Publicacions Matemàtiques

Homogeneous Subsets of a Lipschitz Graph and the Corona Theorem

Brady Max NewDelman

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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

Mathematical Reviews number (MathSciNet)
MR2779577

Zentralblatt MATH identifier
05849329

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. Publ. Mat. 55 (2011), no. 1, 93--121. http://projecteuclid.org/euclid.pm/1298670085.


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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.