Boundedness of the Bergman projector and Bell's duality theorem
type: Item
instance of: scholarly article
title: Boundedness of the Bergman projector and Bell's duality theorem
zbMATH Open document ID: 0533.32003
author: Gen Komatsu
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1984-01-01T00:00:00Z
review text: Let \(W^ sH(\Omega)\) denote the space of holomorphic functions in \(\Omega\) contained in the \(L^ 2\) Sobolev space \(W^ s(\Omega)\) of order \(s>0,\) and let \(W^{-s}\!\!_{cl}H(\Omega)\) denote the closure of \(L^ 2H(\Omega)=W^ 0H(\Omega)\) in \(W^{-s}H(\Omega)\), where \(\Omega\) is a bounded domain in \({\mathbb{C}}^ n\) with smooth boundary. This paper is concerned with the duality \(W^ sH(\Omega)\subset L^ 2H(\Omega)\subset W^ sH(\Omega)^*\) and the isomorphism \(\Lambda^ s:W^ sH(\Omega)\to W^ sH(\Omega)^*.\) Three theorems are contained. - Theorem 1: \(W^ sH(\Omega)^*=W^{-s}\!\!_{cl}H(\Omega)\) if and only if the condition \((R)^ s\!_ 0\) holds, i.e., the Bergman projector K maps \(W^ s\!_ 0(\Omega)\) to \(W^ s(\Omega)\) boundedly. - Theorem 2: If \((R)^ s\!_ 0\) holds, then \(\Lambda^ s\) is given by \(\Lambda^ sg(z)=(g,K(\cdot,z))_ s,\) where \(K(\cdot,\cdot)\) denotes the Bergman kernel and \((\cdot,\cdot)_ s\) stands for the scalar product in \(W^ s(\Omega)\). - Theorem 3. If \((R)^ s\!_ 0\) holds, then every element of \(W^{-s}\!\!_{cl}H(\Omega)\) admits a Fourier series expansion with respect to the eigenfunctions of \(\Lambda^ s\) regarded as a self-adjoint operator in \(L^ 2H(\Omega)\). Analogous results are also valid in a framework of the Szegö projector. In case \(\Omega\) is strictly pseudoconvex, results corresponding to theorems 1 and 2 have been known by \textit{S. R. Bell} [Ill. J. Math. 26, 19-26 (1982; Zbl 0475.32004)]. A similar generalization has been obtained independently by \textit{S. R. Bell} and \textit{H. P. Boas} [Math. Ann. 267, 473-478 (1984)]. It has since been pointed out by E. J. Straube that the boundedness of \(K:W^ s\!_ 0(\Omega)\to W^ s(\Omega)\) is equivalent to that of K in \(W^ s(\Omega)\).
Mathematics Subject Classification ID: 32A35
Mathematics Subject Classification ID: 32A25
Mathematics Subject Classification ID: 32A38
Mathematics Subject Classification ID: 32E35
zbMATH DE Number: 3845948
zbMATH Keywords: Bell duality theorem
zbMATH Keywords: boundedness of Bergman projection
zbMATH Keywords: space of holomorphic functions
zbMATH Keywords: Sobolev space
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178228810
OpenAlex ID: W2057967059
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DOI: 10.2748/TMJ/1178228810
description: scientific article; zbMATH DE number 3845948