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.
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Communicated by Michael Lacey.
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Barton, A., Ward, L.A. A New Class of Harmonic Measure Distribution Functions. J Geom Anal 24, 2035–2071 (2014). https://doi.org/10.1007/s12220-013-9408-7
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DOI: https://doi.org/10.1007/s12220-013-9408-7
Keywords
- Harmonic measure
- Planar domains
- Brownian motion
- Harmonic measure distribution functions
- Simply connected domains