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

Abstract

We study the Gevrey hypoellipticity of the sum of two squares in two variables, satisfying the Hörmander hypothesis: Dx2+f(x,y)Dy2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ D_{x}^{2} + \left( f(x, y) D_{y}\right) ^{2} $$\end{document}assuming that the function f, describing (the space projection of) its characteristic variety, vanishes at x=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x=0$$\end{document}. A hypoellipticity result is given in a Gevrey class, depending on f. In some examples the found regularity is known to be optimal.