Superharmonic functions and bounded point evaluations
type: Item
instance of: scholarly article
title: Superharmonic functions and bounded point evaluations
zbMATH Open document ID: 0691.31001
DOI: 10.1155/S0161171289000566
author: Edwin Wolf
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/46440
review text: Summary: Let E be a compact subset of the complex plane \({\mathbb{C}}\). We denote by \(R_ 0(E)\) the algebra consisting of the (restrictions to E of) rational functions with poles off E. Let m denote the 2-dimensional Lebesgue measure. Let \(R^ 2(E)\) be the closure of \(R_ 0(E)\) in \(L^ 2(E,dm).\) We consider points \(x\in E\) such that ``evaluation at x'' extends from \(R_ 0(E)\) to a continuous linear functional on \(R^ 2(E)\). These points are bounded point evaluations on \(R^ 2(E)\). Hedberg, Fernström and Polking used capacity to identity bounded point evaluations. We use their results to show that the existence of a bounded point evaluation \(x\in E\) is equivalent to the existence of a superharmonic function u(y) that grows sufficiently fast as y approaches x through the complement of E.
Mathematics Subject Classification ID: 31A05
Mathematics Subject Classification ID: 31A15
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 4130750
zbMATH Keywords: bounded point evaluation
zbMATH Keywords: balayage
zbMATH Keywords: Green function
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4130750