An off-diagonal estimate of Bergman kernels
type: Item
instance of: scholarly article
title: An off-diagonal estimate of Bergman kernels
zbMATH Open document ID: 0971.46017
DOI: 10.1016/S0021-7824(99)00031-8
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2000-06-22T00:00:00Z
review text: Let \(\mathbb D\) denote the unit disk in \(\mathbb C\) with area measure \(d\Sigma\). For a nonnegative weight \(\omega\) on \(\mathbb D\), \(L^{2}(\mathbb D,\omega)\) denotes the space of complex-valued functions on \(\mathbb D\) that are square integrable with respect to the measure \(\omega d\Sigma\) with norm \(\|f\|_{\omega}\), and \(A^2(\mathbb D,\omega)\) the closure of the analytic polynomials in \(L^2(\mathbb D,\omega)\). The space \(A^2(\mathbb D,\omega)\) is a weighted Bergman space with reproducing kernel \(K_{\omega}\) provided \(|p(z)|\leq C(X)\|p\|_{\omega}\) (\(z\in X\)) for all polynomials \(p\) and compact subsets \(X\) of \(\mathbb D\). In the paper the author proves that if \(\omega\) is logarithmically subharmonic on \(\mathbb D\) and reproducing for the origin, then the kernel \(K_{\omega}\) satisfies the following inequalities: for all \((z,\zeta)\in\mathbb D\times\mathbb D\), \[ \frac 12K_{\omega}(\zeta,\zeta) \frac{(1-|\zeta|^2)^2}{|1-z\overline\zeta|^2} \leq |K_{\omega}(z,\zeta)|\leq \frac 2{|1-z\overline \zeta|^2}. \]
reviewed by: Manfred Stoll
Mathematics Subject Classification ID: 46E22
Mathematics Subject Classification ID: 30C40
zbMATH DE Number: 1463870
zbMATH Keywords: nonnegative weight
zbMATH Keywords: weighted Bergman space
zbMATH Keywords: reproducing kernel
zbMATH Keywords: logarithmically subharmonic
Wikidata QID: Q59288543
author: Håkan Hedenmalm
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1463870