Complex vector fields and hypoelliptic partial differential operators
[Champs vectoriels complexes et opérateurs aux dérivées partielles hypoelliptiques]
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.

DOI : 10.5802/aif.2545
Classification : 35H20, 35H10, 32V20
Keywords: Complex distribution, subelliptic estimate, hypoellipticity, Levi form, CR manifold, pseudoconcavity, flag manifold
Mot clés : distribution complexe, estimation subelliptique, hypoellipticité, forme de Levi, variété CR, pseudo-concavité
Altomani, Andrea 1 ; Hill, C. Denson 2 ; Nacinovich, Mauro 3 ; Porten, Egmont 4

1 University of Luxembourg Research Unity in Mathematics 162a, avenue de la Faïencerie 1511 Luxembourg (Luxembourg)
2 Stony Brook University Department of Mathematics Stony Brook, NY 11794 (USA)
3 II Università di Roma “Tor Vergata” Dipartimento di Matematica Via della Ricerca Scientifica 00133 Roma (Italy)
4 Sweden University Department of Mathematics 85170 Sundsvall (Sweden)
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Altomani, Andrea; Hill, C. Denson; Nacinovich, Mauro; Porten, Egmont. Complex vector fields and hypoelliptic partial differential operators. Annales de l'Institut Fourier, Tome 60 (2010) no. 3, pp. 987-1034. doi : 10.5802/aif.2545. http://archive.numdam.org/articles/10.5802/aif.2545/

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