Hadamard products with power functions and multipliers of Hardy spaces
type: Item
instance of: scholarly article
title: Hadamard products with power functions and multipliers of Hardy spaces
zbMATH Open document ID: 1028.30028
DOI: 10.1016/S0022-247X(03)00128-8
author: Thomas H. MacGregor
author: M. P. Sterner
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-07-30T00:00:00Z
review text: In this interesting paper the authors study Hadamard products of power functions given by \(P(z)=(1-z)^{-b}\), where \(b\in \mathbf{C}\), with analytic functions in the open unit disk in the complex plane. They obtain an integral representation when \(0<\operatorname {Re} b<2\). Also the authors show that sequences \(\{\mu_n\}\) of the form \(\mu_n=\int_{\overline{\Delta}}\zeta^n d\mu(\zeta)\) are multipliers of the Hardy spaces \(H^p\) for \(1\leq p\leq \infty\), where \(\mu\) is a complex-valued measure on the closed unit disc \(\overline{\Delta}\). Moreover, in the case that the support of \(\mu\) is contained in a finite set of Stolz angles with vertices on the unit circle, then it is proved that the sequence \(\{\mu_n\}\) is a multiplier of \(H^p\) for any \(p>0\). The authors also propose the following conjectures: Conjecture 1. If \(\beta\) is a nonzero real number then \(\{(n+1)^{i\beta}\}\) is a multiplier of \(H^p\) for \(0<p<\infty\). The authors mention that this conjecture is true when \(p=2\). Conjecture 2. If \(p>0\) and \(0<\alpha<1/p\) then the sequence \(\{1/(n+1)^b\}\) is a multiplier of \(H^p\) into \(H^q\), where \(q=p/(1-\alpha p)\) and \(\alpha=\operatorname {Re} b\).
reviewed by: G. Kohr
Mathematics Subject Classification ID: 30D55
Mathematics Subject Classification ID: 46J15
zbMATH DE Number: 1956909
zbMATH Keywords: analytic function
zbMATH Keywords: convolution
zbMATH Keywords: Hadamard products
zbMATH Keywords: Hardy spaces
zbMATH Keywords: Minkowski's inequality
MaRDI profile type: Publication
cites work: On the Multipliers of H p Spaces
cites work: Q5618011
cites work: Some properties of fractional integrals. II
cites work: Q4059363
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description: scientific article; zbMATH DE number 1956909