Hadamard products with power functions and multipliers of Hardy spaces
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 \(\alpha=\operatorname {Re} b\).
scientific article; zbMATH DE number 1956909