The cauchy transform for the hodge/de rham system and some of its properties

Ricardo Abreu-Blaya, Juan Bory-Reyes, Michael Shapiro

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    Abstract

    We study the analogue of the Cauchy transform for the theory of solutions of the Hodge/de Rham system in the case of a rectifiable surface of integration which additionally satisfies an Ahlfors/David regularity condition and we prove the Cauchy integral formula, the Plemelj/Privalov theorem and the Sokhotski/Plemelj theorem for it, as well as the necessary and sufficient condition for the possibility to extend a given k-form from such a surface to a harmonic k-form in the domain. A formula for the square of the singular Cauchy transform is given. The proofs of all these facts are based on a close relation between algebra-valued null-solutions of the Dirac operator in the Euclidean space and hyperholomorphic functions of Clifford analysis. © 2007, Heldermann Verlag. All rights reserved.
    Original languageAmerican English
    Pages (from-to)1-20
    Number of pages2
    JournalGeorgian Mathematical Journal
    DOIs
    StatePublished - 1 Jan 2007

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