Quaternionic Clifford analysis: the Hermitian setting (Q997397)

The authors continue earlier papers [\textit{F. Brackx, H. de Schepper} and \textit{F. Sommen}, A Hermitian setting for wavelet analysis: The basic, to appear in Proc. ICCA7 (Toulouse, 2005)] and [\textit{I. Sabadini} and \textit{F. Sommen}, Math. Methods Appl. Sci. 25, 1395--1413 (2002; Zbl 1013.30033)]. They deal with functions defined in a domain of \(\mathbb{R}_m\), the Clifford algebra generated by \(e_1,...,e_m\), and with values in the tensor product with the quaternions \(\mathbb{H}_m: = \mathbb{H} \otimes_{\mathbb{R}} \mathbb{R}_m\), here \(m=4n\). A Witt basis in \(\mathbb{H}_m\) is defined using the conjugation and main involution in \(\mathbb{R}_m\). This leads to corresponding 4 differential operators and the quaternionic hermitian vector derivative, which gives a quaternionic hermitian Dirac equation. A function \(f\) is called hermitian monogenic if it is in the kernel of all the above mentioned 4 differential operators. For the cases \(m=4\) resp. \(m=8\) (\(n=1\) resp. \(n=2\)) the resolutions of the modules of hermitian monogenic functions are given. At last a Bochner-Martinelli formula is proved using a convenient kernel. Hints for further research are given.