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 in R2. We assume that the vector fields Dx and pj(x,y)Dy 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 pj are quasi-homogeneous of degreemj, that is, that pj(λx, λèy) = λmjpj(x,y), for every positive number λ. 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.