Analytic functions of bounded mean oscillation and the Bloch space. (English) Zbl 0796.46011

Summary: We show that the closure of the space BMOA of analytic functions of bounded mean oscillation in the Bloch space \({\mathcal B}\) is the image \(P({\mathcal U})\) of the space of all continuous functions on the maximal ideal space of \(H^ \infty\) under the Bergman projection \(P\). It is proved that the radial growth of functions in \(P({\mathcal U})\) is slower than the iterated logarithm studied by Makarow. So some geometric conditions are given for functions \(P({\mathcal U})\), which we can easily use to construct many Bloch functions not in \(P({\mathcal U})\).


46E15 Banach spaces of continuous, differentiable or analytic functions
46J15 Banach algebras of differentiable or analytic functions, \(H^p\)-spaces
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[1] L.V. Ahlfors, Quasiconformal reflections, Acta Math. 109 (1963) 291-301. · Zbl 0121.06403
[2] J. M. Anderson, Bloch Functions: The Basic Theory, Operators and Function Theory (ed. S. C. power), NATO ASI Series C, Vol. 152, Reidel, Bordrecht, 1984, 1-17.
[3] J. M. Anderson, J. G. Clunie and Ch. Pommerenke, On Bloch functions and normal functions, J. Reime und Angewandte Math. 270 (1974), 12-37.
[4] S. Axler, Bergman spaces and their operators, Surveys of some recent results in operator theory, (ed. J. Conway and B. Morrel) Pitman Research Notes in Mathematics 171 (1988), 1-50.
[5] S. Axler and P. Gorkin, Algebras on the disk and doubly commuting multiplication operators, Transactions AMS (2) 309 (1988), 711-723. · Zbl 0706.46040
[6] S. Axler and K. Zhu, Boundary behavior of derivatives of analytic function, Michigan Math. J. 39 (1992), 129-143. · Zbl 0769.30020
[7] A. Baernstein II, Analytic functions of bounded mean oscillation, Aspects of Contemporary Complex Analysis (ed. D.A. Brannan and J. G. Clunie), Academic Press, London (1980), 3-36. · Zbl 0492.30026
[8] F. Bonsall, Hankel operators on the Bergman space for the disc, J. London Math. Soc. (2) 33 (1986), 355-364. · Zbl 0604.47014
[9] R. Bañuelos and C. Moore, Mean growth of Bloch functions and Makarov’s law of the iterated logarithm, Proceedings AMS 112 (1991), 851-854. · Zbl 0737.30024
[10] L. Carleson, An interpolating problem for bounded analytic functions, American J. Math. 80 (1958), 921-930. · Zbl 0085.06504
[11] L. Carleson, Interpolations by bounded analytic functions and the corona problem, Annals. of Math. 96 (1962), 547-559. · Zbl 0112.29702
[12] J. Garnett, Bounded analytic functions, Pure and Applied Mathematics 96, Academic Press, New York (1981). · Zbl 0469.30024
[13] P. Ghatage, Lifting Hankel operators from the Hardy space to the Bergman space, Rocky Mountain J. Math. 20 (1990), 433-437. · Zbl 0722.47029
[14] K. Hoffman, Analytic functions and logmodular Banach algebras, Acta Math. 108 (1962), 271-317. · Zbl 0107.33102
[15] K. Hoffman, Banach analytic functions, Prentice-Hall, Englewood N. J., 1962. · Zbl 0117.34001
[16] K. Hoffman, Bounded analytic functions and Gleason parts, Annals of Math. 86 (1967), 74-111. · Zbl 0192.48302
[17] P. W. Jones, Square functions, Cauchy integrals, analytic capacity and harmonic measure, Lecture Notes in Mathematics 1384, Springer-Verlag (1988), 24-68.
[18] N. Makarov, On the distortion of boundary sets under conformal mappings, Proc. London Math. Soc. 51 (1985), 369-384. · Zbl 0573.30029
[19] N. Makarov, Metric properties of Hausdorff measures, Proceedings of the International Congress of Math., Berkeley, California, 1986. · Zbl 0628.35070
[20] Ch. Pommerenke, On Bloch functions, J. London. Math. Soc. 2 (1970) 689-695.
[21] Ch. Pommerenke, Univalent functions, Vanderhoek and Ruprecht, Gottingen, 1975.
[22] Ch. Pommerenke, On univalent functions, Bloch functions and VMOA, Math. Ann. 236 (1978), 199-208. · Zbl 0385.30013
[23] Ch. Pommerenke, The growth of the derivative of a univalent function, J. London Math. Soc. 32 (1985), 254-258. · Zbl 0576.30017
[24] F. Przytycki, On law of iterated logarithm for Bloch functions, Studia Math. 93 (1989), 145-154. · Zbl 0682.30027
[25] F. Przytycki, M. Urbanski and A. Zdunik, Harmonic Hausdroff and Gibbs measures on repels for holomorphic maps, I, Annals of Math. 130 (1989), 1-40. · Zbl 0703.58036
[26] D. Sarason, Function theory on the unit circle, Lecture notes, Virginia Polytechnic Institute and State University. · Zbl 0398.30027
[27] D. Zheng, Toeplitz operators and Hankel operators, Integral Equations and Operator Theory 12 (1989), 280-299. · Zbl 0692.47031
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