Potential Analysis

, Volume 51, Issue 3, pp 469–481 | Cite as

Domains of Existence for Finely Holomorphic Functions

  • Bent FugledeEmail author
  • Alan Groot
  • Jan Wiegerinck
Open Access


We show that fine domains in ℂ with the property that they are Euclidean Fs and Gd, are in fact fine domains of existence for finely holomorphic functions. Moreover regular fine domains are also fine domains of existence. Next we show that fine domains such as ℂ \ ℚ or ℂ \ (ℚ × iℚ), more specifically fine domains V with the properties that their complement contains a non-empty polar set E that is of the first Baire category in its Euclidean closure K and that (K \ E) ⊂ V, are not fine domains of existence.


Finely holomorphic function Domain of existence 

Mathematics Subject Classification (2010)

30G12 30A14 31C40 


  1. 1.
    Armitage, D.H., Gardiner, S.J.: Classical Potential Theory. Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London (2001)CrossRefGoogle Scholar
  2. 2.
    Borel, É.: Sur quelques points de la théorie des fonctions, Thèse, Paris 1894. Annales Scientifiques de l’É.N.S. 12, 9–55 (1895)zbMATHGoogle Scholar
  3. 3.
    Borel, É.: Leçons Sur Les Fonctions Monogènes Uniformes D’une Variable Complexe. Gauthier Villars, Paris (1917)zbMATHGoogle Scholar
  4. 4.
    Doob, J.L.: Classical Potential Theory and Its Probabilistic Counterpart, Grundl. Math. Wiss. 262 Springer (1984)Google Scholar
  5. 5.
    Edlund, T.: Complete pluripolar curves and graphs. Ann. Polon. Math. 84(1), 75–86 (2004)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Edigarian, A., El Marzguioui, S., Wiegerinck, J.: The image of a finely holomorphic map is pluripolar. Ann. Polon. Math. 97(2), 137–149 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    El Marzguioui, S., Wiegerinck, J.: The Pluri-fine topology is locally connected. Potential Anal. 25, 283–288 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Fuglede, B.: Finely Harmonic Functions. Springer LNM, vol. 289. Springer, Berlin (1972)CrossRefGoogle Scholar
  9. 9.
    Fuglede, B.: Asymptotic Paths for Subharmonic Functions and Polygonal Connectedness of Fine Domains, Seminar on Potential Theory, Paris, No. 5, pp. 97—116, LNM, vol. 814. Springer, Berlin (1980)Google Scholar
  10. 10.
    Fuglede, B.: Sur les fonctions finement holomorphes. Ann. Inst. Fourier (Grenoble) 31(4), 57–88 (1981)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fuglede, B.: Finely holomorphic functions. In: 18th Scandinavian Congress of Mathematicians (Aarhus, 1980), pp. 22–38, Progr. Math. 11, Birkhäuser Boston Mass (1981)Google Scholar
  12. 12.
    Fuglede, B.: Finely holomorphic functions. A survey. Rev. Roumaine Math. Pures Appl. 33(4), 283–295 (1988)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gardiner, S.J.: Finely continuously differentiable functions. Math. Z. 266(4), 851–861 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lukeš, J., Malý, J., Netuka, I., Spurný, J.: Integral representation theory, Walter de Gruyter & Co., Berlin de Gruyter Studies in Mathematics 35 (2010)Google Scholar
  15. 15.
    Lyons, T.J.: Finely holomorphic functions. J. Funct. Anal. 37(1), 1–18 (1980)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Pyrih, P.: Finely holomorphic functions and quasi-analytic classes. Potential Anal. 3(3), 273–281 (1994)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ransford, T.: Potential Theory in the Complex Plane London Math. Soc. Student Texts 28 (1995)Google Scholar

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Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of CopenhagenKøbenhavnDenmark
  2. 2.Analysis SectionKU LeuvenLeuvenBelgium
  3. 3.KdV Institute for MathematicsUniversity of AmsterdamAmsterdamThe Netherlands

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