Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-30T01:16:10.205Z Has data issue: false hasContentIssue false
  • English
  • Français

Fine limits of superharmonic functions

Published online by Cambridge University Press:  24 October 2008

Stephen J. Gardiner
Stephen J. Gardiner
Affiliation:
University College, Dublin, Ireland
Get access Rights & Permissions [Opens in a new window]

Extract

Let O denote the origin of ℝn (where n ≥ 2), let B(X, r) denote the open ball in ℝn of centre X and radius r, and define φn: (0, + ∞) → ℝ by φn(t) = t2-n if n ≥ 3 and φ2(t) = log (1/t). A sequence (Xk) in B(O, 1) is said to converge regularly to O if Xk → O and there is a constant k > 1 such that φn(|Xk+1|) < Kφn(|Xk|) for each k ∈ ℕ. If n ≥ 3, this is clearly equivalent to saying that there is a constant k′ ε (0, 1) such that |Xk+1| > k′|Xk| for each k (cf. [18], p. 149).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 2 , September 1990 , pp. 381 - 394
Copyright
Copyright © Cambridge Philosophical Society 1990

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Aikawa, H.. On the behavior at infinity of non-negative superharmonic functions in a half-space. Hiroshima Math. J. 11 (1981), 425–441.CrossRefGoogle Scholar
[2]Armitage, D. H. and Goldstein, M.. Characterizations of thin and semi-thin sets. Analysis 8 (1988), 165–186.CrossRefGoogle Scholar
[3]Azarin, V. S.. Generalization of a theorem of Hayman on subharmonic functions in an m-dimensional cone. Amer. Math. Soc. Transl. (2) 80 (1969), 119–138.Google Scholar
[4]Beardon, A. F.. Montel's Theorem for subharmonic functions and solutions of partial differential equations. Proc. Cambridge Philos. Soc. 69 (1971), 123–150.CrossRefGoogle Scholar
[5]Brelot, M.. Points irréguliers et transformations continues en théorie du potentiel. J. Math. Pures Appl. 19 (1940), 319–337.Google Scholar
[6]Brelot, M.. On Topologies and Boundaries in Potential Theory. Lecture Notes in Math. vol. 175 (Springer-Verlag, 1971).CrossRefGoogle Scholar
[7]Brelot, M. and Doob, J. L.. Limites angulaires et limites fines. Ann. Inst. Fourier (Grenoble) 13 (1963), 395–415.CrossRefGoogle Scholar
[8]Carleson, L.. Selected Problems on Exceptional Sets (Van Nostrand, 1967).Google Scholar
[9]Dellacherie, C.. Ensembles Analytiques. Capacités, Mesures de Hausdorff. Lecture Notes in Math. vol. 295 (Springer-Verlag, 1972).CrossRefGoogle Scholar
[10]Deny, J.. Un théorème sur les ensembles effilés. Ann. Univ. Grenoble Math. Phys. 23 (1948), 139–142.Google Scholar
[11]Doob, J. L.. Classical Potential Theory and its Probabilistic Counterpart (Springer-Verlag, 1983).Google Scholar
[12]Essén, M., Hayman, W. K. and Hcber, A.. Slowly growing subharmonic functions I. Comment. Math. Helv. 52 (1977), 329–356.CrossRefGoogle Scholar
[13]Essén, M. and Jackson, H. L.. A comparison between thin sets and generalized Azarin sets. Canad. Math. Bull. 18 (1975), 335–346.CrossRefGoogle Scholar
[14]Essén, M. and Jackson, H. L.. On the covering properties of certain exceptional sets in a half-space. Hiroshima Math. J. 10 (1980) 233–262.CrossRefGoogle Scholar
[15]Essén, M., Jackson, H. L. and Rippon, P. J.. On minimally thin and rarefied sets in ℕp, p ≥ 2. Hiroshima Math. J. 15 (1985), 393–410.CrossRefGoogle Scholar
[16]Gardiner, S. J.. Representation and growth of subharmonic functions in half-spaces. Proc. London Math. Soc. (3) 48 (1984), 300–318.CrossRefGoogle Scholar
[17]Hayman, W. K. and Kennedy, P. B.. Subharmonic Functions, vol. 1 (Academic Press, 1976).Google Scholar
[18]Lelong-Ferrand, J.. Étude au voisinage de la frontière des fonctions surharmoniques positives dans un demi-espace. Ann. Sci. Ecole Norm. Sup. 66 (1949), 125–159.CrossRefGoogle Scholar
[19]Mizuta, Y.. On the radial limits of potentials and angular limits of harmonic functions. Hiroshima Math. J. 8 (1978), 415–437.CrossRefGoogle Scholar
[20]Mizuta, Y.. On semi-fine limits of potentials. Analysis 2 (1982), 115–139.CrossRefGoogle Scholar
[21]Mizuta, Y.. Minimally semi-fine limits of Green potentials of general order. Hiroshima Math. J. 12 (1982), 505–511.CrossRefGoogle Scholar
[22]Rippon, P. J.. Semi-fine limits of Green potentials. Bull. London Math. Soc. 11 (1979), 259–264.CrossRefGoogle Scholar