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

Cordaro, Paulo D.; Hanges, Nicholas

doi:10.5802/aif.2442

A New Proof of Okaji’s Theorem for a Class of Sum of Squares Operators
[Une Nouvelle Preuve du Théorème d’Okaji pour une Classe d’Opérateurs “Somme de Carrés”]

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)

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

Résumé
Abstract

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.

MR Zbl

DOI : 10.5802/aif.2442

Classification : 35H10, 35H20, 35A17, 35A20, 35A27
Keywords: Analytic hypoelliptic, sum of squares
Mot clés : hypoelliptique analytique, somme de carrés

Affiliations des auteurs :

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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     title = {A {New} {Proof} of {Okaji{\textquoteright}s} {Theorem}  for a {Class} of {Sum} of {Squares} {Operators}},
     journal = {Annales de l'Institut Fourier},
     pages = {595--619},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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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, Tome 59 (2009) no. 2, pp. 595-619. doi : 10.5802/aif.2442. https://aif.centre-mersenne.org/articles/10.5802/aif.2442/

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