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

We prove a sharp Gevrey hypoellipticity for the operator D-x(2) + (x(2n+1)D(y))(2)+ (x(n)y(m)D(y))(2),in omega open neighborhood of the origin in R-2, where n and m are positive integers. The operator is a non trivial generalization of the M & eacute;tivier operator studied in M & eacute;tivier (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.