Complex vector fields and hypoelliptic partial differential operators

Andrea Altomani[1]; C. Denson Hill[2]; Mauro Nacinovich[3]; Egmont Porten[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)

Annales de l’institut Fourier (2010)

  • Volume: 60, Issue: 3, page 987-1034
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Altomani, Andrea, et al. "Complex vector fields and hypoelliptic partial differential operators." Annales de l’institut Fourier 60.3 (2010): 987-1034. <http://eudml.org/doc/116298>.

@article{Altomani2010,
abstract = {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.},
affiliation = {University of Luxembourg Research Unity in Mathematics 162a, avenue de la Faïencerie 1511 Luxembourg (Luxembourg); Stony Brook University Department of Mathematics Stony Brook, NY 11794 (USA); II Università di Roma “Tor Vergata” Dipartimento di Matematica Via della Ricerca Scientifica 00133 Roma (Italy); Sweden University Department of Mathematics 85170 Sundsvall (Sweden)},
author = {Altomani, Andrea, Hill, C. Denson, Nacinovich, Mauro, Porten, Egmont},
journal = {Annales de l’institut Fourier},
keywords = {Complex distribution; subelliptic estimate; hypoellipticity; Levi form; CR manifold; pseudoconcavity; flag manifold; complex distribution; subellipticity; hypoellipticity, Levi form; CR manifolds},
language = {eng},
number = {3},
pages = {987-1034},
publisher = {Association des Annales de l’institut Fourier},
title = {Complex vector fields and hypoelliptic partial differential operators},
url = {http://eudml.org/doc/116298},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Altomani, Andrea
AU - Hill, C. Denson
AU - Nacinovich, Mauro
AU - Porten, Egmont
TI - Complex vector fields and hypoelliptic partial differential operators
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 3
SP - 987
EP - 1034
AB - 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.
LA - eng
KW - Complex distribution; subelliptic estimate; hypoellipticity; Levi form; CR manifold; pseudoconcavity; flag manifold; complex distribution; subellipticity; hypoellipticity, Levi form; CR manifolds
UR - http://eudml.org/doc/116298
ER -

References

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  1. A. Altomani, C. Medori, M. Nacinovich, The CR structure of minimal orbits in complex flag manifolds, J. Lie Theory 16 (2006), 483-530 Zbl1120.32023MR2248142
  2. A. Altomani, C. Medori, M. Nacinovich, Orbits of real forms in complex flag manifolds, (to appear) Zbl1198.53051
  3. S. Araki, On root systems and an infinitesimal classification of irreducible symmetric spaces, J. Math. Osaka City Univ 13 (1962), 1-34 Zbl0123.03002MR153782
  4. T. Bloom, I. Graham, A geometric characterization of points of type m on real submanifolds of C n , J. Differential Geometry 12 (1977), 171-182 Zbl0436.32013MR492369
  5. N. Bourbaki, Éléments de mathématique, (1975), Hermann, Paris Zbl0329.17002MR453824
  6. A. Bove, M. Derridj, J. J. Kohn, D. S. Tartakoff, Sums of squares of complex vector fields and (analytic-) hypoellipticity, Math. Res. Lett. 13 (2006), 683-701 Zbl1220.35021MR2280767
  7. M. Christ, A remark on sums of squares of complex vector fields, (2005) 
  8. C. Fefferman, D. H. Phong, The uncertainty principle and sharp Gȧrding inequalities, Comm. Pure Appl. Math. 34 (1981), 285-331 Zbl0458.35099MR611747
  9. E. Hebey, Sobolev spaces on Riemannian manifolds, 1635 (1996), Springer-Verlag, Berlin Zbl0866.58068MR1481970
  10. S. Helgason, Differential geometry, Lie groups, and symmetric spaces, 80 (1978), Academic Press, New York Zbl0451.53038MR514561
  11. C. D. Hill, M. Nacinovich, Pseudoconcave C R manifolds, Complex Analysis and Geometry 173 (1996), 275-297, Marcel Dekker, Inc, New York Zbl0921.32004MR1365978
  12. C. D. Hill, M. Nacinovich, A weak pseudoconcavity condition for abstract almost C R manifolds, Invent. Math. 142 (2000), 251-283 Zbl0973.32018MR1794063
  13. C. D. Hill, M. Nacinovich, Weak pseudoconcavity and the maximum modulus principle, Ann. Mat. Pura Appl. (4) 182 (2003), 103-112 Zbl1098.35045MR1970466
  14. L. Hörmander, Hypoelliptic second order differential equations, Acta Math. 119 (1967), 147-171 Zbl0156.10701MR222474
  15. L. Hörmander, The analysis of linear partial differential operators. III, 274 (1985), Springer-Verlag, Berlin Zbl0601.35001MR781536
  16. J. J. Kohn, Pseudo-differential operators and hypoellipticity, Partial differential equations (Proc. Sympos. Pure Math., Vol. XXIII, Univ. California, Berkeley, Calif., 1971) (1973), 61-69, Amer. Math. Soc., Providence, R.I. Zbl0262.35007MR338592
  17. J. J. Kohn, Multiplier ideals and microlocalization, Lectures on partial differential equations 2 (2003), 141-151, Int. Press, Somerville, MA Zbl1062.35010MR2055843
  18. J. J. Kohn, Hypoellipticity and loss of derivatives, Ann. of Math. (2) 162 (2005), 943-986 Zbl1107.35044MR2183286
  19. J. J. Kohn, L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965), 443-492 Zbl0125.33302MR181815
  20. C. Medori, M. Nacinovich, Algebras of infinitesimal CR automorphisms, J. Algebra 287 (2005), 234-274 Zbl1132.32013MR2134266
  21. M. Nacinovich, On weakly pseudoconcave CR manifolds, Hyperbolic problems and regularity questions (2007), 137-150, Birkhäuser, Basel Zbl1125.32015MR2298789
  22. C. Parenti, A. Parmeggiani, On the hypoellipticity with a big loss of derivatives, Kyushu J. Math. 59 (2005), 155-230 Zbl1076.35138MR2134059
  23. C. Parenti, A. Parmeggiani, A note on Kohn’s and Christ’s examples, Hyperbolic problems and regularity questions (2007), 151-158, Birkhäuser, Basel Zbl1124.35015MR2298790
  24. J. A. Wolf, The action of a real semisimple group on a complex flag manifold. I. Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc. 75 (1969), 1121-1237 Zbl0183.50901MR251246

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