Regularity for a sum of two squares in two variables with symplectic characteristic manifold

We study the Gevrey hypoellipticity of the sum of two squares in two variables, satisfying the Hörmander hypothesis: Dₓ² + ( f(x, y) Dy) ² D x 2 + f ( x , y ) D y 2 assuming that the function f , describing (the space projection of) its characteristic variety, vanishes at $$x=0$$ x = 0 . A hypoellipticity result is given in a Gevrey class, depending on f . In some examples the found regularity is known to be optimal.