Cluster values of bounded analytic functions
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- by T. W. Gamelin PDF
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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 ${\text {Cl}}(f,\zeta )\backslash f(\Sha _\zeta )$ 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 $\zeta$, $\Sha _\zeta$ is the Shilov boundary of the fiber algebra, and $\lambda$ is the harmonic measure on $\mathfrak {M}(D)$.References
- Corneliu Constantinescu and Aurel Cornea, Ideale Ränder Riemannscher Flächen, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 32, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963 (German). MR 0159935, DOI 10.1007/978-3-642-87031-6 T. W. Gamelin, Lectures on ${H^\infty }(D)$, Notas de Matemática, La Plata, Argentina, 1972.
- T. W. Gamelin, Localization of the corona problem, Pacific J. Math. 34 (1970), 73–81. MR 276742, DOI 10.2140/pjm.1970.34.73
- T. W. Gamelin, Iversen’s theorem and fiber algebras, Pacific J. Math. 46 (1973), 389–414. MR 343039, DOI 10.2140/pjm.1973.46.389
- T. W. Gamelin, The algebra of bounded analytic functions, Bull. Amer. Math. Soc. 79 (1973), 1095–1108. MR 326398, DOI 10.1090/S0002-9904-1973-13345-2
- T. W. Gamelin and John Garnett, Distinguished homomorphisms and fiber algebras, Amer. J. Math. 92 (1970), 455–474. MR 303296, DOI 10.2307/2373334
- S. Ja. Havinson, The analytic capacity of sets related to the non-triviality of various classes of analytic functions, and on Schwarz’s lemma in arbitrary domains, Mat. Sb. (N.S.) 54 (96) (1961), 3–50 (Russian). MR 0136720
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
- A. J. Lohwater, On the theorems of Gross and Iversen, J. Analyse Math. 7 (1959/60), 209–221. MR 123721, DOI 10.1007/BF02787686 R. Ludwig, Approximation of harmonic functions, Ph.D. Thesis, UCLA, 1969.
- Kiyoshi Noshiro, Cluster sets, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 28, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960. MR 0133464, DOI 10.1007/978-3-642-85928-1
- M. Tsuji, Potential theory in modern function theory, Maruzen Co. Ltd., Tokyo, 1959. MR 0114894
- Max L. Weiss, Cluster sets of bounded analytic functions from a Banach algebraic viewpoint, Ann. Acad. Sci. Fenn. Ser. A I No. 367 (1965), 14. MR 0209481
Additional Information
- © Copyright 1977 American Mathematical Society
- 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