Surjectivity criteria for convolution operators in \(A^{-\infty}\)
type: Item
instance of: scholarly article
title: Surjectivity criteria for convolution operators in \(A^{-\infty}\)
zbMATH Open document ID: 1190.32001
author: Alexander V. Abanin
author: Ryuichi Ishimura
author: Le Hai Khoi
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2010-03-29T00:00:00Z
review text: Let \(\Omega\) be a convex bounded domain in \(\mathbb C^n\) and \(d(z) = \inf_{\zeta\in \partial\Omega} |z-\zeta|\), \(z\in\Omega\). Let \({\mathcal O}(\Omega)\) be the space of holomorphic functions on \(\Omega\), and \(A^{-\infty}(\Omega)\) be its subspace of those holomorphic functions, which have polynomial growth near the boundary \(\partial\Omega\). The following criterion of surjectivity of convolution operators is proved. Theorem. Let \(\mu\) be an analytic functional on \(\mathbb C^n\), carried by a compact convex set \(K\), and \(\mu* A^{-\infty}(\Omega+K)\subseteq A^{-\infty}(\Omega)\) for any convex bounded domain \(\Omega\in\mathbb C^n\). The convolution operator \(\mu*: A^{-\infty}(\Omega+K)\to A^{-\infty}(\Omega)\) is surjective for every \(\Omega\) if and only if the regularized radial indicator \(h^*_\mu(\zeta)\) of the Fourier-Borel transformation \(\widehat\mu\) of \(\mu\) coincides with \(H_K\) and \(\widehat\mu\) as a function of exponential type satisfies the following condition: \[ \exists s,\;N>0\;\forall \zeta \in\mathbb C^n,\;|\zeta| > N,\;\exists \zeta'\in\mathbb C,\;|\zeta-\zeta'|< \log(1+|\zeta|): \log|\widehat\mu|\geq |\zeta|h^*_\mu \bigg(\frac\zeta{|\zeta|}\bigg)- s\log |\zeta|. \]
reviewed by: Sergei V. Rogosin
Mathematics Subject Classification ID: 32A10
Mathematics Subject Classification ID: 32A15
Mathematics Subject Classification ID: 47B33
Mathematics Subject Classification ID: 32A37
zbMATH DE Number: 5687606
zbMATH Keywords: holomorphic functions of several complex variables
zbMATH Keywords: convolution operator
zbMATH Keywords: analytic functional
zbMATH Keywords: Fourier-Borel transformation
zbMATH Keywords: entire functions of exponential type
MaRDI profile type: Publication
full work available at URL: j.crma.2010.01.015
OpenAlex ID: W2038791396
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DOI: 10.1016/J.CRMA.2010.01.015
description: scientific article; zbMATH DE number 5687606