A conjecture of L. Carleson and applications
type: Item
instance of: scholarly article
title: A conjecture of L. Carleson and applications
zbMATH Open document ID: 0773.30025
author: Alexandru Aleman
published in: Michigan Mathematical Journal
publication date: 1993-06-29T00:00:00Z
review text: For a meromorphic function \(f\) in \(\{| z|<1\}\) let \(A(r,f)\) be the spherical area of the image of \(\{| z|<r\}\) by \(f\). Let \(T_ \alpha\) be the class of functions for which \[ \int^ 1_ 0 A(r,f)(1- r)^{-\alpha} dr< \infty,\qquad 0\leq \alpha<1. \] \(T_ 1\) is the class of functions \(f\) with the property, that \(A(r,f)\) is bounded. Thus, \(T_ 0\) is the usual Nevanlinna class but for \(\alpha>0\), \(T_ \alpha\) does not contain all bounded analytic functions. Carleson asked whether each function in \(T_ \alpha\) is the quotient of two bounded functions in the same class and this question is answered affirmatively in the paper for more general classes of meromorphic functions in the disc. The technique used for the proof is then applied to study the inner-outer factorization in these classes and to prove that invertible elements in weighted Dirichlet spaces are cyclic for multiplication by \(z\), i.e. their polynomial multiples are dense in the space.
Mathematics Subject Classification ID: 30D30
Mathematics Subject Classification ID: 30H05
zbMATH DE Number: 222036
MaRDI profile type: Publication
Wikidata QID: Q122918421
full work available at URL: https://doi.org/10.1307/mmj/1029004605
OpenAlex ID: W2039698190
DOI: 10.1307/MMJ/1029004605
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description: scientific article; zbMATH DE number 222036