A New Proof of Okaji’s Theorem for a Class of Sum of Squares Operators
Annales de l'Institut Fourier, Volume 59 (2009) no. 2, pp. 595-619.

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.

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.

DOI: 10.5802/aif.2442
Classification: 35H10,  35H20,  35A17,  35A20,  35A27
Keywords: Analytic hypoelliptic, sum of squares
Cordaro, Paulo D. 1; Hanges, Nicholas 2

1 Universidade de São Paulo São Paulo, SP (Brazil)
2 Lehman College Department of Mathematics/CUNY Bronx, New York 10468 (USA)
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Cordaro, Paulo D.; Hanges, Nicholas. A New Proof of Okaji’s Theorem  for a Class of Sum of Squares Operators. Annales de l'Institut Fourier, Volume 59 (2009) no. 2, pp. 595-619. doi : 10.5802/aif.2442. http://www.numdam.org/articles/10.5802/aif.2442/

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