Norm of Berezin transform on \(L^p\) space
type: Item
instance of: scholarly article
title: Norm of Berezin transform on \(L^p\) space
zbMATH Open document ID: 1155.47051
published in: Journal d'Analyse Mathématique
publication date: 2008-09-03T00:00:00Z
review text: Let \(D\) be the unit disc in \(\mathbb C\) equipped with the standard Lebesgue measure \(\mu\). For \(K_\alpha(z, \xi)= (\alpha +1)\frac{(1-| \xi| ^2)^\alpha}{^(1-z\overline{\xi})^{\alpha + 2}}\), the operator \(G_\alpha: L^p(D) \to L^p(D)\), \((G_\alpha f)(z):= \frac1\pi \int_D| K_\alpha(z, \xi)| f(\xi)\,d\mu(\xi)\) is introduced. It is shown that for \(1 \leq p < \infty\), \(\alpha > -1\) and \(p(\alpha + 1) >1\), \[ \| G_\alpha\| _p = \frac{\alpha + 1}{\Gamma^2(1 + \alpha/2)}\Gamma(1/p)\Gamma(\alpha + 1 - 1/p). \] This gives an upper estimate for the norm of corresponding Bergman type operator and, in particular (when \(\alpha=0\)), estimates the norm of classical Bergman projection. By a duality argument, it is shown that for \(1 < p \leqslant \infty\), the norm of Berezin transform \(B\) on \(L^p (D)\) equals \( \frac{p + 1}{p^2 }\frac{\pi}{\sin (\pi /p)}\).
reviewed by: Vladimir Kadets
Mathematics Subject Classification ID: 47G10
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 5320467
zbMATH Keywords: Berezin transform
zbMATH Keywords: Bergman projection
Wikidata QID: Q60060321
author: Milutin R. Dostanić
MaRDI profile type: Publication
full work available at URL: s11854-008-0014-8
OpenAlex ID: W2035619604
cites work: COVARIANT AND CONTRAVARIANT SYMBOLS OF OPERATORS
cites work: Q5621389
cites work: Norm estimate of the Cauchy transform on \(L^p(\Omega)\)
cites work: Q4490700
cites work: Notions of convexity
cites work: Q5494080
DOI: 10.1007/S11854-008-0014-8
description: scientific article; zbMATH DE number 5320467