Gevrey Hypo-Ellipticity for Sums of Squares of Vector Fields in R2 with Quasi-Homogeneous Polynomial Vanishing

Analytic and Gevrey hypo-ellipticity are studied for operators of the form P(x,y,D-x,D-y) = D-x(2) + Sigma(j=1)(P-j(X, Y)D-y)(2), in R-2. We assume that the vector fields D-x and p(j)(x,y)D-y satisfy Hormander's condition, that is, that they as well as their Poisson brackets generate a two-dimensional vector space. It is also assumed that the polynomials p(j) are quasi-homogeneous of degree m(j), that is, that p(j) (lambda x, lambda(theta)y) = lambda(mj) p(j) (x, y), for every positive number lambda. We prove that if the associated Poisson-Treves stratification is not symplectic, then P is Gevrey S hypo-elliptic for an s which can be explicitly computed. On the other hand, if the stratification is symplectic, then P is analytic hypo-elliptic.