Abstract
Let ann-dimensional differential form Ω be defined at points of aC 1-smooth boundary π of a domainG ⊂ ℝn. Under what condition can Ω be represented as Ω = Ω+ + Ω+ + Ω-, where Ω± are forms insideG and outsideG, harmonic in the sense of Hodge? A necessary condition is that both restrictions Ω{inπ and *Ω{inπ be closed in the sense of currents. This condition, with an additional smoothness assumption, turns out to be sufficient as well. This is an analogue of the Cauchy integral decomposition of functions in the plane.
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This research was supported by the fund for the promotion of research at the Technion.
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Dyn’kin, E. Cauchy integral decomposition for harmonic forms. J. Anal. Math. 73, 165–186 (1997). https://doi.org/10.1007/BF02788142
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DOI: https://doi.org/10.1007/BF02788142