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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Cluster values of bounded analytic functions


Author: T. W. Gamelin
Journal: Trans. Amer. Math. Soc. 225 (1977), 295-306
MSC: Primary 46J15; Secondary 30A72
DOI: https://doi.org/10.1090/S0002-9947-1977-0438133-8
MathSciNet review: 0438133
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Abstract: Let D be a bounded domain in the complex plane, and let $ \zeta $ belong to the topological boundary $ \partial D$ of D. We prove two theorems concerning the cluster set $ {\text{Cl}}(f,\zeta )$ of a bounded analytic function f on D. The first theorem asserts that values in $ {\text{Cl}}(f,\zeta )\backslash f({\mathcyr{SH}_\zeta })$ are assumed infinitely often in every neighborhood of $ \zeta $, with the exception of those lying in a set of zero analytic capacity. The second asserts that all values in $ {\text{Cl}}(f,\zeta )\backslash f({\mathfrak{M}_\zeta } \cap {\text{supp}}\;\lambda )$ are assumed infinitely often in every neighborhood of $ \zeta $, with the exception of those lying in a set of zero logarithmic capacity. Here $ {\mathfrak{M}_\zeta }$ is the fiber of the maximal ideal space $ \mathfrak{M}(D)$ of $ {H^\infty }(D)$ lying over $ \zeta $, $ {\mathcyr{SH}_\zeta }$ is the Shilov boundary of the fiber algebra, and $ \lambda $ is the harmonic measure on $ \mathfrak{M}(D)$.


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  • [1] C. Constantinescu and A. Cornea, Ideale Ränder Riemannscher Flächen, Ergebnisse Math. Grenzgebiete, Bd. 32, Springer-Verlag, Berlin, 1963. MR 28 #3151. MR 0159935 (28:3151)
  • [2] T. W. Gamelin, Lectures on $ {H^\infty }(D)$, Notas de Matemática, La Plata, Argentina, 1972.
  • [3] -, Localization of the corona problem, Pacific J. Math. 34 (1970), 73-81. MR 43 #2482. MR 0276742 (43:2482)
  • [4] -, Iversen's theorem and fiber algebras, Pacific J. Math. 46 (1973), 389-414. MR 49 #7783. MR 0343039 (49:7783)
  • [5] -, The algebra of bounded analytic functions, Bull. Amer. Math. Soc. 79 (1973), 1095-1108. MR 48 #4742. MR 0326398 (48:4742)
  • [6] T. W. Gamelin and J. Garnett, Distinguished homomorphisms and fiber algebras, Amer. J. Math. 92 (1970), 455-474. MR 46 #2434. MR 0303296 (46:2434)
  • [7] S. Ya. Havinson, Analytic capacity of sets, joint nontriviality of various classes of analytic functions and the Schwarz lemma in arbitrary domains, Mat. Sb. 54 (96) (1961), 3-50; English transl., Amer. Math. Soc. Transl. (2) 43 (1964), 215-266. MR 25 #182. MR 0136720 (25:182)
  • [8] Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Ser. in Modern Analysis, Prentice-Hall, Englewood Cliffs, N. J., 1962. MR 24 #A2844. MR 0133008 (24:A2844)
  • [9] A. J. Lohwater, On the theorems of Gross and Iversen, J. Analyse Math. 7 (1959/60), 209-221. MR 23 #A1043. MR 0123721 (23:A1043)
  • [10] R. Ludwig, Approximation of harmonic functions, Ph.D. Thesis, UCLA, 1969.
  • [11] K. Noshiro, Cluster sets, Ergebnisse Math. Grenzgebiete, N. F., Heft 28, Springer-Verlag, Berlin, 1960. MR 24 #A3295. MR 0133464 (24:A3295)
  • [12] M. Tsuji, Potential theory in modern function theory, Maruzen, Tokyo, 1959. MR 22#5712. MR 0114894 (22:5712)
  • [13] M. L. Weiss, Cluster sets of bounded analytic functions from a Banach algebraic viewpoint, Ann. Acad. Sci. Fenn. Ser. A I No. 367 (1965), 14 pp. MR 35 #379. MR 0209481 (35:379)

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DOI: https://doi.org/10.1090/S0002-9947-1977-0438133-8
Article copyright: © Copyright 1977 American Mathematical Society

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