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Growth properties of superharmonic functions along rays

Author: Stephen J. Gardiner
Journal: Proc. Amer. Math. Soc. 128 (2000), 1963-1970
MSC (2000): Primary 31B05
Published electronically: November 1, 1999
MathSciNet review: 1646303
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Abstract: This paper gives a precise topological description of the set of rays along which a superharmonic function on $\mathbb{R}^n$ may grow quickly. The corollary that arbitrary growth cannot occur along all rays answers a question posed by Armitage.

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  • 1. D. H. Armitage, Radial limits of superharmonic functions in the plane, Colloq. Math. 67 (1994), no. 2, 245–252. MR 1305216
  • 2. Vazguen Avanissian, Fonctions plurisousharmoniques et fonctions doublement sousharmoniques., Ann. Sci. École Norm. Sup. (3) 78 (1961), 101–161 (French). MR 0132206
  • 3. J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. MR 731258
  • 4. M. R. Essén and S. J. Gardiner, Limits along parallel lines and the classical fine topology, J. London Math. Soc., to appear.
  • 5. Bent Fuglede, Finely harmonic functions, Lecture Notes in Mathematics, Vol. 289, Springer-Verlag, Berlin-New York, 1972. MR 0450590
  • 6. Stephen J. Gardiner, Harmonic approximation, London Mathematical Society Lecture Note Series, vol. 221, Cambridge University Press, Cambridge, 1995. MR 1342298
  • 7. Stephen J. Gardiner, The Lusin-Privalov theorem for subharmonic functions, Proc. Amer. Math. Soc. 124 (1996), no. 12, 3721–3727. MR 1396977,
  • 8. Wolfhard Hansen, Abbildungen harmonischer Räume mit Anwendung auf die Laplace und Wärmeleitungsgleichung, Ann. Inst. Fourier (Grenoble) 21 (1971), no. 3, 203–216 (German, with French and English summaries). MR 0335839
  • 9. Lester L. Helms, Introduction to potential theory, Robert E. Krieger Publishing Co., Huntington, N.Y., 1975. Reprint of the 1969 edition; Pure and Applied Mathematics, Vol. XXII. MR 0460666
  • 10. N. S. Landkof, Foundations of modern potential theory, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy; Die Grundlehren der mathematischen Wissenschaften, Band 180. MR 0350027
  • 11. A. A. Shaginyan, Uniform and tangential harmonic approximation of continuous functions on arbitrary sets, Mat. Zametki 9 (1971), 131-142; English translation in Math. Notes 9 (1971), 78-84.

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Additional Information

Stephen J. Gardiner
Affiliation: Department of Mathematics, University College Dublin, Dublin 4, Ireland

Received by editor(s): April 1, 1998
Received by editor(s) in revised form: August 13, 1998
Published electronically: November 1, 1999
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2000 American Mathematical Society