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

Bove, Antonio; Treves, François

doi:10.5802/aif.2055

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

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)

Annales de l'Institut Fourier, Tome 54 (2004) no. 5, pp. 1443-1475.

Résumé
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.

MR Zbl

DOI : 10.5802/aif.2055

Classification : 35H05, 35A20

Affiliations des auteurs :

Bove, Antonio ¹ ; Treves, François

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     title = {On the {Gevrey} hypo-ellipticity of sums of squares of vector fields},
     journal = {Annales de l'Institut Fourier},
     pages = {1443--1475},
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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, Tome 54 (2004) no. 5, pp. 1443-1475. doi : 10.5802/aif.2055. https://aif.centre-mersenne.org/articles/10.5802/aif.2055/

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