The Bergman projection as a singular integral operator
type: Item
instance of: scholarly article
title: The Bergman projection as a singular integral operator
zbMATH Open document ID: 0804.32015
DOI: 10.1007/BF02921594
published in: The Journal of Geometric Analysis
publication date: 1994-07-10T00:00:00Z
review text: A Calderón-Zygmund type kernel is a kernel \((x,y\) are variables in a metric space) such that \[ \bigl | K(x,y) \bigr | < {K \over \mu \biggl( P(x, \rho \bigl( x,y) \bigr) \biggr)}, \qquad \int_{\rho (x,y)>T \rho(x,y)} \bigl | K(x,y) - K(z,y) \bigr | dy \leq K. \] \((\rho\) a metric, \(P\) a \(\rho\)-ball). This can be extended for a pseudometric \((\rho (x,z) \leq C(\rho (x,y) + \rho (y,z))\) if the measure \(\mu\) satisfies certain conditions (leading to a so-called space of homogeneous type). Starting from a defining function for a domain \(D\) (i.e. \(D = \{q : r(q) < 0\})\) of \(\mathbb{C}^ n\) several pseudometrics are constructed using the derivatives of \(r\), or local subharmonic functions. Using estimates previously obtained, mostly be the same author, it is then proved that under certain conditions the Bergman kernel is a Calderón-Zygmund type operator for these pseudometrics (the measure \(\mu\) being ordinary Lebesgue measure). As a consequence, the Bergman projection is, in these cases, a bounded operator in \(L_ p(D)\), \(1<p<\infty\).
Mathematics Subject Classification ID: 32A25
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 32A40
zbMATH DE Number: 579776
zbMATH Keywords: pseudometrics
zbMATH Keywords: Bergman kernel
zbMATH Keywords: Calderón-Zygmund type operator
author: J. D. McNeal
reviewed by: J. Cnops
MaRDI profile type: Publication
cites work: Q2548398
cites work: Boundary behavior of the Bergman kernel function in \({\mathbb{C}}^ 2\)
cites work: Q3973872
cites work: Estimates on the Bergman kernels of convex domains
cites work: Estimates for the Bergman and Szegö kernels in \({\mathbb{C}}^ 2\)
cites work: Q5607484
description: scientific article; zbMATH DE number 579776