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

Bove, Antonio; Treves, François

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

On étudie un opérateur différentiel du second ordre du type $-L=$ $X_1^2+\cdots +X_r^2$, 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 $\sigma \left(L\right)$ 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 $-L=$ $X_1^2+\cdots +X_r^2$, 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 $\sigma \left(L\right)$ to $2D$ or $4D$ symplectic manifolds associated to each bicharateristic curve in a nonsymplectic stratum.

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