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On bounded analytic functions in finitely connected domains. (English) Zbl 0629.32003
Das cet important article l’A. continue ses recherches sur le problème des sélections des fonctions analytiques multiformes convexes, avec applications au problème de la couronne, qu’il avait commencées dans Trans. Am. Math. Soc. 294, 367-377 (1986; Zbl 0594.32008).
D’après le résultat principal de cet article (Theorem 1.4) à chaque fonction analytique multiforme, définie sur un domaine ayant un nombre fini de trous, est associée une fonction méromorphe ayant ses pôles aux points critiques de la fonction de Green associée à ce domaine. Ce résultat est même amélioré dans le cas où le domaine est une couronne.
Tout cela est utilisé pour donner une nouvelle démonstration du théorème de la couronne dans le cas d’un domaine avec un nombre fini de trous. Cette nouvelle démonstration, contrairement à celles de Forelli, Stout et Gamelin, a l’avantage d’être directe, c’est-à-dire de ne pas ramener le cas d’un domaine avec un nombre fini de trous au cas d’un disque.
Les mêmes méthodes permettent d’améliorer légèrement certains résultats de T. Wolff, à propos de savoir quand une puissance d’une fonction donnée de \(H^{\infty}(G)\) appartient à l’idéal engendré par n fonctions données dans \(H^{\infty}(G)\).
Reviewer: B.Aupetit

MSC:
32A10 Holomorphic functions of several complex variables
32A20 Meromorphic functions of several complex variables
30G30 Other generalizations of analytic functions (including abstract-valued functions)
32E30 Holomorphic, polynomial and rational approximation, and interpolation in several complex variables; Runge pairs
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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[1] Herbert Alexander and John Wermer, Polynomial hulls with convex fibers, Math. Ann. 271 (1985), no. 1, 99 – 109. · Zbl 0538.32011
[2] B. Berndtsson and T. J. Ransford, Analytic multifunctions, the \overline\partial -equation, and a proof of the corona theorem, Pacific J. Math. 124 (1986), no. 1, 57 – 72. · Zbl 0602.32002
[3] André Boivin, On Carleman approximation by meromorphic functions, Analytic functions, Błażejewko 1982 (Błażejewko, 1982) Lecture Notes in Math., vol. 1039, Springer, Berlin, 1983, pp. 9 – 15.
[4] Stephen D. Fisher, Function theory on planar domains, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1983. A second course in complex analysis; A Wiley-Interscience Publication. · Zbl 0511.30022
[5] Frank Forelli, Bounded holomorphic functions and projections, Illinois J. Math. 10 (1966), 367 – 380. · Zbl 0141.31401
[6] John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. · Zbl 0469.30024
[7] T. W. Gamelin, Localization of the corona problem, Pacific J. Math. 34 (1970), 73 – 81. · Zbl 0199.18801
[8] T. W. Gamelin, Uniform algebras and Jensen measures, London Mathematical Society Lecture Note Series, vol. 32, Cambridge University Press, Cambridge-New York, 1978. · Zbl 0418.46042
[9] P. W. Jones and D. E. Marshall, Critical points of Green’s function, harmonic measure and the corona problem, Institut Mittag-Leffler, Report No. 2, 1984.
[10] K. V. Rajeswara Rao, On a generalized corona problem, J. Analyse Math. 18 (1967), 277 – 278. · Zbl 0176.44103
[11] T. J. Ransford, Interpolation and extrapolation of analytic multivalued functions, Proc. London Math. Soc. (3) 50 (1985), no. 3, 480 – 504. · Zbl 0535.30035
[12] H. L. Royden, The boundary values of analytic and harmonic functions, Math. Z. 78 (1962), 1 – 24. · Zbl 0196.33701
[13] Walter Rudin, Analytic functions of class \?_{\?}, Trans. Amer. Math. Soc. 78 (1955), 46 – 66. · Zbl 0064.31203
[14] S. Scheinberg, Thesis, Princeton Univ., 1962.
[15] Zbigniew Słodkowski, Analytic set-valued functions and spectra, Math. Ann. 256 (1981), no. 3, 363 – 386. · Zbl 0452.46028
[16] -, Analytic set-valued functions and spectra, Inst. of Math., Polish Academy of Sciences, Preprint, October 1980. · Zbl 0471.46034
[17] Zbigniew Slodkowski, Analytic multifunctions: \?-plurisubharmonic functions and uniform algebras, Proceedings of the conference on Banach algebras and several complex variables (New Haven, Conn., 1983) Contemp. Math., vol. 32, Amer. Math. Soc., Providence, RI, 1984, pp. 243 – 258. · Zbl 0596.46047
[18] Zbigniew Slodkowski, Polynomial hulls with convex sections and interpolating spaces, Proc. Amer. Math. Soc. 96 (1986), no. 2, 255 – 260. · Zbl 0588.32017
[19] Zbigniew Slodkowski, An analytic set-valued selection and its applications to the corona theorem, to polynomial hulls and joint spectra, Trans. Amer. Math. Soc. 294 (1986), no. 1, 367 – 377. · Zbl 0594.32008
[20] E. L. Stout, Bounded holomorphic functions on finite Reimann surfaces, Trans. Amer. Math. Soc. 120 (1965), 255 – 285. · Zbl 0154.32903
[21] John Wermer, Subalgebras of the algebra of all complex-valued continuous functions on the circle, Amer. J. Math. 78 (1956), 225 – 242. · Zbl 0072.12303
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