On the sharp Gevrey regularity for a generalization of the Métivier operator

Abstract

We prove a sharp Gevrey hypoellipticity for the operator $$\begin{aligned} D_{x}^{2}+\left( x^{2n+1}D_{y}\right) ^{2}+\left( x^{n}y^{m}D_{y}\right) ^{2}, \end{aligned}$$in \(\Omega \) open neighborhood of the origin in \({\mathbb {R}}^{2}\), where n and m are positive integers. The operator is a non trivial generalization of the Métivier operator studied in Métivier (C R Acad Sci Paris 292:401–404, 1981). However it has a symplectic characteristic manifold and a non symplectic stratum according to the Poisson–Treves stratification. According to Treves conjecture it turns out not to be analytic hypoelliptic.