NDLI: A New Class of Harmonic Measure Distribution Functions

Content Provider	Springer Nature Link
Author	Barton, Ariel Ward, Lesley A.
Copyright Year	2013
Abstract	Let Ω be a planar domain containing 0. Let h $_{ Ω }$(r) be the harmonic measure at 0 in Ω of the part of the boundary of Ω within distance r of 0. The resulting function h $_{ Ω }$ is called the harmonic measure distribution function of Ω. In this paper we address the inverse problem by establishing several sets of sufficient conditions on a function f for f to arise as a harmonic measure distribution function. In particular, earlier work of Snipes and Ward shows that for each function f that increases from zero to one, there is a sequence of multiply connected domains X $_{ n }$ such that $h_{X_{n}}$ converges to f pointwise almost everywhere. We show that if f satisfies our sufficient conditions, then f=h $_{ Ω }$, where Ω is a subsequential limit of bounded simply connected domains that approximate the domains X $_{ n }$. Further, the limit domain is unique in a class of suitably symmetric domains. Thus f=h $_{ Ω }$ for a unique symmetric bounded simply connected domain Ω.
Starting Page	2035
Ending Page	2071
Page Count	37
File Format	PDF
ISSN	10506926
Journal	Journal of Geometric Analysis
Volume Number	24
Issue Number	4
e-ISSN	1559002X
Language	English
Publisher	Springer US
Publisher Date	2013-03-29
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Harmonic measure Planar domains Brownian motion Harmonic measure distribution functions Simply connected domains Capacity and harmonic measure in the complex plane Conformal mappings of special domains Potentials and capacity, harmonic measure, extremal length Differential Geometry Convex and Discrete Geometry Fourier Analysis Abstract Harmonic Analysis Dynamical Systems and Ergodic Theory Global Analysis and Analysis on Manifolds
Content Type	Text
Resource Type	Article
Subject	Geometry and Topology

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