Bergman projections and duality in weighted mixed-norm spaces of analytic functions
type: Item
instance of: scholarly article
title: Bergman projections and duality in weighted mixed-norm spaces of analytic functions
zbMATH Open document ID: 0787.32012
author: Dangsheng Gu
published in: Michigan Mathematical Journal
publication date: 1992-09-27T00:00:00Z
review text: The Bergman operator \(T_ \alpha\) on the unit ball \(U\) of \(C^ N\) is given by \[ T_ \alpha f(z)={N+\alpha \choose N} \int_ U(1-\langle z,w \rangle)^{-N-1-\alpha} f(w) c_ \alpha(1-| w |^ 2)^ \alpha dm \] where \(c_ \alpha\) is a normalising constant. The operator \(T^*_ \alpha\) is obtained by replacing \(1-\langle z,w \rangle\) with its norm. Békollé studied the continuity of these operators on weighted \(L_ p\)-spaces. In this article his results are extended to mixed norm spaces with radial symmetry, i.e. with norms of the form \[ \| f \|=\int_ 0^ 1\| f_ r \|^ q_ pw (r)dr \] where \(w\) is a weight function on (0,1) and \(\| f_ r \|_ p\) is the unweighted \(L_ p\) norm on the unit sphere. This generalised result is used to obtain some information on mixed norm spaces of holomorphic functions, the most important being the characterisation of the dual of some of these spaces.
Mathematics Subject Classification ID: 32A35
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 63012
zbMATH Keywords: Bergman projections
zbMATH Keywords: duality
zbMATH Keywords: mixed norm spaces
reviewed by: J. Cnops
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029004455
OpenAlex ID: W1994251341
DOI: 10.1307/MMJ/1029004455
description: scientific article; zbMATH DE number 63012