On a classification of plane domains for BMOA
type: Item
instance of: scholarly article
title: On a classification of plane domains for BMOA
zbMATH Open document ID: 0543.30023
author: Shōji Kobayashi
published in: Kodai Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: The space of analytic functions of bounded mean oscillation has been studied by many authors over the past few years. Here, using an idea due to the reviewer [Can. J. Math. 33, 1255-1260 (1981; Zbl 0442.30030)], the author studies BMOA(G) for arbitrary domains G. Specifically, an analytic function f(z) is said to belong to BMOA(G) if \[ \sup_{a\in G}\iint_{G}| f'(z)|^ 2g(z,a)dxdy<\infty, \] where \(g(\cdot,a)\) is the Green's function for G with pole at a. If u(z) is a subharmonic function on G then û(z) is defined to be the least harmonic majorant of u(z) in G. The author proves that f(z) belongs to BMOA(G) if and only if \(\sup_{a\in G}\hat u_ a(a)<\infty\) where for \(u_ a(z)\) one can take either \(| f(z)-f(a)|^ p\) for some \(p>0\) or \(\log^+| f(z)-f(a)|.\) In the second half of the paper it is shown that the classification results \(\cup_{p>0}O_ p\subset O_{BMOA}\subset O_{AB},\) originally proved by the reviewer (op. cit.) are all strict containments by constructing planar regions which satisfy the conditions needed to prove the stated conclusions.
Mathematics Subject Classification ID: 30D50
zbMATH DE Number: 3863551
zbMATH Keywords: bounded mean oscillation
zbMATH Keywords: Green's function
zbMATH Keywords: subharmonic function
reviewed by: Thomas A. Metzger
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138036860
OpenAlex ID: W2028531173
cites work: Q3954045
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DOI: 10.2996/KMJ/1138036860
description: scientific article; zbMATH DE number 3863551