Note on the Fourier-Laplace transform of \(\overline{\partial}\)-cohomology classes
type: Item
instance of: scholarly article
title: Note on the Fourier-Laplace transform of \(\overline{\partial}\)-cohomology classes
zbMATH Open document ID: 0924.43004
DOI: 10.4171/ZAA/858
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1999-05-25T00:00:00Z
review text: This paper derives from \textit{B. Berndtsson}'s treatment [Several complex variables, Proc. Mittag-Leffler Inst., Stockholm/Swed. 1987-88, Math. Notes 38, 160-187 (1993; Zbl 0786.32003)] of the Pólya-Martineau theorem, which says that the Fourier-Laplace transform establishes a bijection between the set of analytic functionals carried by a compact, convex set \(K\) in \(\mathbf{C}^n\) and the space \(A_K(\mathbf{C}^n)\) of entire functions \(f\) of exponential type with the property that for every positive \(\epsilon\) there exists a constant \(C_\epsilon\) such that \(| f(z)| \leq C_\epsilon \exp(\epsilon| z| + \sup_{\zeta\in K}\Re\langle \zeta,z\rangle)\). Moreover, there is an explicit formula for the inverse transform. The author's main observation is that by thinking of integrating differential forms on an arbitrary smooth surface surrounding \(K\), one can recast the theorem as establishing a linear isomorphism between \(A_K(\mathbf{C}^n)\) and the space \(H^{(n,n-1)}(\mathbf{C}^n \setminus K)\) of \(\overline{\partial}\)-cohomology classes of \((n,n-1)\) forms in the complement of \(K\).
Mathematics Subject Classification ID: 43A30
Mathematics Subject Classification ID: 32A25
Mathematics Subject Classification ID: 46F15
zbMATH DE Number: 1270169
zbMATH Keywords: Borel transform
zbMATH Keywords: analytic functional
zbMATH Keywords: entire functions of exponential type
zbMATH Keywords: Martineau's theorem
zbMATH Keywords: Fourier-Laplace transform
zbMATH Keywords: differential forms
author: T. Hatziafratis
reviewed by: Harold P. Boas
MaRDI profile type: Publication
cites work: Q4694587
cites work: Q3204740
cites work: Notions of convexity
cites work: Sur les fonctionnelles analytiques et la transformation de Fourier-Borel
full work available at URL: https://doi.org/10.4171/zaa/858
OpenAlex ID: W2032724496
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description: scientific article; zbMATH DE number 1270169