On bounded analytic functions in finitely connected domains

Slodkowski, Zbigniew

doi:10.1090/S0002-9947-1987-0876475-9

On bounded analytic functions in finitely connected domains
HTML articles powered by AMS MathViewer

by Zbigniew Slodkowski

Trans. Amer. Math. Soc. 300 (1987), 721-736

DOI: https://doi.org/10.1090/S0002-9947-1987-0876475-9

PDF | Request permission

Abstract:

A new proof of the corona theorem for finitely connected domains is given. It is based on a result on the existence of a meromorphic selection from an analytic set-valued function. The latter fact is also applied to the study of finitely generated ideals of ${H^\infty }$ over multiply connected domains.

References

Herbert Alexander and John Wermer, Polynomial hulls with convex fibers, Math. Ann. 271 (1985), no. 1, 99–109. MR 779607, DOI 10.1007/BF01455798
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. MR 850666
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. MR 739334, DOI 10.1007/BFb0073355
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. MR 694693
Frank Forelli, Bounded holomorphic functions and projections, Illinois J. Math. 10 (1966), 367–380. MR 193534
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
T. W. Gamelin, Localization of the corona problem, Pacific J. Math. 34 (1970), 73–81. MR 276742
T. W. Gamelin, Uniform algebras and Jensen measures, London Mathematical Society Lecture Note Series, vol. 32, Cambridge University Press, Cambridge-New York, 1978. MR 521440

Critical points of Green’s function, harmonic measure and the corona problem

K. V. Rajeswara Rao, On a generalized corona problem, J. Analyse Math. 18 (1967), 277–278. MR 210910, DOI 10.1007/BF02798049
T. J. Ransford, Interpolation and extrapolation of analytic multivalued functions, Proc. London Math. Soc. (3) 50 (1985), no. 3, 480–504. MR 779400, DOI 10.1112/plms/s3-50.3.480
H. L. Royden, The boundary values of analytic and harmonic functions, Math. Z. 78 (1962), 1–24. MR 138747, DOI 10.1007/BF01195147
Walter Rudin, Analytic functions of class $H_p$, Trans. Amer. Math. Soc. 78 (1955), 46–66. MR 67993, DOI 10.1090/S0002-9947-1955-0067993-3
Zbigniew Słodkowski, Analytic set-valued functions and spectra, Math. Ann. 256 (1981), no. 3, 363–386. MR 626955, DOI 10.1007/BF01679703

Analytic set-valued functions and spectra

Zbigniew Slodkowski, Analytic multifunctions: $q$-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. MR 769513, DOI 10.1090/conm/032/769513
Zbigniew Slodkowski, Polynomial hulls with convex sections and interpolating spaces, Proc. Amer. Math. Soc. 96 (1986), no. 2, 255–260. MR 818455, DOI 10.1090/S0002-9939-1986-0818455-X
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. MR 819954, DOI 10.1090/S0002-9947-1986-0819954-1
E. L. Stout, Bounded holomorphic functions on finite Reimann surfaces, Trans. Amer. Math. Soc. 120 (1965), 255–285. MR 183882, DOI 10.1090/S0002-9947-1965-0183882-4
John Wermer, Subalgebras of the algebra of all complex-valued continuous functions on the circle, Amer. J. Math. 78 (1956), 225–242. MR 84729, DOI 10.2307/2372513