Global Gevrey Hypoellipticity of Involutive Systems on Non-Compact Manifolds

TL;DR

This paper studies the conditions under which certain differential operators on non-compact manifolds exhibit Gevrey regularity, using geometric and analytical tools to establish sharp criteria for hypoellipticity.

Contribution

It introduces a new approach to analyze Gevrey hypoellipticity on non-compact manifolds with involutive structures, including constructing specialized metrics and criteria based on Liouville behavior.

Findings

Established a sharp criterion for Gevrey hypoellipticity based on rationality and Liouville conditions.

Constructed a Gevrey-regular scattering metric on non-compact manifolds.

Applied Hodge theory to obtain Gevrey regularity of harmonic forms.

Abstract

We investigate the global Gevrey hypoellipticity of a class of first-order differential operators associated with tube-type involutive structures on $M\times\mathbb{T}^m$, where $M$ is a non-compact manifold diffeomorphic to the interior of a compact manifold with boundary and $\mathbb{T}^m$ is the $m$-dimensional torus. For $s>1$, we work in Gevrey classes of Roumieu and Beurling type. A key step is the construction, on $M$, of a scattering metric whose coefficients are Gevrey of order $s$ in every analytic chart; this allows us to use Hodge theory and obtain Gevrey regularity for the harmonic forms. Under a natural condition on the defining closed $1$-forms, we obtain a sharp criterion for global Gevrey hypoellipticity in terms of rationality and (Roumieu/Beurling) exponential Liouville behavior.