Siberian Mathematical Journal

, Volume 44, Issue 4, pp 611–628 | Cite as

On Construction of Exact Complexes Connected with the Dolbeault Complex

  • A. M. Kytmanov
  • S. G. Myslivets


We consider some exact subcomplex of the Dolbeault complex which is constructed by means of the Koppelman integral representation. We give its intrinsic description. Using this subcomplex, we construct a double exact complex one of whose components is the Dolbeault complex.

Dolbeault complex Koppelman integral representation Cauchy–Riemann operator 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Stein E. M., “Singular integrals and estimates for Cauchy-Riemann equations,” Bull. Amer. Math. Soc., 79, No. 2, 440–445 (1973).Google Scholar
  2. 2.
    Calderon A. P., “Boundary value problem for elliptic equations,” Outlines of the Joint Soviet-American Sympos. on PDE's, Novosibirsk, 1963, pp. 303–304.Google Scholar
  3. 3.
    Atiyah M. F., Patodi V. K., and Singer I. M., “Spectral asymmetry and Riemann geometry. I,” Math. Proc. Cambridge Philos. Soc., 77, 43–69 (1975).Google Scholar
  4. 4.
    Kondrat'ev V. A., “Boundary value problems for elliptic equations in domains with conic or angular points,” Trudy Moskov. Mat. Obshch., 16, 209–292 (1967).Google Scholar
  5. 5.
    Schulze B.-W., “An algebra of boundary value problems not requiring Shapiro-Lopatinskj conditions,” J. Funct. Anal., 179, 374–408 (2001).Google Scholar
  6. 6.
    Venugopalkrishna V., “Fredholm operators associated with strongly pseudoconvex domains in Cn,” J. Funct. Anal., 9, 349–373 (1972).Google Scholar
  7. 7.
    Dynin A. S., “Boundary elliptic problems for pseudodifferential complexes,” Funktsional. Anal. i Prilozhen., 6, No. 1, 75–76 (1972).Google Scholar
  8. 8.
    Koppelman W., “The Cauchy integral for differential forms,” Bull. Amer. Math. Soc., 73, No. 4, 554–556 (1967).Google Scholar
  9. 9.
    Wells R., Differential Analysis on Complex Manifolds [Russian translation], Mir, Moscow (1976).Google Scholar
  10. 10.
    A?zenberg L. A. and Dautov Sh. A., Differential Forms Orthogonal to Holomorphic Functions or Forms, and Their Properties [in Russian], Nauka, Novosibirsk (1975).Google Scholar
  11. 11.
    Tarkhanov N. N., The Parametrix Method in the Theory of Differential Complexes [in Russian], Nauka, Novosibirsk (1989).Google Scholar
  12. 12.
    Egorov Yu. V. and Shubin M. A., “Linear partial differential operators. The fundamentals of the classical theory,” in: Contemporary Problems of Mathematics. Fundamental Trends [in Russian], VINITI, Moscow, 1988, 30, pp. 1–262. (Itogi Nauki i Tekhniki.)Google Scholar
  13. 13.
    A?zenberg L. A. and Yuzhakov A. P., Integral Representations and Residues in Multidimensional Complex Analysis [in Russian], Nauka, Novosibirsk (1979).Google Scholar
  14. 14.
    Kohn J. J. and Rossi H., “On extension of holomorphic functions from the boundary of a complex manifold,” Ann. Math., 81, No. 3, 451–472 (1965).Google Scholar
  15. 15.
    Kytmanov A. M., The Bochner-Martinelli Integral and Its Applications [in Russian], Nauka, Novosibirsk (1992).Google Scholar
  16. 16.
    Khenkin G. M., “The method of integral representations in complex analysis,” in: Contemporary Problems of Mathematics. Fundamental Trends [in Russian], VINITI, Moscow, 1985, 7, pp. 23–124. (Itogi Nauki i Tekhniki.) 628Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • A. M. Kytmanov
    • 1
  • S. G. Myslivets
    • 1
  1. 1.Krasnoyarsk State UniversityRussia

Personalised recommendations