A New Proof of Okaji’s Theorem for a Class of Sum of Squares Operators

Cordaro, Paulo D.; Hanges, Nicholas

Annales de l'Institut Fourier, Tome 59 (2009) no. 2, pp. 595-619.

Soit $P$ un opérateur différentiel analytique, de la forme “somme de carrés”, avec la condition d’Hörmander réalisée. Soit $q$ un point caractéristique de $P$. On suppose que $q$ est un point d’un “symplectic Poisson stratum” de codimension deux (au sens de Treves). D’après le théorème d’Okaji, $P$ est hypoelliptique analytique en $q$. Autrement dit, la conjecture de Treves est vraie en codimension deux. On donne dans ce travail une preuve élémentaire de ce fait.

Let $P$ be a linear partial differential operator with analytic coefficients. We assume that $P$ is of the form “sum of squares”, satisfying Hörmander’s bracket condition. Let $q$ be a characteristic point for $P$. We assume that $q$ lies on a symplectic Poisson stratum of codimension two. General results of Okaji show that $P$ is analytic hypoelliptic at $q$. Hence Okaji has established the validity of Treves’ conjecture in the codimension two case. Our goal here is to give a simple, self-contained proof of this fact.

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