Mathematische Annalen

, Volume 265, Issue 2, pp 221–251 | Cite as

On integral representations and a priori Lipschitz estimates for the canonical solution of the\(\bar \partial \)-equation

  • Ingo Lieb
  • R. Michael Range


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ahern, P., Schneider, R.: Holomorphic Lipschitz functions in pseudoconvex domains. Am. J. Math.101, 543–563 (1979)Google Scholar
  2. 2.
    Bochner, S.: Analytic and meromorphic continuation by means of Green's formula. Ann. Math.44, 652–673 (1943)Google Scholar
  3. 3.
    Fefferman, Ch.: The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Invent. Math.26, 1–65 (1974)Google Scholar
  4. 4.
    Fischer, W., Lieb, I.: Lokale Kerne und beschränkte Lösungen für den\(\bar \partial \) aufq-konvexen Gebieten. Math. Ann.208, 249–265 (1974)Google Scholar
  5. 5.
    Folland, G., Stein, E.: Estimates for the\(\bar \partial _b \) and analysis on the Heisenberg group. Commun. Pure Appl. Math.27, 429–522 (1974)Google Scholar
  6. 6.
    Grauert, H., Lieb, I.: Das Ramireszsche Integral und die Gleichung\(\bar \partial f = \alpha \) im Bereich der beschränkten Formen. Rice Univ. Studies56, 29–50 (1970)Google Scholar
  7. 7.
    Greiner, P., Stein, E.: Estimates for the\(\bar \partial \) problem. Princeton: Princeton University Press 1977Google Scholar
  8. 8.
    Harvey, R., Polking, J.: The\(\bar \partial \)-Neumann solution to the inhomogeneous Cauchy-Riemann equation in the ball in ℂn. Trans Am. Math. Soc. (to appear)Google Scholar
  9. 9.
    Henkin, G.M.: Integral representations of functions holomorphic in strictly pseudoconvex domains and applications to the\(\bar \partial \)-problem. Mat. Sb.82 (124), 300–308 (1970); Math. USSR Sb.11, 273–281 (1970)Google Scholar
  10. 10.
    Hörmander, L.:L 2-estimates and existence theorems for the\(\bar \partial \)-opeartor. Acta math.113, 89–152 (1965)Google Scholar
  11. 11.
    Kerzman, N., Stein, E.: The Szegö kernel in terms of Cauchy-Fantappié kernels. Duke Math. J.45, 197–224 (1978)Google Scholar
  12. 12.
    Kohn, J.J.: Harmonic integrals on strongly pseudoconvex manifolds. I. Ann. Math.78, 112–148 (1963); II.79, 450–472 (1964)Google Scholar
  13. 13.
    Koppelman, W.: The Cauchy integral for differential forms. Bull. Am. Math. Soc.73, 554–556 (1967)Google Scholar
  14. 14.
    Lieb, I.: Die Cauchy-Riemannschen Differentialgleichungen auf streng pseudokonvexen Gebieten. Math. Ann.190, 6–44 (1970)Google Scholar
  15. 15.
    Ligocka, E.: Lecture at Oberwolfach, Sept. 1980Google Scholar
  16. 16.
    Øvrelid, N.: Pseudodifferential operators and the\(\bar \partial \)-equation. In: Lecture Notes in Mathematics, Vol. 512, pp. 185–192. Berlin, Heidelberg, New York: Springer 1976Google Scholar
  17. 17.
    Phong, D.H.: On integral representations for the Neumann operator. Proc. Natl. Acad. Sci. USA76, 1554–1558 (1978)Google Scholar
  18. 18.
    Phong, D.H., Stein, E.: Estimates for the Bergman and Szegö projections on strongly pseudoconvex domains. Duke Math. J.44, 695–704 (1977)Google Scholar
  19. 19.
    Range, R.M.: An elementary integral solution operator for the Cauchy-Riemann equations on pseudoconvex domains in ℂn. Trans. Am. Math. Soc.274, 809–816 (1982)Google Scholar
  20. 20.
    Range, R.M.: The\(\bar \partial \)-Nuemann operator on the unit ball in ℂn (to appear)Google Scholar
  21. 21.
    Rothschild, L., Stein, E.: Hypoelliptic differential operators and nilpotent groups. Acta math.137, 247–320 (1976)Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Ingo Lieb
    • 1
  • R. Michael Range
    • 2
  1. 1.Mathematisches Institut der UniversitätBonnFederal Republic of Germany
  2. 2.Department of MathematicsState University of New YorkAlbanyUSA

Personalised recommendations