Global analytic hypoellipticity and solvability of certain operators subject to group actions

TL;DR

This paper investigates the relationship between global hypoellipticity and analytic hypoellipticity of certain invariant differential operators on product manifolds involving compact Lie groups, establishing conditions under which hypoellipticity implies stronger regularity.

Contribution

It proves that, under mild conditions, global hypoellipticity of invariant elliptic operators on $T imes G$ implies their global analytic and Gevrey hypoellipticity, extending previous results.

Findings

Global hypoellipticity implies Gevrey hypoellipticity for the operators studied.

The paper establishes a connection between hypoellipticity and solvability in a broader setting.

Conditions under which hypoellipticity leads to stronger regularity properties are identified.

Abstract

On $T \times G$, where $T$ is a compact real-analytic manifold and $G$ is a compact Lie group, we consider differential operators $P$ which are invariant by left translations on $G$ and are elliptic in $T$. Under a mild technical condition, we prove that global hypoellipticity of $P$ implies its global analytic-hypoellipticity (actually Gevrey of any order $s \geq 1$). We also study the connection between the latter property and the notion of global analytic (resp. Gevrey) solvability, but in a much more general setup.