Global Gevrey Hypoellipticity of Involutive Systems on Non-Compact Manifolds

TL;DR

Let $M$ be the interior of a compact manifold with boundary and consider the operator $\mathbb{L}$ on $M\times\mathbb{T}^m$ given by $\mathbb{L}u = \mathrm{d}_t u + \sum_{k=1}^m \omega_k \wedge \partial_{x_k} u$ with a family of closed $1$-forms $\omega_k$ and $s>1$. The authors construct a Gevrey scattering metric $g$ on $M$ so that the Laplace–Beltrami operator $\Delta_g$ is elliptic with Gevrey coefficients, which, via a Gevrey Hodge theorem, yields Gevrey regularity for harmonic forms and a reduction to de Rham cohomology. They then analyze $\mathbb{L}$ through partial Fourier series on $\mathbb{T}^m$, linking global regularity to a Diophantine condition encoded by the matrix of cycles $A(\boldsymbol{\omega})$ and small-denominator phenomena. The main result provides a sharp criterion: $\mathbb{L}$ is $[s]$-globally hypoelliptic (and likewise in the Beurling sense) if and only if the $\omega_k$ are not rational and not $[s]$-exponential Liouville. This work extends global Gevrey hypoellipticity results to non-compact scattering manifolds and clarifies the arithmetic needed to obtain Gevrey regularity for tube-type involutive systems.

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.

Theorems & Definitions (61)

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