Publicacions Matemàtiques

Homogeneous Subsets of a Lipschitz Graph and the Corona Theorem

Brady Max NewDelman

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

This paper proves the Corona Theorem to be affirmative for domains in the complex plane bounded by thick subsets of a Lipschitz graph. Specifically, the boundary of these domains $E_0$ has a Carleson lower density:

$$ \Lambda\left(B(z,r) \cap E_0\right) > \epsilon_0 r \quad\text{for all } z\in E_0, \quad \text{and all } r>0. $$

Article information

Source
Publ. Mat. Volume 55, Number 1 (2011), 93-121.

Dates
First available in Project Euclid: 25 February 2011

Permanent link to this document
http://projecteuclid.org/euclid.pm/1298670085

Zentralblatt MATH identifier
05849329

Mathematical Reviews number (MathSciNet)
MR2779577

Subjects
Primary: 30-XX: FUNCTIONS OF A COMPLEX VARIABLE {For analysis on manifolds, see 58-XX}

Keywords
Corona harmonic measure homogeneous Lipschitz

Citation

NewDelman, Brady Max. Homogeneous Subsets of a Lipschitz Graph and the Corona Theorem. Publicacions Matemàtiques 55 (2011), no. 1, 93--121. http://projecteuclid.org/euclid.pm/1298670085.


Export citation

References

  • N. L. Alling, A proof of the corona conjecture for finite open Riemann surfaces, Bull. Amer. Math. Soc. 70 (1964), 110\Ndash12.
  • N. L. Alling, Extensions of meromorphic function rings over noncompact Riemann surfaces. I, Math. Z. 89 (1965), 273\Ndash299.
  • M. Behrens, The corona conjecture for a class of infinitely connected domains, Bull. Amer. Math. Soc. 76 (1970), 387\Ndash391.
  • L. Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547\Ndash559.
  • L. Carleson, On $H^{\infty }$ in multiply connected domains, in: “Conference on harmonic analysis in honor of Antoni Zygmund”, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 349\Ndash372.
  • C. J. Earle and A. Marden, Projections to automorphic functions, Proc. Amer. Math. Soc. 19 (1968), 274\Ndash278.
  • F. Forelli, Bounded holomorphic functions and projections, Illinois J. Math. 10 (1966), 367\Ndash380.
  • T. W. Gamelin, Wolff's proof of the corona theorem, Israel J. Math. 37(1–2) (1980), 113\Ndash119.
  • T. W. Gamelin, Localization of the corona problem, Pacific J. Math. 34 (1970), 73\Ndash81.
  • J. B. Garnett, “Bounded analytic functions”, Revised first edition, Graduate Texts in Mathematics 236, Springer, New York, 2007.
  • J. B. Garnett and P. W. Jones, The corona theorem for Denjoy domains, Acta Math. 155(1–2) (1985), 27\Ndash40.
  • M. J. González, Uniformly perfect sets, Green's function, and fundamental domains, Rev. Mat. Iberoamericana 8(2) (1992), 239\Ndash269.
  • M. J. González and A. Nicolau, Quasiconformal mappings preserving interpolating sequences, Ann. Acad. Sci. Fenn. Math. 23(2) (1998), 283\Ndash290.
  • J. Handy, The corona theorem on the complement of certain square Cantor sets, J. Anal. Math. 108 (2009), 1\Ndash18.
  • L. Hörmander, Generators for some rings of analytic functions, Bull. Amer. Math. Soc. 73 (1967), 943\Ndash949.
  • P. W. Jones, Carleson measures and the Fefferman-Stein decomposition of $\operatorname{BMO}({\mathbf R})$, Ann. of Math. (2) 111(1) (1980), 197\Ndash208.
  • P. W. Jones, $L^{\infty}$ estimates for the $\bar \partial $ problem in a half-plane, Acta Math. 150(1–2) (1983), 137\Ndash152.
  • P. W. Jones, Some problems in complex analysis, in: “The Bieberbach conjecture” (West Lafayette, Ind., 1985), Math. Surveys Monogr. 21, Amer. Math. Soc., Providence, RI, 1986, pp. 105\Ndash108.
  • P. W. Jones and D. E. Marshall, Critical points of Green's function, harmonic measure, and the corona problem, Ark. Mat. 23(2) (1985), 281\Ndash314.
  • C. E. Kenig, Weighted $H^{p}$ spaces on Lipschitz domains, Amer. J. Math. 102(1) (1980), 129\Ndash163.
  • B. Muckenhoupt, The equivalence of two conditions for weight functions, Studia Math. 49 (1973/74), 101\Ndash106.
  • Ch. Pommerenke, Uniformly perfect sets and the Poincaré metric, Arch. Math. (Basel) 32(2) (1979), 192\Ndash199.
  • Z. Slodkowski, On bounded analytic functions in finitely connected domains, Trans. Amer. Math. Soc. 300(2) (1987), 721\Ndash736.
  • E. L. Stout, Two theorems concerning functions holomorphic on multiply connected domains, Bull. Amer. Math. Soc. 69 (1963), 527\Ndash530.
  • E. L. Stout, Bounded holomorphic functions on finite Reimann surfaces, Trans. Amer. Math. Soc. 120 (1965), 255\Ndash285.
  • E. L. Stout, On some algebras of analytic functions on finite open Riemann surfaces, Math. Z. 92 (1966), 366\Ndash379; Corrections to: “On some algebras of analytic functions on finite Riemann surfaces”, Math. Z. 95 (1967), 403\Ndash404.
  • M. Tsuji, “Potential theory in modern function theory”, Maruzen Co., Ltd., Tokyo, 1959.
  • Th. Varopoulos, Ensembles pics et ensembles d'interpolation pour les algèbres uniformes, C. R. Acad. Sci. Paris Sér. A-B 272 (1971), A866\NdashA867.