On a spatial generalization of the complex \(\Pi\)-operator
type: Item
instance of: scholarly article
title: On a spatial generalization of the complex \(\Pi\)-operator
zbMATH Open document ID: 0864.30039
DOI: 10.4171/ZAA/700
author: K. Gürlebeck
author: Uwe Kähler
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 1996-06-13T00:00:00Z
review text: The complex Pompeiu or \(T\)-operator can be generalized to the Clifford setting without any difficulties. It is very useful in the Clifford just as in the complex case. The shifting of the \(\Pi\)-operator from the complex plane to the Clifford algebra \(C\) was used sometimes, e.g. \textit{W. Sprößig} [Dissertation TH Karl-Marx-Stadt (1979)]. The authors analyze very carefully the properties of this Clifford \(\Pi\)-operator, defining it by \(\Pi f=\overline DTf\), where \(D\) is the usual Cauchy-Riemann operator in \(C\), \(Tf\) defined as usually by the domain integral over the Cauchy kernel multiplied by the function \(f\). \(\Pi\) is a strongly singular integral operator, its mapping and invertibility properties are examined, an application to the solution of the Clifford-Beltrami equation is given.
reviewed by: Klaus Habetha
Mathematics Subject Classification ID: 30G35
Mathematics Subject Classification ID: 30G20
Mathematics Subject Classification ID: 35A22
Mathematics Subject Classification ID: 30E20
zbMATH DE Number: 888997
zbMATH Keywords: Clifford analysis
zbMATH Keywords: Pompeiu operator
zbMATH Keywords: \(\Pi\)-operator
Wikidata QID: Q58038967
MaRDI profile type: Publication
cites work: Q3310867
cites work: Q4382876
cites work: Q4003806
cites work: A theorem about the representation of linear combinations in Clifford algebras
cites work: Quaternionic analysis and elliptic boundary value problems
cites work: Definition and properties of a hypercomplex singular integral operator
cites work: Q4306511
description: scientific article; zbMATH DE number 888997