Paracomplex Hermitean Clifford Analysis

Abstract

Substituting the complex structure by the paracomplex structure plays an important role in para-geometry and para-analysis. In this article we shall introduce the paracomplex structure into the realm of Clifford analysis and establish paracomplex Hermitean Clifford analysis by constructing a paracomplex Hermitean Dirac operator \({\mathcal {D}}\) and establishing the corresponding Cauchy integral formula. The theory of paracomplex Hermitean Clifford analysis turns out to be similar to that of complex Hermitean Clifford analysis which recently emerged as a refinement of the theory of several complex variables. It deserves to be pointed out that the introduction of a single operator \({\mathcal {D}}\) in the paracomplex setting has an advantage over the complex setting where complex Hermitean monogenic functions are described by a system of equations instead of being given as null-solution of a single Dirac operator as in the case of classic monogenic functions.