Littlewood's one-circle problem, revisited
type: Item
instance of: scholarly article
title: Littlewood's one-circle problem, revisited
zbMATH Open document ID: 1160.31002
author: Wolfhard Hansen
published in: Expositiones Mathematicae
publication date: 2008-11-06T00:00:00Z
review text: Let \(U\) be the open unit disc in the plane, and let \(f\) be a continuous, bounded function on \(U\). Suppose that, for each \(x\in U\), there exists \textit{one} radius \(r(x)\leq 1-|x|\) such that \[ f(x)=\frac{1}{2\pi}\int_{0}^{2\pi}f\big(x+r(x)(\cos t,\sin t)\big)\,dt. \] Littlewood's one-circle problem is to decide whether \(f\) is necessarily harmonic. In an earlier paper [J. Lond. Math. Soc., II. Ser. 50, No. 2, 349--360 (1994; Zbl 0804.31001)], the author and N. Nadirashvili gave an example to show that \(f\) need not be harmonic. In the present paper, a simpler construction is given, which depends on the work of \textit{M. Talagrand} [Bull. Sci. Math., II. Ser. 104, 225--231 (1980; Zbl 0437.28004)].
reviewed by: Neil A. Watson
Mathematics Subject Classification ID: 31A05
Mathematics Subject Classification ID: 28A75
Mathematics Subject Classification ID: 60J10
zbMATH DE Number: 5363193
zbMATH Keywords: harmonic function
zbMATH Keywords: mean value property
zbMATH Keywords: length
zbMATH Keywords: area
zbMATH Keywords: Markov chain
MaRDI profile type: Publication
full work available at URL: j.exmath.2008.04.001
OpenAlex ID: W2076260442
cites work: A Strong Version of Liouville's Theorem
cites work: Liouville's theorem and the restricted mean value property in the plane
cites work: A Liouville property for spherical averages in the plane
cites work: A converse to the mean value theorem for harmonic functions
cites work: Littlewood's One Circle Problem
cites work: Liouville's theorem and the restricted mean value property
cites work: Harmonic functions and averages on shells
cites work: Q5577283
cites work: Q3877902
DOI: 10.1016/J.EXMATH.2008.04.001
description: scientific article; zbMATH DE number 5363193