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

Abstract

Abstract We study the Gevrey hypoellipticity of the sum of two squares in two variables, satisfying the Hörmander hypothesis: $$ D_{x}^{2} + \left( f(x, y) D_{y}\right) ^{2} $$ 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.