Abstract
Let Ω be a bounded open and connected subset of ℝm which has a C ∞-boundary Σ and let F k ∊ C ∞(Σ) be a k-multi-vector valued function on Σ. Under which conditions can F k be decomposed as F k = F k + + F k− where F k +- are extendable to harmonic k-multi-vector fields in Ω± with Ω+ = Ω and \(\Omega \_ = \mathbb{R}^m \backslash \bar \Omega ?\) This question is answered by proving a set of equivalent assertions, including a conservation law on F k and conditions on the Cauchy transform C Σ F k and on the Hilbert transform H Σ F k} of F k.
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The first two authors were supported by the FWO Research Network WO. 003. 01N and the last author was supported by the FWO “Krediet aan Navorsers: 1.5.065.04, 1.5.106.02”.
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Blaya, R.A., Reyes, J.B., Delanghe, R. et al. Cauchy Integral Decomposition of Multi-Vector Valued Functions on Hypersurfaces. Comput. Methods Funct. Theory 5, 111–134 (2005). https://doi.org/10.1007/BF03321089
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DOI: https://doi.org/10.1007/BF03321089