On the Gevrey hypo-ellipticity of sums of squares of vector fields
Annales de l'Institut Fourier, Volume 54 (2004) no. 5, pp. 1443-1475.

The article studies a second-order linear differential operator of the type -L= X 1 2 ++X r 2 , i. e., a sum of squares of real, and real-analytic, vector fields X i . The conjectured necessary and sufficient condition for analytic hypo-ellipticity, based on the Poisson stratification of the characteristic variety, is recalled in simple analytic and geometric terms. It is conjectured that the microlocal Gevrey hypo-ellipticity of L depends on the restrictions of the principal symbol σL to 2D or 4D symplectic manifolds associated to each bicharateristic curve in a nonsymplectic stratum.

On étudie un opérateur différentiel du second ordre du type -L= X 1 2 ++X r 2 , où les X i sont des champs vectoriels réels et analytiques. On décrit, en termes analytiques et géométriques simples, la stratification de Poisson de la variété caractéristique de L et on rappelle la conjecture selon laquelle une condition nécessaire et suffisante pour l’hypo-ellipticité analytique de L serait que chaque strate de Poisson soit symplectique. Les auteurs formulent une conjecture nouvelle sur l’hypo-ellipticité Gevrey de L selon laquelle cette propriété dépendrait de la restriction du symbole principal σL à certaines sous-variétés bi- ou quadri-dimensionnelles contenant une courbe bicaratéristique d’une strate non symplectique.

DOI: 10.5802/aif.2055
Classification: 35H05,  35A20
Bove, Antonio 1; Treves, François 

1 Università di Bologna, Dipartimento di Matematica, Piazza di porta S. Donato 5, 40127 Bologna (Italy), Rutgers University, Department of Mathematics, 110 Frelinghuysen RD, Piscataway, N.J. 08854-8019 (USA)
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Bove, Antonio; Treves, François. On the Gevrey hypo-ellipticity of sums of squares of vector fields. Annales de l'Institut Fourier, Volume 54 (2004) no. 5, pp. 1443-1475. doi : 10.5802/aif.2055. http://www.numdam.org/articles/10.5802/aif.2055/

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