Computational Methods and Function Theory

, Volume 7, Issue 1, pp 91–104 | Cite as

A Note on Harmonic Measure



Let Ω be a subregion of {z : |z| < 1} for which the Dirichlet problem is solvable, assume that 0 ∈ Ω and let ω Ω denote harmonic measure on ∂Ω for evaluation at 0. If E is a Borel subset of {z : |z| = 1} and ω Ω (E) > 0, then we find a simply connected region G, where 0 ∈ G ⊆ {z : |z| < 1}, ∂G ⊆ Ω ∪ E and ω G (E) > 0, such that U := G ∪ Ω has the property that w U and w Ω are boundedly equivalent on ∂U. We mention consequences of this in function theory.


Harmonic measure Green’s potential 

2000 MSC

31A15 31A05 46E15 


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  1. 1.
    J. R. Akeroyd, Champagne subregions of the disk whose bubbles carry harmonic measure, Math. Ann. 323 2002) No.2, 267–279.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    J. R. Akeroyd, Another look at some index theorems for the shift, Indiana Univ. Math. J. 50 2001) No.2, 705–718.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    A. Aleman, S. Richter and C. Sundberg, Nontangential limits in P t (μ)-spaces and the index of invariant subspaces, preprint.Google Scholar
  4. 4.
    C. J. Bishop, L. Carleson, J. B. Garnett and P. W. Jones, Harmonic measures supported on curves, Pacific J. Math. 138 1989) No.2, 233–236.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    J. B. Conway, Functions of One Complex Variable, Springer-Verlag, 1973.Google Scholar
  6. 6.
    —, The Theory of Subnormal Operators, Math. Surveys Monographs, Vol. 36 (1991), Amer. Math. Soc., Providence, RI.Google Scholar
  7. 7.
    J. Ortega-Cerda and K. Seip, Harmonic measure and uniform densities, Indiana Univ. Math. J. 53 2004) No.3, 905–924.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    J. B. Garnett, Bounded Analytic Functions, Academic Press, 1981.Google Scholar
  9. 9.
    J. B. Garnett, F. W. Gehring and P. W. Jones, Conformally invariant length sums, Indiana Univ. Math. J. 32 1983) No.6, 809–829.MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Ch. Pommerenke, Boundary Behavior of Conformal Maps, Springer-Verlag, 1992.Google Scholar
  11. 11.
    T. Ransford, Potential Theory in the Complex Plane, London Mathematical Society Student Texts 28, Cambridge University Press, Cambridge, 1995.Google Scholar

Copyright information

© Heldermann  Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ArkansasFayettevilleUSA

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