Norm of the Bergman projection
type: Item
instance of: scholarly article
zbMATH Open document ID: 1302.47017
author: David Kalaj
author: Marijan Marković
publication date: 2014-10-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1203.6009
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 46B10
Mathematics Subject Classification ID: 30H30
Mathematics Subject Classification ID: 32A25
zbMATH DE Number: 6357913
MaRDI profile type: Publication
OpenAlex ID: W2964206196
arXiv ID: 1203.6009
DOI: 10.7146/MATH.SCAND.A-18007
title: Norm of the Bergman projection
published in: Mathematica Scandinavica
review text: The Bloch space \(\mathfrak{B}\) contains all holomorphic functions \(f\) on the unit ball \(\mathbb{B}\) for which the semi-norm \(\displaystyle \|f\|_{\beta} = \sup_{z\in \mathbb{B}} (1-|z|^2)|\nabla f(z)|\) is finite. For \(f\in\mathfrak{B}\), the \(\mathfrak{B}\)-norm is defined as \(\displaystyle \|f\|_{\mathfrak{B}}= |f(0)|+ \|f\|_{\beta}\). The Bloch space \(\mathfrak{B}\), with invariant norm, contains all holomorphic functions \(f\in H(\mathbb{B})\) for which the \(\tilde{\mathfrak{B}}\)-norm \(\displaystyle \|f\|_{\tilde{\mathfrak{B}}}= |f(0)|+ \|f\|_{\tilde{\beta}}\) is finite, where the \(\tilde{\beta}\)-norm is defined as \(\displaystyle \|f\|_{\tilde{\beta}} = \sup_{z\in \mathbb{B}} |\tilde{\nabla} f(z)|\) and the invariant gradient \(\tilde{\nabla}f(z) = \nabla(f\circ \phi_z)(0)\). For \(1\leq p\leq\infty\) and \(\alpha>-1\), the Bergman projection operator \(P_{\alpha}\) is defined by \(\displaystyle P_{\alpha}g(z)=\int_{\mathbb{B}} K_{\alpha}(z, w) g(w) \,dv_{\alpha}(w)\), for \(g\in L^{p}\) and \(z, w \in \mathbb{B}\). The authors consider the \(\beta\)-norm and \(\mathfrak{B}\)-norm of the Bergman projection \(P_{\alpha}: L^{\infty} \rightarrow \mathfrak{B}\) which are defined as \(\displaystyle \|P_{\alpha}\|_{\beta}= \sup_{\|g\|_{\infty}<1}\|P_{\alpha}g\|_{\beta}\) and \(\displaystyle \|P_{\alpha}\|_{\mathfrak{B}}= \sup_{\|g\|_{\infty}<1}\|P_{\alpha}g\|_{\mathfrak{B}}\); similarly, they define the \(\tilde{\beta}\)-norm and \(\tilde{\mathfrak{B}}\)-norm of the Bergman projection.NEWLINENEWLINEIn this paper, the authors present the exact \(\beta\)-norm and an estimate for the \(\tilde{\beta}\)-norm of \(P_{\alpha}\); namely, NEWLINE\[NEWLINE\|P_{\alpha}\|_{\beta} =C_{\alpha},\quad \|P_{\alpha}\|_{\tilde{\beta}} =\max_{0\leq t\leq \pi/2} \ell(t),NEWLINE\]NEWLINE and NEWLINE\[NEWLINE \frac{\pi}{2} C_{\alpha}\leq \|P_{\alpha}\|_{\tilde{\beta}} \leq \frac{\sqrt{\pi^2 +4}}{2} C_{\alpha}.NEWLINE\]NEWLINE Here, \(\displaystyle \theta= n+1+\alpha \), \(\…
reviewed by: Waleed Al-Rawashdeh
description: scientific article; zbMATH DE number 6357913