On Cauchy and Martinelli-Bochner integral formulae in Hermitean Clifford analysis. (English) Zbl 1182.30081

The first-order differential operator (called Dirac operator) \[ \partial_{\underline X}= \sum^n_{j=1} e_j\partial_{X_j} \] and the so-called twisted Dirac operator \[ \partial_{\underline X}= \sum^n_{j=1} (e_j\partial_{y_j}- e_{n+j}\partial_{x_j}) \] are simultaneously considered within the theory of Hermitean Clifford analysis. Hermitean monogenic functions are introduced as common null functions of the Dirac operator and the twisted Dirac operator. In this very general setting, Clifford-Stokes theorems and Hermitean Cauchy-Pompeiu formulae are proved. For spinor valued functions, a Martinelli-Bochner formula for holomorphic functions of several complex variables can be deduced from the Hermitean Cauchy integral formula.


30G35 Functions of hypercomplex variables and generalized variables
32A26 Integral representations, constructed kernels (e.g., Cauchy, Fantappiè-type kernels)
46F10 Operations with distributions and generalized functions
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