Analytic Hypoellipticity at Non-Symplectic Poisson-Treves Strata for Sums of Squares of Vector Fields

Abstract

We consider an operator $P$ which is a sum of squares of vector fields with analytic coefficients. The operator has a non-symplectic characteristic manifold, but the rank of the symplectic form $\sigma$ is not constant on $ \Char P $. Moreover the Hamilton foliation of the non symplectic stratum of the Poisson-Treves stratification for $P$ consists of closed curves in a ring-shaped open set around the origin. We prove that then $P$ is analytic hypoelliptic on that open set. And we note explicitly that the local Gevrey hypoellipticity for $P$ is $G^{k+1}$ and that this is sharp.

20pp