The dual of a Bergman space on simply connected domains.
type: Item
instance of: scholarly article
title: The dual of a Bergman space on simply connected domains.
zbMATH Open document ID: 1043.46024
DOI: 10.1007/BF02786580
published in: Journal d'Analyse Mathématique
publication date: 2003-12-15T00:00:00Z
review text: Let \(\Omega\) be a domain (open connected subset) in the complex plane. For \(1\leq p < \infty\), the Bergman \(p\)-space, denoted \(A^p(\Omega)\), consists of all holomorphic functions on \(\Omega\) such that \[ \| f\| _{A^p(\Omega)}= (\int_\Omega | f(z)| ^p\, dA(z))^{1/p}<\infty, \] where \(dA(z)\) is normalized area measure in the plane. The question studied in this interesting paper is whether one can represent the dual space of \(A^p(\Omega)\) as \(A^{p'}(\Omega)\) with respect to the dual action induced by \(A^2(\Omega)\): \[ \langle f,g\rangle= \int_{\Omega}f(z) \overline g(z) \,dA(z), \quad f\in A^p(\Omega),\;g\in A^{p'}(\Omega), \] where \(p'\) is the dual exponent to \(p\) defined by \(1/p + 1/p'=1.\) Unless \(p=2\), some kind of geometric requirement on \(\Omega\) is needed to have duality. The above question can be reformulated in the following form: For which \(\Omega\) and \(p\) does the Bergman projection \(\Pi: L^2(\Omega)\to A^2(\Omega)\) extend continuously to an operator \(L^p(\Omega) \to A^p(\Omega)\)? The author obtains the following main result: Theorem. Let \(\Omega\) be a simply connected domain in the complex plane other than the plane itself. There exists a universal constant \(p_0\), \(4/3 \leq p_0 < 2\), independent of \(\Omega\), such that the Bergman projection \(\Pi: L^2(\Omega)\to A^2(\Omega)\) extends to a bounded projection \(L^p(\Omega) \to A^p(\Omega)\) for all \(p\) in the interval \(p_0 < p < p'_0\). Moreover, the boundedness of the Bergman projection fails in general outside the closed interval \([p_0,p'_0]\). From this theorem follows that \(A^p(\Omega)^*= A^{p'}(\Omega)\) holds for all \(p\) with \(p_0 < p < p_0'\) and all simply connected domains. Moreover, this identification of the dual space fails generally outside the interval \([p_0, p_0']\). The exact value of the constant \(p_0\) is not known, but the estimate \(p_0 \leq p_1 \approx 1,413\) is obtained. Further, it is proved that the statement \(p_0 = 4/3\) is equiva…
reviewed by: Mikael Lindström
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 32D99
zbMATH DE Number: 1998323
zbMATH Keywords: Bergman space
zbMATH Keywords: simply connected
zbMATH Keywords: duality
Wikidata QID: Q59288538
author: Håkan Hedenmalm
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02786580
OpenAlex ID: W2086139761
description: scientific article; zbMATH DE number 1998323