Computational Methods and Function Theory

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

A Note on Harmonic Measure

  • John R. Akeroyd


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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Copyright information

© Heldermann  Verlag 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ArkansasFayettevilleUSA

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