Title

Harmonic measure distributions of planar domains: a survey

Document Type

Article

Publication Date

2016

Abstract

Given a domain Ω" role="presentation" style="box-sizing: inherit; display: inline-table; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">ΩΩ in the complex plane and a basepoint z0∈Ω" role="presentation" style="box-sizing: inherit; display: inline-table; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">z0∈Ωz0∈Ω, the harmonic measure distribution functionh:(0,∞)→[0,1]" role="presentation" style="box-sizing: inherit; display: inline-table; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">h:(0,∞)→[0,1]h:(0,∞)→[0,1] of the pair (Ω,z0)" role="presentation" style="box-sizing: inherit; display: inline-table; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">(Ω,z0)(Ω,z0) maps each radius r>0" role="presentation" style="box-sizing: inherit; display: inline-table; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">r>0r>0 to the harmonic measure of the part of the boundary ∂Ω" role="presentation" style="box-sizing: inherit; display: inline-table; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">∂Ω∂Ω within distance r of the basepoint. This function was first introduced by Walden and Ward, inspired by a question posed by Stephenson, as a signature that encodes information about the geometry of Ω" role="presentation" style="box-sizing: inherit; display: inline-table; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">ΩΩ. It has subsequently been studied in various works. Two main goals of harmonic measure distribution studies are (1) to understand precisely what can be determined about a domain from its h-function, and (2) given a function f:(0,∞)→[0,1]" role="presentation" style="box-sizing: inherit; display: inline-table; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">f:(0,∞)→[0,1]f:(0,∞)→[0,1], to determine whether there exists a pair (Ω,z0)" role="presentation" style="box-sizing: inherit; display: inline-table; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">(Ω,z0)(Ω,z0) that has f as its h-function. In this survey paper, we present key examples of h-functions and summarize results related to these two goals. In particular, we discuss what is known about uniqueness of domains that generate h-functions, necessary conditions and sufficient conditions for a function to be an h-function, asymptotic behavior of h-functions, and convergence results involving h-functions. We also highlight current open problems.

Journal

The Journal of Analysis

First Page

1

Last Page

38