Abstract
The hyperholomorphic Bergmann kernel function ψß for a domain Ω is introduced as the special quaternionic “derivative” of the Green function for Ω. It is shown that ψß is hyperholomorphic, Hermitian symmetric and reproduces hyperholomorphic functions.
We obtain an integral representation of ψß as a sum of two integrals. One of them gives a smooth function, and the other describes the behaviour of ψß near a boundary. To investigate the hyperholomorphic Bergmann function for some fixed class of hyperholomorphic functions we have to use not only the properties of just this class but also those of some other classes. The second fact is completely unpredictable from the complex analysis point of view.
The connection between the hyperholomorphic Bergmann projector (the integral operator with the kernel ψß) and some classical multidimensional singular integral operators is established.
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Shapiro, M.V., Vasilevski, N.L. On the Bergmann Kernel Function in Hyperholomorphic Analysis. Acta Applicandae Mathematicae 46, 1–27 (1997). https://doi.org/10.1023/A:1017916828448
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DOI: https://doi.org/10.1023/A:1017916828448