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A geometric characterization of $ N\sp{+}$ domains

Authors: Patrick Ahern and William Cohn
Journal: Proc. Amer. Math. Soc. 88 (1983), 454-458
MSC: Primary 30D50; Secondary 31A15
MathSciNet review: 699413
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Abstract: A connected open set $ \mathcal{O} \subseteq {\mathbf{C}}$ is called an $ {N^ + }$ domain if every holomorphic function defined in the unit disc and taking values in $ \mathcal{O}$ is necessarily in the Smirnov class $ {N^ + }$. We show that $ \mathcal{O}$ is an $ {N^ + }$ domain if and only if $ \infty $ is a regular point for the solution of the Dirichlet problem for $ \mathcal{O}$. We get a similar characterization when $ {N^ + }$ is replaced by the class of outer functions.

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  • [1] Matts Essén, On analytic functions which are in 𝐻^{𝑝} for some positive 𝑝, Ark. Mat. 19 (1981), no. 1, 43–51. MR 625536,
  • [2] O. Frostman, Potentials d'équilibre et capacité des ensembles avec quelques applications à la théorie des fonctions, Med. Lunds Univ. Mat. Sem. 3 (1935), 1-118.
  • [3] W. K. Hayman, Values and growth of functions regular in the unit disk, Complex analysis (Proc. Conf., Univ. Kentucky, Lexington, Ky., 1976), Springer, Berlin, 1977, pp. 68–75. Lecture Notes in Math., Vol. 599. MR 0492219
  • [4] W. K. Hayman and Ch. Pommerenke, On analytic functions of bounded mean oscillation, Bull. London Math. Soc. 10 (1978), no. 2, 219–224. MR 500932,
  • [5] L. L. Helms, Introduction to potential theory, Pure and Applied Mathematics, Vol. XXII, Wiley-Interscience A Division of John Wiley & Sons, New York-London-Sydney, 1969. MR 0261018
  • [6] Rolf Nevanlinna, Eindeutige analytische Funktionen, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, Bd XLVI, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1953 (German). 2te Aufl. MR 0057330
  • [7] Walter Rudin, Real and complex analysis, 2nd ed., McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1974. McGraw-Hill Series in Higher Mathematics. MR 0344043
  • [8] David A. Stegenga, A geometric condition which implies BMOA, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 427–430. MR 545283
  • [9] M. Tsuji, Potential theory in modern function theory, Maruzen Co., Ltd., Tokyo, 1959. MR 0114894

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Article copyright: © Copyright 1983 American Mathematical Society

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