Complex vector fields and hypoelliptic partial differential operators

Altomani, Andrea; Hill, C. Denson; Nacinovich, Mauro; Porten, Egmont

Annales de l'Institut Fourier, Tome 60 (2010) no. 3, pp. 987-1034.

On prouve une estimation subelliptique pour les systèmes de champs vectoriels complexes sous certaines hypothèses, qui généralisent la condition de pseudoconcavité essentielle pour les variétés CR, introduite pour la première fois par deux des auteurs, et la condition de commutation d’Hörmander pour des champs vectoriels réels.

On donne des applications afin de démontrer l’hypoellipticité de systèmes de premier ordre et d’opérateurs aux dérivées partielles de second ordre.

Finalement, on décrit une classe de variétés CR compactes homogènes pour lesquelles la distribution des champs vectoriels de type $(0,1)$ satisfait une estimation subelliptique.

We prove a subelliptic estimate for systems of complex vector fields under some assumptions that generalize the essential pseudoconcavity for CR manifolds, that was first introduced by two of the authors, and the Hörmander’s bracket condition for real vector fields.

Applications are given to prove the hypoellipticity of first order systems and second order partial differential operators.

Finally we describe a class of compact homogeneous CR manifolds for which the distribution of $(0,1)$ vector fields satisfies a subelliptic estimate.

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