Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

An elementary proof of Benedicks's and Carleson's estimates of harmonic measure of linear sets


Author: Mikhail Sodin
Journal: Proc. Amer. Math. Soc. 121 (1994), 1079-1085
MSC: Primary 30C85; Secondary 31A15
DOI: https://doi.org/10.1090/S0002-9939-1994-1189752-0
MathSciNet review: 1189752
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Benedicks and Carleson proved sophisticated estimates of harmonic measure of linear sets which have found applications in approximation theory and harmonic analysis. We give an elementary proof of these estimates. This proof allows us to relax the assumptions and strengthen estimates.


References [Enhancements On Off] (What's this?)

  • [AL] N. I. Akhiezer and B. Ya. Levin, Generalizations of Bernstein inequality for derivatives of entire function, Studies in the Modern Problems of Complex Variables, Fizmatgiz, Moscow. (Russian)
  • [Ba] Albert Baernstein II, An extremal problem for certain subharmonic functions in the plane, Rev. Mat. Iberoamericana 4 (1988), no. 2, 199–218. MR 1028739, https://doi.org/10.4171/RMI/71
  • [Be1] Michael Benedicks, Positive harmonic functions vanishing on the boundary of certain domains in 𝑅ⁿ, Ark. Mat. 18 (1980), no. 1, 53–72. MR 608327, https://doi.org/10.1007/BF02384681
  • [Be2] -, Weighted polynomial approximation on subsets of the real line, preprint, Uppsala Univ. Math. Dept., 1981.
  • [C] Lennart Carleson, Estimates of harmonic measures, Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (1982), no. 1, 25–32. MR 667737, https://doi.org/10.5186/aasfm.1982.0704
  • [F] A. E. Fryntov, An extremal problem of potential theory, Dokl. Akad. Nauk SSSR 300 (1988), no. 4, 819–820 (Russian); English transl., Soviet Math. Dokl. 37 (1988), no. 3, 754–755. MR 950785
  • [K] Paul Koosis, The logarithmic integral. I, Cambridge Studies in Advanced Mathematics, vol. 12, Cambridge University Press, Cambridge, 1988. MR 961844
  • [LLS] B. Ya. Levin, V. N. Logvinenko, and M. L. Sodin, Subharmonic functions of finite degree bounded on subsets of the “real hyperplane”, Entire and subharmonic functions, Adv. Soviet Math., vol. 11, Amer. Math. Soc., Providence, RI, 1992, pp. 181–197. MR 1188008
  • [R] L. I. Ronkin, Introduction to the theory of entire functions of several variables, American Mathematical Society, Providence, R.I., 1974. Translated from the Russian by Israel Program for Scientific Translations; Translations of Mathematical Monographs, Vol. 44. MR 0346175
  • [S] A. C. Schaeffer, Entire functions and trigonometric polynomials, Duke Math. J. 20 (1953), 77–88. MR 0052512

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30C85, 31A15

Retrieve articles in all journals with MSC: 30C85, 31A15


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1189752-0
Keywords: Harmonic measure, positive harmonic function in half-plane, capacity
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society