Analytic Hypoellipticity for Sums of Squares and the Treves Conjecture

We are concerned with the problem of real analytic regularity of the solutions of sums of squares with real analytic coefficients. Treves conjecture states that an operator of this type is analytic hypoelliptic if and only if all the strata in the Poisson-Treves stratification are symplectic. We produce a model operator, $P_{1}$, having a single symplectic stratum and prove that it is Gevrey $s_{0}$ hypoelliptic and not better. See Theorem \ref{th:1} for a definition of $s_{0}$. We also show that this phenomenon has a microlocal character. We point out explicitly that this is a counterexample to the sufficient part of Treves conjecture and not to the necessary part, which is still an open problem.