Polyharmonic Bergman spaces and Bargmann type transforms
type: Item
instance of: scholarly article
title: Polyharmonic Bergman spaces and Bargmann type transforms
zbMATH Open document ID: 1360.46019
author: Luís V. Pessoa
author: Ana Moura Santos
published in: Journal of Mathematical Analysis and Applications
publication date: 2017-01-17T00:00:00Z
review text: Let \(dA\) denote the area measure on the upper half-plane \(\Pi\). For \(j=1,2,\dots\), the poly-Bergman space \(\mathcal{A}_j^2(\Pi)\) consists of those complex-valued smooth functions \(f\in L^2(\Pi, dA)\) for which \(\overline{\partial}{}^j f =0\). The polyharmonic Bergman space \(\mathcal{H}_j^2(\Pi)\) consists of \(u\in L^2(\Pi, dA)\) such that \(\Delta^j u = 0\), where \(\Delta\) is the Euclidean Laplacian. The authors show that \[ \mathcal{H}_j^2(\Pi) = \mathcal{A}_j^2(\Pi) \oplus \overline{\mathcal{A}_j^2}(\Pi), \quad j=1,2,\dots. \] The proof of the above decomposition is based on certain properties of the two-sided compression of the Beurling-Ahlfors transform to \(L^2(\Pi, dA)\). Also, the authors establish relations between \(\mathcal{H}_j^2(\Pi)\), its true polyharmonic Bergman spaces and true poly-Bergman spaces. They construct isometric isomorphisms from \(L^2(\mathbb{R})\) onto the true polyharmonic Bergman spaces. Finally, using the classical Paley-Wiener theorem, the authors prove that \(\mathcal{H}_j^2(\Pi)\) is isometrically isomorphic to \(2j\) copies of the Hardy space.
reviewed by: Evgueni Doubtsov
Mathematics Subject Classification ID: 46E20
Mathematics Subject Classification ID: 30H20
Mathematics Subject Classification ID: 31A30
zbMATH DE Number: 6675609
zbMATH Keywords: polyharmonic Bergman space
zbMATH Keywords: poly-Bergman space
zbMATH Keywords: Beurling-Ahlfors transform
zbMATH Keywords: kernel function
MaRDI profile type: Publication
full work available at URL: j.jmaa.2016.12.034
OpenAlex ID: W2562085945
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DOI: 10.1016/J.JMAA.2016.12.034
description: scientific article; zbMATH DE number 6675609