On properties of multipliers of Cauchy transforms
type: Item
instance of: scholarly article
title: On properties of multipliers of Cauchy transforms
zbMATH Open document ID: 0928.30023
author: Krzysztof Samotij
author: David J. Hallenbeck
published in: Rocky Mountain Journal of Mathematics
publication date: 2000-01-09T00:00:00Z
full work available at URL: CONT281.html
review text: For \(\alpha>0\), \({\mathcal F}_\alpha\) denotes the set of functions \(g\) on \(\Delta= \{z:| z|<1\}\) such that \(g(z)=\int_\Gamma(1-xz)^{-\alpha} d\mu(x)\) for some complex-valued Borel measure \(\mu\) on the unit circle \(\Gamma\). \(M_\alpha\), the set of multipliers of \({\mathcal F}_\alpha\), is the set of functions \(f\) on \(\Delta\) such that \(fg\) is in \({\mathcal F}_\alpha\) for every \(g\) in \({\mathcal F}_\alpha\). Similar definitions are given for \({\mathcal F}_0\) and \(M_0\). Norms are defined on these spaces, under which they become Banach spaces. The present paper continues the work of the authors and of their collaborators. Their main result gives a bound on the length of the arc \(f\circ\gamma\) for \(f\) in \(M_\alpha\), and \(\gamma\) a rectifiable arc in a Stolz angle. This length is at most \(C\| f\|_{M_\alpha}\), where \(C\) depends on the Stolz angle and the geometry of \(\gamma\). In particular if \(\alpha>0\) and \(\gamma\) is a chord in the unit disk of length \(\ell\) at most 2, then the length of \(f\circ\gamma\) is at most \((C_3/ \ell)^{\alpha+1} \| f\|_{M_\alpha}\), where \(C_3\) is an absolute constant. Another consequence is a bound on the area of the image of the Stolz angle. These results are shown to be sharp. For instance, for \(\alpha >0\) and \(p>1\), there is a function \(f\) in \(M_\alpha\) such that \(\int^1_0 | f'(p)|^p dp=\infty\). Finally, they show that strong restrictions apply for all derivatives \(f^{(n)}\) of analytic functions which are multipliers of \({\mathcal F}_\alpha\), \(\alpha>0\).
reviewed by: Francis W. Carroll
Mathematics Subject Classification ID: 30E20
Mathematics Subject Classification ID: 30D99
zbMATH DE Number: 1286535
MaRDI profile type: Publication
OpenAlex ID: W2071365751
cites work: Fractional Cauchy Transforms, Inner Functions and Multipliers
cites work: On Cauchy integrals of logarithmic potentials and their multipliers
cites work: Multipliers of Families of Cauchy-Stieltjes Transforms
cites work: On the coefficients of close-to-convex functions
cites work: Properties of multipliers of Cauchy-Stieltjes integrals and some factorization problems for analytic functions
DOI: 10.1216/RMJM/1181071830
description: scientific article; zbMATH DE number 1286535