The Teodorescu operator in Clifford analysis

Abstract

Euclidean Clifford analysis is a higher dimensional function theory centred around monogenic functions, i.e., null solutions to a first order vector valued rotation invariant differential operator \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\partial }\) called the Dirac operator. More recently, Hermitian Clifford analysis has emerged as a new branch, offering yet a refinement of the Euclidean case; it focuses on the simultaneous null solutions, called Hermitian monogenic functions, to two Hermitian Dirac operators \(\partial _{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z} }\) and \(\partial _{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z} ^\dag }\) which are invariant under the action of the unitary group. In Euclidean Clifford analysis, the Teodorescu operator is the right inverse of the Dirac operator \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\partial }\). In this paper, Teodorescu operators for the Hermitian Dirac operators \(\partial _{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z} }\) and \(\partial _{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{z} ^\dag }\) are constructed. Moreover, the structure of the Euclidean and Hermitian Teodorescu operators is revealed by analyzing the more subtle behaviour of their components. Finally, the obtained inversion relations are still refined for the differential operators issuing from the Euclidean and Hermitian Dirac operators by splitting the Clifford algebra product into its dot and wedge parts. Their relationship with several complex variables theory is discussed.