On the Makarov law of the iterated logarithm
Authors:
Hå kan Hedenmalm and Ilgiz Kayumov
Journal:
Proc. Amer. Math. Soc. 135 (2007), 22352248
MSC (2000):
Primary 35R35, 35Q35; Secondary 31A05, 31C12, 53B20, 76D27
Published electronically:
February 6, 2007
MathSciNet review:
2299501
Fulltext PDF Free Access
Abstract 
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Additional Information
Abstract: We obtain considerable improvement of Makarov's estimate of the boundary behavior of a general conformal mapping from the unit disk to a simply connected domain in the complex plane. We apply the result to improve Makarov's comparison of harmonic measure with Hausdorff measure on simply connected domains.
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 1.
 L. Carleson, On the distortion of sets on a Jordan curve under conformal mapping, Duke Math. J. 40 (1973), 547559. MR 0330430 (48:8767)
 2.
 H. Hedenmalm, S. Shimorin, Weighted Bergman spaces and the integral means spectrum of conformal mappings, Duke Math. J. 127 (2005), 341393. MR 2130416 (2005m:30010)
 3.
 H. Hedenmalm, S. Shimorin, On the universal integral means spectrum of conformal mappings near the origin, Proc. Amer. Math. Soc., to appear.
 4.
 I. R. Kayumov, The integral means spectrum for lacunary series, Ann. Acad. Sci. Fenn. Math. 26 (2001), 447453. MR 1833250 (2002g:30002)
 5.
 I. R. Kayumov, The law of the iterated logarithm for locally univalent functions, Ann. Acad. Sci. Fenn. Math. 27 (2002), 357364. MR 1921312 (2003g:30029)
 6.
 I. R. Kayumov, On the law of the iterated logarithm for conformal mappings, Math. Notes 79 (2006), 139142.
 7.
 I. R. Kayumov, Lower estimates for integral means of univalent functions, Ark. Mat. (to appear).
 8.
 N. G. Makarov, On the distortion of boundary sets under conformal mappings, Proc. London Math. Soc. 51 (1985), no 3, 369384. MR 0794117 (87d:30012)
 9.
 Ch. Pommerenke, Boundary Behaviour of Conformal Maps, Grundlehren der mathematischen Wissenschaften 299, SpringerVerlag, Berlin, 1992. MR 1217706 (95b:30008)
 10.
 F. Przytycki, M. Urbanski, A. Zdunik, Harmonic, Gibbs, and Hausdorff measures on repellers for holomorphic maps. I, Ann. Math. 2 (1989), 140. MR 1005606 (91i:58115)
 11.
 F. Przytycki, M. Urbanski, A. Zdunik, Harmonic, Gibbs, and Hausdorff measures on repellers for holomorphic maps. II, Studia Math. 97 (1991), 189225. MR 1100687 (93d:58140)
 12.
 W. Smith, D. A. Stegenga, Exponential integrability of the quasihyperbolic metric on Hölder domains, Ann. Acad. Sci. Fenn. Ser. A I Math. 16 (1991), no. 2, 345360. MR 1139802 (93b:30016)
 13.
 M. Weiss, On the law of the iterated logarithm for lacunary trigonometric series, Trans. Amer. Math. Soc. 91 (1959), 444469. MR 0108681 (21:7396)
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Additional Information
Hå kan Hedenmalm
Affiliation:
Department of Mathematics, The Royal Institute of Technology, S – 100 44 Stockholm, Sweden
Email:
haakanh@math.kth.se
Ilgiz Kayumov
Affiliation:
Institute of Mathematics and Mechanics, Kazan State University, 420008 Kazan, Russia
Email:
ikayumov@ksu.ru
DOI:
http://dx.doi.org/10.1090/S0002993907087722
PII:
S 00029939(07)087722
Keywords:
Conformal mapping,
law of the iterated logarithm
Received by editor(s):
October 26, 2005
Received by editor(s) in revised form:
March 29, 2006
Published electronically:
February 6, 2007
Additional Notes:
Research supported by the Göran Gustafsson Foundation and by the Russian Fund of Basic Research (050100523, 030100015).
Communicated by:
Juha M. Heinonen
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
