Bergman kernel function for Hartogs domains over bounded homogeneous domains
type: Item
instance of: scholarly article
title: Bergman kernel function for Hartogs domains over bounded homogeneous domains
zbMATH Open document ID: 1376.32004
author: Hideyuki Ishi
author: Jong-Do Park
author: Atsushi Yamamori
published in: The Journal of Geometric Analysis
publication date: 2017-08-03T00:00:00Z
review text: The aim of the paper is to give an explicit closed form of the Bergman kernel of the Hartogs-type domain \[ D_{m,s}=\big\{(z,\zeta)\in D\times\mathbb C^m:\|\zeta\|^2<K_D(z,z)^{-s}\big\},\quad m\in\mathbb N,\quad s\in\mathbb R, \] where \(K_D\) denotes the Bergman kernel of the bounded homogeneous domain \(D\), and to give sufficient conditions for the Bergman kernel \(K_{m,s}\) of \(D_{m,s}\) to be zero-free. First, the authors obtain an explicit form of the weighted Bergman kernel \(K_s\) for the weighted Bergman space \(L^2_a(D,K_D(z,z)^{-s}\,dV(z))\). Next, using the virtual Bergman kernel, they express the Bergman kernel \(K_{m,s}\) as a rational function in the variable \(\|\zeta\|^2K_D(z,z)^s\). Finally, the authors apply their results to study the Lu Qi-Keng problem for the domains \(D_{m,s}\). They show that \(K_{m,s}\) is zero-free for large \(m\) (i.e., \(D_{m,s}\) is a Lu Qi-Keng domain). Moreover, they find the number \(m_0(D)\) such that \(D_{m,s}\) is a Lu Qi-Keng domain for all \(s\in\mathbb R\) and \(m\geq m_0(D)\), when \(D\) has dimension \(4\) or \(5\).
Mathematics Subject Classification ID: 32A25
Mathematics Subject Classification ID: 32A07
Mathematics Subject Classification ID: 32M10
zbMATH DE Number: 6756141
zbMATH Keywords: Bergman kernel
zbMATH Keywords: Hartogs domains
zbMATH Keywords: bounded homogeneous domains
zbMATH Keywords: Siegel domain
reviewed by: Paweł Zapałowski
MaRDI profile type: Publication
full work available at URL: s12220-016-9737-4
OpenAlex ID: W2519255516
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DOI: 10.1007/S12220-016-9737-4
description: scientific article; zbMATH DE number 6756141