On the Gevrey hypo-ellipticity of sums of squares of vector fields  [ Hypo-ellipticité Gevrey de sommes de carrés de champs vectoriels ]
Annales de l'Institut Fourier, Tome 54 (2004) no. 5, pp. 1443-1475.

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.

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.

DOI : https://doi.org/10.5802/aif.2055
Classification : 35H05,  35A20
Mots clés: stratification, symplectique, sommes de carrés de champs vectoriels, hypo-ellipticité analytique et Gevrey
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     author = {Bove, Antonio and Treves, Fran\c cois},
     title = {On the Gevrey hypo-ellipticity of sums of squares of vector fields},
     journal = {Annales de l'Institut Fourier},
     pages = {1443--1475},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {54},
     number = {5},
     year = {2004},
     doi = {10.5802/aif.2055},
     zbl = {1073.35067},
     mrnumber = {2127854},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2004__54_5_1443_0/}
}
Bove, Antonio; Treves, François. On the Gevrey hypo-ellipticity of sums of squares of vector fields. Annales de l'Institut Fourier, Tome 54 (2004) no. 5, pp. 1443-1475. doi : 10.5802/aif.2055. https://aif.centre-mersenne.org/item/AIF_2004__54_5_1443_0/

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