Random series and bounded mean oscillation
type: Item
instance of: scholarly article
title: Random series and bounded mean oscillation
zbMATH Open document ID: 0565.30001
author: Peter Duren
published in: Michigan Mathematical Journal
publication date: 1985-01-01T00:00:00Z
review text: It has long been known that if \(\sum | a_ n|^ 2<\infty\), then for almost every choice of signs \(\epsilon_ n=\pm 1\), the function \(f(z)=\sum \epsilon_ na_ nz^ n\) belongs to \(H^ p\) for every \(p<\infty\). It was recently discovered that if \(\sum | a_ n|^ 2\log n<\infty,\) then \(f\in BMOA\) almost surely. The author explores the effect of the intermediate condition \(\sum | a_ n|^ 2(\log n)^{\beta}<\infty,\) where \(0\leq \beta \leq 1\). The main result is that this implies \[ \int^{1}_{0}(1-r)| \log (1-r)|^{\beta - 1}[M_{\infty}(r,f')]^ 2dr<\infty \quad a.s. \]
Mathematics Subject Classification ID: 30B20
Mathematics Subject Classification ID: 30D55
zbMATH DE Number: 3901097
zbMATH Keywords: bounded mean oscillation
zbMATH Keywords: Bloch functions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029003134
OpenAlex ID: W2089801854
DOI: 10.1307/MMJ/1029003134
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description: scientific article; zbMATH DE number 3901097