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Potential Analysis

, Volume 43, Issue 3, pp 531–545 | Cite as

A Note on the Hyperconvexity of Pseudoconvex Domains Beyond Lipschitz Regularity

  • Benny Avelin
  • Lisa Hed
  • Håkan Persson
Article

Abstract

We show that bounded pseudoconvex domains that are Hölder continuous for all α < 1 are hyperconvex, extending the well-known result by Demailly (Math. Z. 194(4) 519–564, 1987) beyond Lipschitz regularity.

Keywords

Plurisubharmonic functions Continuous boundary Hyperconvexity Bounded exhaustion function Hölder for all exponents Log-Lipschitz Boundary regularity Reinhardt domains 

Mathematics Subject Classifications (2010)

Primary 32U05 32U10 Secondary 31B25 31C10 

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References

  1. 1.
    Avelin, B., Hed, L., Persson, H.: Approximation of Plurisubharmonic Functions, Complex Variables and Elliptic Equations. doi: 10.1080/17476933.2015.1053473
  2. 2.
    Błocki, Z.: The complex Monge-Ampère operator in hyperconvex domains. Ann.Scuola Norm. Sup. Pisa 23, 721–747 (1996)MATHGoogle Scholar
  3. 3.
    Błocki, Z.: Bergman kernel and pluripotential theory, Proceedings of the Conference in honor of Duong Phong, Contemporary Mathematics, American Mathematical Society (to appear)Google Scholar
  4. 4.
    Cegrell, U.: The general definition of the complex Monge-Ampère operator. Ann. Inst. Fourier (Grenoble) 54(1), 159–179 (2004)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Demailly, J.: Mesures de Monge-Ampère et mesures pluriharmoniques. Math. Z 194(4), 519–564 (1987)CrossRefMathSciNetMATHGoogle Scholar
  6. 6.
    Demailly, J.: Complex analytic and differential geometry, Monograph Grenoble (1997)Google Scholar
  7. 7.
    Diederich, K., Fornæss, J.E.: Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions. Invent. Math. 39(2), 129–141 (1977)CrossRefMathSciNetMATHGoogle Scholar
  8. 8.
    Fu, S.: On completeness of invariant metrics of Reinhardt domains. Arch. Math. (Basel) 63(2), 166–172 (1994)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Hai, L.M., Dieu, N.Q., Tuyen, N.H.: Some properties of Reinhardt domains. Ann. Polon. Math. 82(3), 203–217 (2003)CrossRefMathSciNetMATHGoogle Scholar
  10. 10.
    Harrington, P.S.: The order of plurisubharmonicity on pseudoconvex domains with Lipschitz boundaries. Math. Res. Lett. 15(3), 485–490 (2008)CrossRefMathSciNetMATHGoogle Scholar
  11. 11.
    Jarnicki, M., Pflug, P.: First steps in several complex variables: Reinhardt domains, EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zrich (2008)Google Scholar
  12. 12.
    Kerzman, N., Rosay, J.: Fonctions plurisousharmoniques d’exhaustion bornées et domaines taut. Math. Ann. 257(2), 171–184 (1981)CrossRefMathSciNetMATHGoogle Scholar
  13. 13.
    Lévy, P.P.: Théorie de l’addition des variables aléatoires, Gauthier-Villars, Paris (1937)Google Scholar
  14. 14.
    McMullen, C.T.: Renormalization and 3-manifolds which fiber over the circle, vol. 142, p 253 (1996)Google Scholar
  15. 15.
    Ohsawa, T., Sibony, N.: Bounded p.s.h. functions and pseudoconvexity in Kähler manifold. Nagoya Math. J. 149, 1–8 (1998)MathSciNetMATHGoogle Scholar
  16. 16.
    Oka, K.: Sur les fonctions analytiques de plusieurs variables. IX. Domaines finis sans point critique intérieur (French). Jap. J. Math. 23, 97–155 (1953)MathSciNetMATHGoogle Scholar
  17. 17.
    Poletsky, E.A., Stessin, M.I.: Hardy and Bergman spaces on hyperconvex domains and their composition operators. I.diana Univ. Math. J. 57(5), 2153–2201 (2008)CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Range, R.M.: A remark on bounded strictly plurisubharmonic exhaustion functions. Proc. Amer. M.th. Soc. 81(2), 220–222 (1981)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Richberg, R.: Stetige streng pseudokonvexe Funktionen. Math. Ann. 175, 257–86 (1968)CrossRefMathSciNetMATHGoogle Scholar
  20. 20.
    Stehle, J.-L.: Fonctions plurisousharmoniques et convexité holomorphe de certain fibrés analytiques. Lect. Notes Math.. 474, 155–180Google Scholar
  21. 21.
    Wang, X.: Hyperconvexity of non-smooth pseudoconvex domains. Ann. Polon. Math. 111(1), 1–11 (2014)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Zwonek, W.: Completeness, Reinhardt domains and the method of complex geodesics in the theory of invariant functions. Dissertationes Math. (Rozprawy Mat.) 388, 103 (2000)MathSciNetGoogle Scholar
  23. 23.
    Zygmund, A.: Trigonometric series. Vol. I, II, (2002) Cambridge University PressGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of JyväskyläJyväskyläFinland
  2. 2.Institute of MathematicsAalto UniversityAaltoFinland
  3. 3.Department of Mathematics and Mathematical StatisticsUmeå UniversityUmeåSweden
  4. 4.Department of MathematicsUppsala UniversityUppsalaSweden

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