On a new method of proving Gevrey hypoellipticity for certain sums of squares

We consider an operator being a sum of squares of vector fields. It has the form, p,r∈Np,r∈N,P(x,Dx,Dy,Dt)=Dx2+x2(p−1)(Dy−xrDt)2. This type of operator is C∞C∞ hypoelliptic by Hörmander's theorem, [18]. Its analytic or Gevrey hypoellipticity has then been studied by a number of authors and is relevant in relation to the Treves conjecture. The Poisson–Treves stratification of P includes both symplectic and non-symplectic strata.In this paper we show that P   is Gevrey (p+r)/p(p+r)/p hypoelliptic, by constructing a parametrix whose symbol belongs to some exotic classes. One can also show that this number is optimal.