Fundaments of Hermitean Clifford analysis part II: splitting of<b><i>h</i></b>-monogenic equations

Abstract

Hermitean Clifford analysis focuses on h-monogenic functions taking values in a complex Clifford algebra or in a complex spinor space, where h-monogenicity is expressed by means of two complex and mutually adjoint Dirac operators, which are invariant under the action of a Clifford realization of the unitary group. In part 1 of the article the fundamental elements of the Hermitean setting have been introduced in a natural way, i.e., by introducing a complex structure on the underlying vector space, eventually extended to the whole complex Clifford algebra . The two Hermitean Dirac operators are then shown to originate as generalized gradients when projecting the gradient on invariant subspaces. In this part of the article, the aim is to further unravel the conceptual meaning of h-monogenicity, by studying possible splittings of the corresponding first-order system into independent parts without changing the properties of the solutions. In this way further connections with holomorphic functions of several complex variables are established. As an illustration, we give a full characterization of h-monogenic functions for the case n = 2.