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The Lusin-Privalov theorem
for subharmonic functions

Author: Stephen J. Gardiner
Journal: Proc. Amer. Math. Soc. 124 (1996), 3721-3727
MSC (1991): Primary 31B25
MathSciNet review: 1396977
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Abstract: This paper establishes a generalization of the Lusin-Privalov radial uniqueness theorem which applies to subharmonic functions in all dimensions. In particular, it answers a question of Rippon by showing that no subharmonic function on the upper half-space can have normal limit $-\infty $ at every boundary point.

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  • 1. M. G. Arsove, The Lusin-Privalov theorem for subharmonic functions, Proc. London Math. Soc. (3) 14 (1964), 260-270. MR 28:4136
  • 2. K. F. Barth and W. J. Schneider, An asymptotic analogue of the F. and M. Riesz radial uniqueness theorem, Proc. Amer. Math. Soc. 22 (1969), 53-54. MR 40:364
  • 3. R. D. Berman, Analogues of radial uniqueness theorems for subharmonic functions in the unit disk, J. London Math. Soc. (2) 29 (1984), 103-112. MR 85d:31003
  • 4. E. F. Collingwood and A. J. Lohwater, The theory of cluster sets, Cambridge University Press, 1966. MR 38:325
  • 5. J. Deny and P. Lelong, Étude des fonctions sousharmoniques dans un cylindre ou dans un cône, Bull. Soc. Math. France 75 (1947), 89-112. MR 9:352e
  • 6. J. L. Doob, A non-probabilistic proof of the relative Fatou theorem, Ann. Inst. Fourier (Grenoble) 9 (1959), 293-300. MR 22:8233
  • 7. J. L. Doob, Some classical function theory theorems and their modern versions, Ann. Inst. Fourier (Grenoble) 15 (1965), 113-136; 17 (1967), 469. MR 34:2923;MR 36:4013
  • 8. J. L. Doob, Classical potential theory and its probabilistic counterpart, Springer, New York, 1983. MR 85k:31001
  • 9. S. J. Gardiner, Integrals of subharmonic functions over affine sets, Bull. London Math. Soc. 19 (1987), 343-349. MR 88g:31005
  • 10. S. J. Gardiner, Harmonic approximation, London Math. Soc. Lecture Note Ser., no. 221, Cambridge University Press, 1995. CMP 95:15
  • 11. J. Lelong-Ferrand, Étude au voisinage de la frontière des fonctions surharmoniques positives dans un demi-espace, Ann. Sci. École Norm. Sup. (3) 66 (1949), 125-159. MR 11:176f
  • 12. N. Lusin and I. Privalov, Sur l'unicité et la multiplicité des fonctions analytiques, Ann. Sci. École Norm. Sup (3) 42 (1925), 143-191.
  • 13. P. J. Rippon, The boundary cluster sets of subharmonic functions, J. London Math. Soc. (2) 17 (1978), 469-479. MR 81h:30036
  • 14. 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. MR 45:2375

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Additional Information

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

Received by editor(s): May 10, 1995
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1996 American Mathematical Society

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