.~Ragognette [J. Anal. Math. 148, No. 1, 85-118, 2022] for compact manifolds.