A new class of harmonic measure distribution functions

TL;DR

This paper characterizes when a function can be realized as the harmonic measure distribution function of a planar domain, establishing conditions for uniqueness and construction of such domains.

Contribution

It provides sufficient conditions for a function to be a harmonic measure distribution function and proves the uniqueness of the associated domain under symmetry constraints.

Findings

Conditions under which a function is a harmonic measure distribution function.

Existence of a domain corresponding to a given harmonic measure distribution function.

Uniqueness of the domain in a symmetric class.

Abstract

Let D be a planar domain containing 0. Let h_D(r) be the harmonic measure at 0 in D of the part of the boundary of D within distance r of 0. The resulting function h_D is called the harmonic measure distribution function of D. 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_D, where D 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_D for a unique symmetric bounded simply connected domain D.