Smooth Pseudoconvex Domains in ℂ2 For Which the Corona Theorem and Lp Estimates for ∂̄ Fail

  • John Erik Fornaess
  • Nessim Sibony
Part of the The University Series in Mathematics book series (USMA)


Let Ω be a bounded domain in Cn and let H 8(Ω) denote the algebra of bounded holomorphic functions in Ω. The problem of whether Ω is dense in the spectrum of H 8(Ω) (corona problem) has attracted some attention. The answer is known to be affirmative for many open sets in C ; see Ref. 4 for a discussion. The answer is not known in ℂ n n ≥ 2 even for the ball or the polydisk.


Holomorphic Function Smooth Boundary Pseudoconvex Domain Subharmonic Function Reinhardt Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    B. Bemdtsson, A smooth pseudoconvex domain in C2 for which L8-estimates for ∂0304 do not hold, preprint.Google Scholar
  2. 2.
    A. Bonami and N. Sibony, Sobolev embedding in Cn and ∂0304-equation, J. Geom. Anal, to appear.Google Scholar
  3. 3.
    J. E. Fornaess and N. Sibony, Ifestimates for d, Proc. Symp. Pure Math. 52, Part 3,129–163 (1991).CrossRefMathSciNetGoogle Scholar
  4. 4.
    J. Garnett, Bounded Analytic Functions, Academic Press, New York (1981).zbMATHGoogle Scholar
  5. 5.
    J. L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes, Dunod, Paris (1968).zbMATHGoogle Scholar
  6. 6.
    N. Sibony, Prolongement de fonctions holomorphes bornées et métrique de Carathéodory, Invent. Math. 29, 205–230 (1975).zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    N. Sibony, Problème de la couronne pour les domaines faiblement pseudoconvexes à bord lisse, Ann. of Math. 126, 675–682 (1987).zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    N. Sibony, Some aspects of weakly pseudoconvex domains, Proc. Symp. Pure Math. 52, Part1, 199–231 (1991).CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1993

Authors and Affiliations

  • John Erik Fornaess
    • 1
  • Nessim Sibony
    • 2
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.C.N.R.S. UA 57Université de Paris-SudOrsay CedexFrance

Personalised recommendations