Some recent results on estimates for the -equation

  • Bo Berndtsson
Part of the Aspects of Mathematics book series (ASMA, volume E 26)


We survey some recent work by Sibony, Fornaess-Sibony, and the author concerning L P and Holder estimates for solutions to the -equation. Two new results about Holder estimates, and L P -estimates for are also included.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Amar E, LP-estimates for ô in C. In Complex Analysis, ed K Diederich. Vieweg Verlag 1991Google Scholar
  2. [2]
    Boas H and Shaw M, Sobolev estimates for the Lewy operator on weakly pseudoconvex boundaries. Math Ann 274 (1986)Google Scholar
  3. [3]
    Berndtsson B, Weighted estimates for the 5-equation in domains in C. Duke Math J 1992.Google Scholar
  4. [4]
    Berndtsson B, A smoothly bounded pseudoconvex domain in C2 where L°° don’t hold. To appear in Arkiv för MatematikGoogle Scholar
  5. [5]
    Diederich K and Bonneau P, Integral solution operators for the Cauchy Riemann equations on pseudoconvex domains. Math Ann 286 (1990)Google Scholar
  6. [6]
    Christ M, On the 5-equation in weighted L2-norms in Cl. J Geom Anal 1 (1991)Google Scholar
  7. [7]
    Christ M, Precise analysis of c) and ôb on domains of finite type in C2. Proceedings of the ICM Kyoto 1990. Springer Verlag 1991Google Scholar
  8. [8]
    Fornaess J and Sibony N, LP-estimates for ô. Proc Symp Pure Math 52.3, A M S 1990Google Scholar
  9. [9]
    Fornaess J and Sibony N, Pseudoconvex domains in C2, where the Corona Theorem and LP-estimates for 8 don’ hold. PreprintGoogle Scholar
  10. [10]
    Henkin G M and Leiterer J, Theory of Functions on Complex Manifolds. Akademie-Verlag Berlin 1984.Google Scholar
  11. [11]
    Hörmander L, An Introduction to complex analysis in several variablesGoogle Scholar
  12. [12]
    Hörmander L, L2-estimates and existence theorems for the 8- operator, Acta Math 113 (1965)Google Scholar
  13. [13]
    Kohn J and Folland G, The Neumann Problem for the Cauchy-Riemann complex. Annals of Mathematics Studies, Princeton 1972zbMATHGoogle Scholar
  14. [14]
    Kohn J, The range of the tangential Cauchy-Riemann operator. Duke Math J 53 (1986)Google Scholar
  15. [15]
    Polking J, The Cauchy-Riemann equations on convex sets. Proc Symp Pure Math 52.3, A M S 1990Google Scholar
  16. [16]
    Range R M, Holomorphic Functions and integral representations in several complex variables. Springer Verlag 1986Google Scholar
  17. [17]
    Range R M, On Hölder and BMO estimates for ô on convex domains in C2, to appearGoogle Scholar
  18. [18]
    Sibony N, Prolongement analytique des fonctions holomorph bornées et metrique de Caratheodory. Inv Math 29 (1975).Google Scholar
  19. [19]
    Sibony N, Un example de domaine pseudoconvexe regulier ou l’ equation Ou = f n’admet pas de solution bornée pour f bornée. Inv Math 62 (1980)Google Scholar
  20. [20]
    Sibony N, On Hölder estimates for â,Annals of Math Studies, to appearGoogle Scholar
  21. [21]
    Sibony N, Some aspects of weakly pseudoconvex domains. Preprint, abbreviated version in proc ICM Kyoto 1990.Google Scholar
  22. [22]
    Wu H, The Bochner technique, Proc 1980 Beijing conference on Differential geometry and Differential equations, Science Press 1982Google Scholar

Copyright information

© Springer Fachmedien Wiesbaden 1994

Authors and Affiliations

  • Bo Berndtsson

There are no affiliations available

Personalised recommendations