A holomorphic extension theorem using Clifford analysis
type: Item
instance of: scholarly article
title: A holomorphic extension theorem using Clifford analysis
zbMATH Open document ID: 1222.30041
author: Ricardo Abreu-Blaya
author: Juan Bory-Reyes
author: Dixan Peña Peña
author: Frank Sommen
published in: Complex Analysis and Operator Theory
publication date: 2011-05-20T00:00:00Z
full work available at URL: https://biblio.ugent.be/publication/1246305
review text: Let \(\Omega\) be a simply connected bounded and open set in \(\mathbb{C}^n\) and suppose that \(f\) is a \(\mathbb{C}\)-valued continuous function on \(\partial \Omega\). It is a well-known fact that the Bochner-Martinelli integral \(M_1f\) of \(f\) is a harmonic function in \(\mathbb{C}^n \setminus \partial \Omega\) and vanishes at infinity. \textit{A. M. Aronov} and \textit{A. M. Kytmanov} [Funct. Anal. Appl. 9, 254--255 (1976); translation from Funkts. Anal. Prilozh. 9, No. 3, 83--84 (1975; Zbl 0319.32008)] provided the following two characterizations. If \(\partial \Omega\) is a smooth surface and \(f\) is a continuously differentiable function in \(\partial \Omega\), then a necessary and sufficient condition for \(f\) to have a holomorphic extension to \(\Omega\) is that \(M_1f(\underline{x}) = 0\) for \(\underline{x} \in \mathbb{C}^n \setminus \overline{\Omega}\). If \(\partial \Omega\) is a piecewise smooth surface and \(F\) is a continuously differentiable function in \(\overline{\Omega}\), then \(F\) is holomorphic in \(\Omega\) if and only if \(F(\underline{x})=M_1 F(\underline{x})\) for \(\underline{x} \in \Omega\). In this paper alternative characterizations are presented using Clifford analysis, the latter offering a function theory which is a higher dimensional analogue of the theory of the holomorphic functions of one complex variable. The main techniques in obtaining these characterizations are the isotonic functions and the isotonic Cauchy type integral. The results of Aronov and Kytmanov are also re-established for the particular case of \(n=2\) using these techniques.
reviewed by: Nele De Schepper
Mathematics Subject Classification ID: 30G35
Mathematics Subject Classification ID: 32A40
zbMATH DE Number: 5898471
zbMATH Keywords: Clifford analysis
zbMATH Keywords: isotonic functions
zbMATH Keywords: holomorphic extension
zbMATH Keywords: Bohner-Martinelli integral
MaRDI profile type: Publication
OpenAlex ID: W2046741582
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DOI: 10.1007/S11785-009-0037-X
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description: scientific article; zbMATH DE number 5898471