On the Gevrey hypo-ellipticity of sums of squares of vector fields

Bove, Antonio; Treves, François

On the Gevrey hypo-ellipticity of sums of squares of vector fields
[Hypo-ellipticité Gevrey de sommes de carrés de champs vectoriels]

Bove, Antonio ; Treves, François

Annales de l'Institut Fourier, Tome 54 (2004), p. 1443-1475 / Harvested from Numdam

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Abstract

On étudie un opérateur différentiel du second ordre du type $- L =$ $X_{1}^{2} + \dots + X_{r}^{2}$ , 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 $σ (L)$ 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 $- L =$ $X_{1}^{2} + \dots + X_{r}^{2}$ , 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 $σ (L)$ to $2 D$ or $4 D$ symplectic manifolds associated to each bicharateristic curve in a nonsymplectic stratum.

Publié le : 2004-01-01

MR 2127854 | Zbl 1073.35067

DOI : https://doi.org/10.5802/aif.2055
Classification: 35H05, 35A20

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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},
     volume = {54},
     year = {2004},
     pages = {1443-1475},
     doi = {10.5802/aif.2055},
     mrnumber = {2127854},
     zbl = {1073.35067},
     language = {en},
     url = {http://dml.mathdoc.fr/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) pp. 1443-1475. doi : 10.5802/aif.2055. http://gdmltest.u-ga.fr/item/AIF_2004__54_5_1443_0/

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