Hausdorff operators on weighted Banach spaces of type \(H^\infty\)
type: Item
instance of: scholarly article
title: Hausdorff operators on weighted Banach spaces of type \(H^\infty\)
author: José Bonet
published in: Complex Analysis and Operator Theory
publication date: 2022-01-04T00:00:00Z
review text: Let \(\mu\) a positive measure on \((0,\infty)\) and \(K\) be a \(\mu\)-measurable function on \(\mathbb{D}\). The paper is devoted to study the continuity of Hausdorff operators induced by the measure \(\mu\), \(\mathcal{H}_{\mu}\), and associated with \(\mu\) and \(K\), \(\mathcal{H}_{K,\mu}\), on weighted Banach spaces of holomorphic functions of type \(H^{\infty}\) which are \begin{align*} H_v^{\infty} (G) = & \{ f \in H(G): ||f||_v = \sup_{z \in \mathbb{D}} v(z) |f(z)| < \infty \}, \\ H_v^0 (G) = & \{ f \in H(G): v|f| \text{ vanishes at infinity on }G \}, \end{align*} where \(G\) is \(\mathbb{D}\) or \(\mathbb{C}\) and \(v\) is a weight function with some conditions. Among other results, it was proved that the continuity of \(\mathcal{H}_{\mu} : H_v^{\infty} (\mathbb{C}) \rightarrow H_v^{\infty} (\mathbb{C})\) implies the continuity of \(\mathcal{H}_{\mu} : H_v^{0} (\mathbb{C}) \rightarrow H_v^{0} (\mathbb{C})\) and both imply that \[ \sup_{n \in \mathbb{N}_0} \int_0^{\infty} \frac{1}{t^{n+1}} d \mu (t) \leq ||H_{\mu}|| < \infty. \]
reviewed by: Mostafa Hassanlou
Mathematics Subject Classification ID: 47B91
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 30H20
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 7451890
zbMATH Keywords: weighted Banach spaces of holomorphic functions
zbMATH Keywords: Hausdorff operator
zbMATH Keywords: entire functions
MaRDI profile type: Publication
full work available at URL: s11785-021-01189-1
OpenAlex ID: W4200439355
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DOI: 10.1007/S11785-021-01189-1
description: scientific article; zbMATH DE number 7451890