Tangential Cauchy-Riemann equations on quadratic manifolds

Marco M. Peloso; Fulvio Ricci

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2002)

  • Volume: 13, Issue: 3-4, page 285-294
  • ISSN: 1120-6330

Abstract

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We study the tangential Cauchy-Riemann equations ∂ ¯ b ⁢ u = ω for 0 , q -forms on quadratic C ⁢ R manifolds. We discuss solvability for data ω in the Schwartz class and describe the range of the tangential Cauchy-Riemann operator in terms of the signatures of the scalar components of the Levi form.

How to cite

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Peloso, Marco M., and Ricci, Fulvio. "Tangential Cauchy-Riemann equations on quadratic manifolds." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 13.3-4 (2002): 285-294. <http://eudml.org/doc/252369>.

@article{Peloso2002,
abstract = {We study the tangential Cauchy-Riemann equations $\bar\{\partial\}_\{b\} u = \omega$ for $(0,q)$-forms on quadratic $CR$ manifolds. We discuss solvability for data $\omega$ in the Schwartz class and describe the range of the tangential Cauchy-Riemann operator in terms of the signatures of the scalar components of the Levi form.},
author = {Peloso, Marco M., Ricci, Fulvio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Tangential Cauchy-Riemann complex; Kohn Laplacian; CR manifolds; Global solvability; Hypoellipticity; tangential Cauchy-Riemann complex; global solvability; hypoellipticity},
language = {eng},
month = {12},
number = {3-4},
pages = {285-294},
publisher = {Accademia Nazionale dei Lincei},
title = {Tangential Cauchy-Riemann equations on quadratic manifolds},
url = {http://eudml.org/doc/252369},
volume = {13},
year = {2002},
}

TY - JOUR
AU - Peloso, Marco M.
AU - Ricci, Fulvio
TI - Tangential Cauchy-Riemann equations on quadratic manifolds
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2002/12//
PB - Accademia Nazionale dei Lincei
VL - 13
IS - 3-4
SP - 285
EP - 294
AB - We study the tangential Cauchy-Riemann equations $\bar{\partial}_{b} u = \omega$ for $(0,q)$-forms on quadratic $CR$ manifolds. We discuss solvability for data $\omega$ in the Schwartz class and describe the range of the tangential Cauchy-Riemann operator in terms of the signatures of the scalar components of the Levi form.
LA - eng
KW - Tangential Cauchy-Riemann complex; Kohn Laplacian; CR manifolds; Global solvability; Hypoellipticity; tangential Cauchy-Riemann complex; global solvability; hypoellipticity
UR - http://eudml.org/doc/252369
ER -

References

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  1. Airapetyan, R.A. - Khenkin, G.M., Integral represantation of differential forms on Cauchy-Riemann manifolds and the theory of C ⁢ R -functions. Russian Math. Surveys, 39:3, 1984, 41-118. Zbl0589.32035MR747791
  2. Boggess, A., C ⁢ R Manifolds and the Tangential Cauchy-Riemann Complex. CRC Press, Boca Raton1991. Zbl0760.32001MR1211412
  3. Chen, S. - Shaw, M., Partial Differential Equations in Several Complex Variables. International Press, Providence2001. Zbl0963.32001
  4. Folland, G.B. - Kohn, J.J., The Neumann problem for the Cauchy-Riemann complex. Annals of Math. Studies, 57, Princeton U. Press, Princeton1972. Zbl0247.35093MR461588
  5. Folland, G.B. - Stein, E.M., Estimates for the ∂ ¯ b complex and analysis on the Heisenberg group. Comm. Pure Appl. Math., 27, 1974, 429-522. Zbl0293.35012MR367477
  6. Lewy, H., An example of a smooth differential operator without solution. Ann. Math., 66, 1957, 155-158. Zbl0078.08104MR88629
  7. Peloso, M.M. - Ricci, F., Analysis of the Kohn Laplacian on quadratic C ⁢ R manifolds. Preprint 2001. Zbl1043.32021MR2003351DOI10.1016/S0022-1236(03)00176-9
  8. Rossi, H. - Vergne, M., Group representation on Hilbert spaces defined in terms of ∂ ¯ b -cohomology on the Silov boundary of a Siegel domain. Pac. J. Math., 6, 1976, 193-207. Zbl0354.22018MR422517
  9. Andreotti, A. - Fredricks, G. - Nacinovich, M., On the absence of Poincaré lemma in tangential Chauchy-Riemann complexes. Ann. Scuola Norm. Sup. Pisa, 8, 1981, 365-404. Zbl0482.35061MR634855
  10. Kohn, J.J., Boundary of complex manifolds. Proc. Conf. on Complex Manifolds (Minneapolis, 1964). Springer-Verlag, New York1965, 81-94. Zbl0166.36003MR175149
  11. Treves, F., A remark on the Poincaré lemma in analytic complexes with nondegenerate Levi form. Comm. PDE, 7, 1982, 1467-1482. Zbl0535.32005MR679951DOI10.1080/03605308208820259

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