Martinelli-Bochner type formulae in complex Clifford analysis
Let \({\mathcal A}\) be the Clifford algebra over \({\mathbb{C}}^ m,\) ẕ\(=\underline x+i\underline y\in {\mathbb{C}}^ m \)and \(D_ x=\sum^{m}_{i=1}e_ j\partial /\partial x_ j,\) \(D_ y\) analogously defined, where \(\{e_ 1,...,e_ m\}\) is an orthonormal base of \({\mathbb{C}}^ m.\) Using the kernel \(\omega_{2m}E(\underline z)=- \underline z\quad *| \underline z|^{-2m}\) \((\omega_{2m}\) is the area of \(S_{2m-1}\), * means complex conjugation) the author proves a very general theorem of Bochner-Martinelli type for solutions of \((D_ x+iD_ y)f=0\), f continuously differentiable with respect to \(x_ j\), \(y_ j\) in a domain in \({\mathbb{C}}^ m.\) In various special cases this gives generalizations of known theorems and interesting connections to several parts of complex variable theory.
scientific article; zbMATH DE number 4061617