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Harmonic multivector fields and the Cauchy integral decomposition in Clifford analysis. (English) Zbl 1063.30045

The authors deal with the problem to decompose a Hölder continuous \(k\)-grade multivector field \(F_k\) on the boundary \(\Gamma\) of a bounded open set \(\Omega \subset \mathbb{R}^n\) into a sum \(F_k = F_k^+ + F_k^-\) of harmonic \(k\)-grade multivector fields in \(\Omega^+ = \Omega\) and \(\Omega^- = \mathbb{R}^n \setminus (\Omega \cup \Gamma)\) respectively.This is equivalent to the analogue problem for harmonic forms dealt with by E. Dyn’kin [Complex Variables, Theory Appl. 31, 165–176 (1996; Zbl 0865.30056); J. Anal. Math. 73, 165–186 (1997; Zbl 0899.58001)]. Conditions for the field \(F_k\) are given, example giving its Cauchy transform has to be both left and right monogenic in \(\mathbb{R}^n \setminus \Gamma\).

MSC:

30G35 Functions of hypercomplex variables and generalized variables
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
31B25 Boundary behavior of harmonic functions in higher dimensions
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