On a spatial generalization of the complex \(\Pi\)-operator
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.
scientific article; zbMATH DE number 888997