Essential norms of integration operators on weighted Bergman spaces
type: Item
instance of: scholarly article
title: Essential norms of integration operators on weighted Bergman spaces
zbMATH Open document ID: 1370.30025
author: Santeri Miihkinen
author: Pekka J. Nieminen
author: Wen Xu
published in: Journal of Mathematical Analysis and Applications
publication date: 2017-02-14T00:00:00Z
review text: Let \(\omega\) be a radial weight on the open unit disc \(\mathbb{D}\subset \mathbb{C}\) satisfying the doubling property \(\omega\left([r,1)\right)\leq C\omega\left([(1+r)/2,1)\right)\), \(0\leq r<1\). Let \(A^p_\omega\) be its induced weighted Bergman space. The authors obtain operator norm and essential operator norm estimates for the Volterra integral operator \(T_g: A^p_\omega\to A^q_\omega\), \(0<p\leq q<\infty\), defined by \(T_g(f)(z)=\int_0^z f(\zeta)g'(\zeta)d\zeta\). The main results prove that if \(0\leq \frac 1p-\frac 1q<1\) then these operators satisfy \[ \|T_g\|^2\simeq \sup_{a\in \mathbb{D}} \frac{ \int_{S_a}|g'|^2\omega^*} {(\int_{S_a}\omega)^{2(1/p-1/q)+1}} \qquad \text{and} \qquad \|T_g\|_e^2\simeq\limsup_{|a|\to 1^-} \frac{ \int_{S_a}|g'|^2\omega^*} {(\int_{S_a}\omega)^{2(1/p-1/q)+1}}, \] where \(\omega^*(z)=\int_{|z|}^1 s\omega(s)\log\frac{s}{|z|}ds\), \(z\neq 0\), and \(S_a\) is a Carleson square with a basis point \(a\in\mathbb{D}\). If, in addition, \(p<q\), then \[ \|T_g\|\simeq \sup_{a\in \mathbb{D}} \frac{|g'(a)|(1-|a|)}{\omega^*(a)^{1/p-1/q}} \quad\text{and}\quad \|T_g\|_e\simeq \limsup_{|a|\to 1^-} \frac{|g'(a)|(1-|a|)}{\omega^*(a)^{1/p-1/q}}. \]
reviewed by: Joan Fàbrega
Mathematics Subject Classification ID: 30H20
Mathematics Subject Classification ID: 47G10
zbMATH DE Number: 6684635
zbMATH Keywords: weighted Bergman spaces
zbMATH Keywords: Volterra integral operators
zbMATH Keywords: operator norm
zbMATH Keywords: essential norm
MaRDI profile type: Publication
full work available at URL: j.jmaa.2017.01.006
OpenAlex ID: W2578872220
cites work: Q5317928
cites work: Q5385115
cites work: Spectra of integration operators on weighted Bergman spaces
cites work: Integration operators on Bergman spaces
cites work: Q4458484
cites work: Essential norms and weak compactness of integration operators
cites work: Contributions to the theory of the classical Banach spaces
cites work: On generalized Bergman spaces
cites work: Q4528635
cites work: Strict singularity of a Volterra-type integral operator on \(H^p\)
cites work: Weighted Bergman spaces induced by rapidly increasing weights
cites work: Q4183861
cites work: Q4340096
cites work: Q5421749
cites work: $H_{p}$-spaces, p≤1, and spline systems
DOI: 10.1016/J.JMAA.2017.01.006
description: scientific article; zbMATH DE number 6684635