Potential Analysis

, Volume 34, Issue 1, pp 89–100 | Cite as

A Montel Type Result for Super-Polyharmonic Functions on R N

  • Toshihide Futamura
  • Keiji Kitaura
  • Yoshihiro Mizuta
Article

Abstract

Our aim in this paper is to discuss a Montel type result for a family \({\mathcal F}\) of super-polyharmonic functions on R N . We give a condition on spherical means to assure that \({\mathcal F}\) contains a sequence converging outside a set of capacity zero.

Keywords

Polyharmonic functions Super-polyharmonic functions Spherical means Riesz decomposition 

Mathematics Subject Classifications (2010)

Primary 31B30 31B05 31B15 

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References

  1. 1.
    Abkar, A., Hedenmalm, H.: A Riesz representation formula for super-biharmonic functions. Ann. Acad. Sci. Fenn., Math. 26, 305–324 (2001)MATHMathSciNetGoogle Scholar
  2. 2.
    Anderson, M., Baernstein, A.: The size of the set on which a meromorphic function is large. Proc. Lond. Math. Soc. 36, 518–539 (1978)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Aronszajn, N., Creese, T.M., Lipkin, L.J.: Polyharmonic Functions. Clarendon Press (1983)Google Scholar
  4. 4.
    Axler, S., Bourdon, P., Ramey, W.: Harmonic Function Theory, 2nd edn. Springer, New York (2001)MATHGoogle Scholar
  5. 5.
    Futamura, T., Kitaura, K., Mizuta, Y.: Isolated singularities, growth of spherical means and Riesz decomposition for superbiharmonic functions. Hiroshima Math. J. 38, 231–241 (2008)MATHMathSciNetGoogle Scholar
  6. 6.
    Futamura, T., Mizuta, Y.: Isolated singularities of super-polyharmonic functions. Hokkaido Math. J. 33, 675–695 (2004)MATHMathSciNetGoogle Scholar
  7. 7.
    Hayman, W.K., Kennedy, P.B.: Subharmonic Functions, vol. 1. Academic Press, London (1976)Google Scholar
  8. 8.
    Hayman, W.K., Korenblum, B.: Representation and uniqueness theorems for polyharmonic functions. J. Anal. Math. 60, 113–133 (1993)MATHMathSciNetGoogle Scholar
  9. 9.
    Kitaura, K., Mizuta, Y.: Spherical means and Riesz decomposition for superbiharmonic functions. J. Math.Soc. Jpn. 58, 521–533 (2006)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kondratyuk, A.A., Tarasyuk, S.I.: Compact Operators and Normal Families of Subharmonic Functions. Function Spaces, Differential Operators and Nonlinear Analysis (Paseky nad Jizerou, 1995), pp. 227–231 . Prometheus, Prague (1996)Google Scholar
  11. 11.
    Landkof, N.S.: Foundations of Modern Potential Theory. Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer-Verlag, New York-Heidelberg (1972)Google Scholar
  12. 12.
    Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Cambridge Studies in Advanced Mathematics, vol. 44. Cambridge University Press, Cambridge (1995)Google Scholar
  13. 13.
    Mizuta, Y.: An integral representation and fine limits at infinity for functions whose Laplacians iterated m times are measures. Hiroshima Math. J. 27, 415–427 (1997)MATHMathSciNetGoogle Scholar
  14. 14.
    Mizuta, Y.: Potential Theory in Euclidean Spaces. Gakkōtosyo, Tokyo (1996)Google Scholar
  15. 15.
    Pizetti, P.: Sulla media deivalori che una funzione dei punti dello spazio assume alla superficie di una sfera. Rend. Lincei 5, 309–316 (1909)Google Scholar
  16. 16.
    Schiff, J.L.: Normal Families. Springer, New York (1993)MATHGoogle Scholar
  17. 17.
    Supper, R.: Subharmonic functions of order less than one, Potential Anal. 23, 165–179 (2005)Google Scholar
  18. 18.
    Supper, R.: A Montel type result for subharmonic functions. Boll. Unione Mat. Ital. 2, 423–444 (2009)MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  • Toshihide Futamura
    • 1
  • Keiji Kitaura
    • 2
  • Yoshihiro Mizuta
    • 3
  1. 1.Department of MathematicsDaido UniversityNagoyaJapan
  2. 2.Department of Mathematics, Graduate School of ScienceHiroshima UniversityHigashi-HiroshimaJapan
  3. 3.Department of MathematicsHiroshima UniversityHigashi-HiroshimaJapan

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