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The Corona theorem for Denjoy domains. (English) Zbl 0578.30043
The authors prove the famous corona theorem for the space of bounded analytic functions on any Denjoy domain, i.e. a connected open subset $$\Omega$$ of the extended complex plane $${\mathcal C}^*$$ such that the complement $${\mathcal C}^*\setminus \Omega$$ is a subset of the real axis. The proof utilizes the symmetry of the Denjoy domain and the relations between length, harmonic measure, relative to the upper half plane, and analytic capacity of linear sets.
Reviewer: A.Zabulionis

##### MSC:
 30H05 Spaces of bounded analytic functions of one complex variable 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 46J20 Ideals, maximal ideals, boundaries
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##### References:
 [1] Ahlfors, L. &Beurling, A., Conformal invariants and function-theoretic null-sets.Acta Math., 83 (1950), 101–129. · Zbl 0041.20301 [2] Carleson, L. An interpolation problem for bounded analytic functions.Amer. J. Math., 80 (1958), 921–930. · Zbl 0085.06504 [3] –, Interpolation by bounded analytic functions and the corona theorem.Ann of Math., 76 (1962), 547–559. · Zbl 0112.29702 [4] –, OnH in multiply connected domains.Conference on harmonic analysis in honor of Antoni Zygmund, Vol. 2. Wadsworth Inc., 1983, pp. 349–372. [5] Carleson, L. &Garnett, J., Interpolating sequences and separation properties.J. Analyse Math., 28 (1975), 273–299. · Zbl 0347.30032 [6] Gamelin, T., Localization of the corona problem.Pacific J. Math., 34 (1970), 73–81. · Zbl 0199.18801 [7] Gamelin, T.,Uniform algebras and Jensen measures. London Math. Soc. Lecture Note Series, No. 32, 1978. · Zbl 0418.46042 [8] Garnett, J.,Analytic capacity and measure. Springer-Verlag, Lecture Notes in Mathematics, 297, 1972. · Zbl 0253.30014 [9] Garnett, J. Bounded analytic functions. Academic Press, 1981. · Zbl 0469.30024 [10] Hoffman, K., Bounded analytic functions and Gleason parts.Ann. of Math., 86 (1967), 74–111. · Zbl 0192.48302 [11] Hörmander, L., Generators for some rings of analytic functions,Bull. Amer. Math. Soc., 73 (1967), 943–949. · Zbl 0172.41701 [12] Jones, P. W., Carleson measures and the Fefferman-Stein decomposition of BMO (R).Ann. of Math., 111 (1980), 197–208. · Zbl 0416.30030 [13] –,L estimates for the $$\bar \partial$$ problem in a half-plane.Acta Math., 150 (1983), 137–152. · Zbl 0516.35060 [14] Jones, P. W., OnL solutions of $$\bar \partial$$ in domains with thick boundary. To appear. [15] Jones, P. W. & Marshall, D. E.,Critical points of Green’s function, harmonic measure, and the corona problem. To appear inArk. Mat., 23 (1985). · Zbl 0589.30028 [16] Varopoulos, N. Th., ”Ensembles pics et ensembles d’interpolation d’une algèbre uniforme.C. R. Acad. Sci. Paris, Ser. A., 272 (1971), 866–867. · Zbl 0214.13903 [17] Ziskind, S., Interpolating sequences and the Shilov boundary ofH ({$$\Delta$$}).J. Funct. Anal., 21 (1976), 380–388. · Zbl 0323.46053
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