Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Uniqueness theorems for subharmonic functions in unbounded domains

Author: S. J. Gardiner
Journal: Proc. Amer. Math. Soc. 99 (1987), 437-444
MSC: Primary 31B05
MathSciNet review: 875377
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A theorem of Carlson says that a holomorphic function of exponential growth in the half-plane cannot approach zero exponentially along the boundary unless it vanishes identically. This paper presents a generalization of this result for subharmonic functions in a Greenian domain $ \Omega $, using the Martin boundary, minimal fine topology and PWB solution to the $ h$-Dirichlet problem. Applications of the general theorem to specific choices of $ \Omega $, such as the half-space and strip, are presented in later sections.

References [Enhancements On Off] (What's this?)

  • [1] D. H. Armitage, A strong type of regularity for the PWB solution of the Dirichlet problem, Proc. Amer. Math. Soc. 61 (1976), 285-289. MR 0427658 (55:689)
  • [2] -, A Phragmén-Lindelöf theorem for subharmonic functions, Bull. London Math. Soc. 13 (1981), 421-428. MR 631101 (82k:31004)
  • [3] D. H. Armitage and T. B. Fugard, Subharmonic functions in strips, J. Math. Anal. Appl. 89 (1982), 1-27. MR 672185 (83m:31004)
  • [4] F. T. Brawn, The Martin boundary of $ {\mathbb{R}^n} \times \left] {0,1} \right[$, J. London Math. Soc. (2) 5 (1972), 59-66. MR 0296323 (45:5384)
  • [5] -, Positive harmonic majorization of subharmonic functions in strips, Proc. London Math. Soc. (3) 27 (1973), 261-289. MR 0330482 (48:8819)
  • [6] M. Brelot, On topologies and boundaries in potential theory, Lecture Notes in Math., vol. 175, Springer-Verlag, Berlin, 1971. MR 0281940 (43:7654)
  • [7] R. P. Boas, Jr., Entire functions, Academic Press, New York, 1954. MR 0068627 (16:914f)
  • [8] J. L. Doob, Classical potential theory and its probabilistic counterpart, Springer-Verlag, New York, 1984. MR 731258 (85k:31001)
  • [9] S. J. Gardiner, Generalized means of subharmonic functions, Doctoral thesis, Queen's University of Belfast, 1982.
  • [10] -, Harmonic majorization of subharmonic functions in unbounded domains, Ann. Acad. Sci. Fenn. Ser A I Math. 8 (1983), 43-54. MR 698836 (84f:31005)
  • [11] D. S. Jerison and C. E. Kenig, Boundary behaviour of harmonic functions in non-tangentially accessible domains, Adv. in Math. 46 (1982), 80-147. MR 676988 (84d:31005b)
  • [12] Ü. Kuran, On half-spherical means of subharmonic functions in half-spaces, J. London Math. Soc. (2) 2 (1970), 305-317. MR 0262531 (41:7137)
  • [13] L. Naïm, Sur le rôle de la frontière de R. S. Martin dans la théorie du potentiel, Ann. Inst. Fourier (Grenoble) 7 (1957), 183-281.
  • [14] G. N. Watson, A treatise on the theory of Bessel functions, 2nd ed., Cambridge Univ. Press, London, 1944. MR 0010746 (6:64a)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 31B05

Retrieve articles in all journals with MSC: 31B05

Additional Information

Article copyright: © Copyright 1987 American Mathematical Society

American Mathematical Society