Global Gevrey Hypoellipticity of Involutive Systems on Non-Compact Manifolds

Let ω 1 , . . . , ω m be real-valued closed 1-forms on M , belonging either to Λ 1 G s (M ) or to Λ 1 G (s) (M ). We study the operator

where T m is the m-dimensional torus, d t is the exterior derivative on M , and [s] stands for s or (s).

Operators of the form (1.1) arise naturally in the study of tube-type involutive structures and may be viewed locally as systems of first-order linear partial differential equations. In the compact case, global properties of (1.1) have been extensively investigated, see, for instance, [1-6, 12, 13]. For the general theory of involutive systems we refer to [7,21].

We recall the notions of global Gevrey hypoellipticity.

Here D ′ (M × T m ) denotes the space of distributions on M × T m .

Throughout the paper we assume s > 1 and that M is diffeomorphic to the interior of a compact manifold with boundary M . Let ϱ : M → [0, +∞) be a boundary defining function on M . A scattering metric on M is a Riemannian metric g which, in a collar neighbourhood of ∂M , has the form

, where g ′ is a smooth symmetric 2-cotensor restricting to a Riemannian metric on ∂M . In this case, (M, g) is said to be a scattering manifold.

The existence of a scattering metric is important in a key step: it provides a suitable version of the Hodge theorem (see [19,Theorem 6.2]), identifying the space of square-integrable harmonic 1-forms with the image of the compactly supported de Rham cohomology inside H 1 dR (M ). With this, in [8] the authors were able to extend the classic results for compact manifolds to this class of non-compact manifolds. By combining an analytic atlas with Gevrey partitions of unity (hence the restriction s > 1), we construct a boundary defining function and a scattering metric whose coefficients are Gevrey of order s. The associated Laplace-Beltrami operator is then elliptic with Gevrey coefficients, which yields Gevrey regularity of harmonic forms and allows us to represent certain cohomology classes by Gevrey harmonic forms.

In the compact case treated in [3], the authors rely on a theorem by Grauert [11] to endow M with an analytic metric, which also permits the treatment of the case s = 1. In the present non-compact setting, however, one needs a metric that simultaneously has Gevrey coefficients and is compatible with the scattering geometry near the boundary in order to apply the Hodge theorem, and Grauert’s theorem does not provide this additional structure. However, assuming the existence of an analytic scattering metric, our results naturally extend to the case s = 1.

Our analysis follows the strategy introduced in [3]. Expanding u in partial Fourier series on T m reduces (1.1) to a family of twisted operators on M , and the obstruction to global Gevrey regularity is governed by small-denominator phenomena. These are encoded by the notions of rationality and of exponential Liouville behavior for the family ω = (ω 1 , . . . , ω m ), expressed in terms of periods along a basis of cycles associated with the image of compactly supported cohomology. We denote by Λ 1 G

[s] ∂M (M ) the space of real-valued closed Gevrey 1-forms whose cohomology class lies in this distinguished subspace of H 1 dR (M ). Our main result is the following characterization. Theorem 1.2. Fix s > 1 and let ω = (ω 1 , . . . , ω m ) be a family of real-valued closed 1-forms in Λ 1 G

[s] ∂M (M ). Then the operator L defined in (1.1) is [s]-globally hypoelliptic if and only if ω is neither rational nor [s]-exponential Liouville.

The paper is organized as follows. In Section 2 we recall the basic properties of Gevrey classes on open subsets of R n and introduce the corresponding Roumieu and Beurling spaces on analytic manifolds, including Gevrey 1-forms and the relevant boundedness criteria. In Section 3 we construct a Gevrey scattering metric on the interior of a compact manifold with boundary and derive Gevrey regularity for harmonic forms via elliptic regularity for the associated Laplace-Beltrami operator. Section 4 develops the cohomological framework needed in the non-compact setting: we define the space Λ 1 G

[s] ∂M (M ), associate to a family of closed forms its matrix of cycles, and relate rationality and exponential Liouville conditions to suitable Diophantine estimates. In Section 5 we study the global Gevrey hypoellipticity of L using partial Fourier series on T m , reducing the problem to a family of twisted equations on M and proving the characterization stated in the main theorem. Finally, for the reader’s convenience, Appendix A collects th