Global Solvability for Involutive Systems on Non-Compact Manifolds

TL;DR

This work characterizes when first-order involutive-differential systems on non-compact product manifolds $M\times\mathbb{T}^m$ have closed range, by linking global solvability to a Diophantine condition on the period matrix $A(\boldsymbol{\omega})$ associated with $H^1_{\partial M}(M)$. It extends prior compact-manifold results to non-compact settings using scattering metrics, partial Fourier expansions, and a robust FS/DFS functional-analytic framework, including projective limits and Hormander-type a priori estimates. A key contribution is the equivalence between global solvability in various topologies and the non-Liouville property of the forms $\boldsymbol{\omega}$, plus a frequency-decomposition approach that reduces the problem to well-understood components such as the exterior derivative on $M$. The paper also develops an abstract closed-range criterion for maps of pairs, enabling a broad application to differential operators on paracompact manifolds with minimal geometric assumptions.

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.~Araújo, I.~Ferra, and L.~Ragognette [J. Anal. Math. 148, No. 1, 85-118, 2022] for compact manifolds.

Theorems & Definitions (75)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• Proposition 2.2
• proof
• Lemma 2.3
• proof
• Remark 2.4
• Theorem 2.5