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.

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

[1] Altomani, A.; Medori, C.; Nacinovich, M. The CR structure of minimal orbits in complex flag manifolds, J. Lie Theory, Volume 16 (2006) no. 3, pp. 483-530 | MR | Zbl

[2] Altomani, A.; Medori, C.; Nacinovich, M. Orbits of real forms in complex flag manifolds (to appear) (Ann. Scuola Norm. Sup. Pisa Cl. Sci., preprint, arXiv:math/0611755 )

[3] Araki, S. On root systems and an infinitesimal classification of irreducible symmetric spaces, J. Math. Osaka City Univ, Volume 13 (1962), pp. 1-34 | MR | Zbl

[4] Bloom, T.; Graham, I. A geometric characterization of points of type $m$ on real submanifolds of ${\mathbf{C}}^n$, J. Differential Geometry, Volume 12 (1977) no. 2, pp. 171-182 | MR | Zbl

[5] Bourbaki, N. Éléments de mathématique, Hermann, Paris, 1975 (Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Actualités Scientifiques et Industrielles, No. 1364) | MR

[6] Bove, A.; Derridj, M.; Kohn, J. J.; Tartakoff, D. S. Sums of squares of complex vector fields and (analytic-) hypoellipticity, Math. Res. Lett., Volume 13 (2006) no. 5-6, pp. 683-701 | MR

[7] Christ, M. A remark on sums of squares of complex vector fields, 2005 http://www.citebase.org/abstract?id=oai:arXiv.org:math/0503506

[8] Fefferman, C.; Phong, D. H. The uncertainty principle and sharp Gȧrding inequalities, Comm. Pure Appl. Math., Volume 34 (1981) no. 3, pp. 285-331 | DOI | MR | Zbl

[9] Hebey, E. Sobolev spaces on Riemannian manifolds, Lecture Notes in Mathematics, 1635, Springer-Verlag, Berlin, 1996 | MR | Zbl

[10] Helgason, S. Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, 80, Academic Press, New York, 1978 | MR | Zbl

[11] Hill, C. D.; Nacinovich, M.; V. Ancona and E. Ballico and A. Silva Pseudoconcave $CR$ manifolds, Complex Analysis and Geometry (Lecture Notes in Pure and Applied Mathematics), Volume 173, Marcel Dekker, Inc, New York (1996), pp. 275-297 | MR | Zbl

[12] Hill, C. D.; Nacinovich, M. A weak pseudoconcavity condition for abstract almost $CR$ manifolds, Invent. Math., Volume 142 (2000), pp. 251-283 | DOI | MR | Zbl

[13] Hill, C. D.; Nacinovich, M. Weak pseudoconcavity and the maximum modulus principle, Ann. Mat. Pura Appl. (4), Volume 182 (2003) no. 1, pp. 103-112 | DOI | MR | Zbl

[14] Hörmander, L. Hypoelliptic second order differential equations, Acta Math., Volume 119 (1967), pp. 147-171 | DOI | MR | Zbl

[15] Hörmander, L. The analysis of linear partial differential operators. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 274, Springer-Verlag, Berlin, 1985 (Pseudodifferential operators) | MR | Zbl

[16] Kohn, J. J. Pseudo-differential operators and hypoellipticity, Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971), Amer. Math. Soc., Providence, R.I., 1973, pp. 61-69 | MR | Zbl

[17] Kohn, J. J. Multiplier ideals and microlocalization, Lectures on partial differential equations (New Stud. Adv. Math.), Volume 2, Int. Press, Somerville, MA, 2003, pp. 141-151 | MR | Zbl

[18] Kohn, J. J. Hypoellipticity and loss of derivatives, Ann. of Math. (2), Volume 162 (2005) no. 2, pp. 943-986 (With an appendix by Makhlouf Derridj and David S. Tartakoff) | DOI | MR | Zbl

[19] Kohn, J. J.; Nirenberg, L. Non-coercive boundary value problems, Comm. Pure Appl. Math., Volume 18 (1965), pp. 443-492 | DOI | MR | Zbl

[20] Medori, C.; Nacinovich, M. Algebras of infinitesimal CR automorphisms, J. Algebra, Volume 287 (2005) no. 1, pp. 234-274 | DOI | MR | Zbl

[21] Nacinovich, M. On weakly pseudoconcave CR manifolds, Hyperbolic problems and regularity questions (Trends Math.), Birkhäuser, Basel, 2007, pp. 137-150 | MR | Zbl

[22] Parenti, C.; Parmeggiani, A. On the hypoellipticity with a big loss of derivatives, Kyushu J. Math., Volume 59 (2005) no. 1, pp. 155-230 | DOI | MR | Zbl

[23] Parenti, C.; Parmeggiani, A. A note on Kohn’s and Christ’s examples, Hyperbolic problems and regularity questions (Trends Math.), Birkhäuser, Basel, 2007, pp. 151-158 | MR | Zbl

[24] Wolf, J. A. The action of a real semisimple group on a complex flag manifold. I. Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc., Volume 75 (1969), pp. 1121-1237 | DOI | MR | Zbl