Deriving harmonic functions in higher dimensional spaces
type: Item
instance of: scholarly article
title: Deriving harmonic functions in higher dimensional spaces
zbMATH Open document ID: 1052.30055
DOI: 10.4171/ZAA/1145
author: Tao Qian
author: Frank Sommen
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2004-09-06T00:00:00Z
review text: A well-known result of R. Fueter says how to construct a quaternionic monogenic function starting from a holomorphic function in an open set of the upper half complex plane, then making an appropriate change of variable and applying, finally, the Laplace operator. A later result by Sce involves certain powers of the Laplace operator, and it was extended by Sommen. The present paper deals with harmonic functions using Clifford analysis. The statements of the new results are contained in Section 2, and they are given in terms of identities in relation to multiple powers of the Laplacian on induced functions from lower dimensional harmonic functions, the identities show explicitly the harmonicity of the induced functions. The proofs of the theorems one can find in Section 3, and Section 4 presents applications, in particular, it is shown that almost all commonly used kernels, as well as generalizations of Fueter's theorem, are deducible from the results of Section 2.
Mathematics Subject Classification ID: 30G35
Mathematics Subject Classification ID: 31A05
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 2096446
zbMATH Keywords: harmonic functions
zbMATH Keywords: Clifford monogenic functions
zbMATH Keywords: singular integrals
zbMATH Keywords: Fourier multipliers
reviewed by: M. Shapiro
MaRDI profile type: Publication
cites work: Q3310867
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cites work: Q5607484
full work available at URL: https://doi.org/10.4171/zaa/1145
OpenAlex ID: W1974576704
description: scientific article; zbMATH DE number 2096446