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.
Abreu-Blaya, R., Bory-Reyes, J., & Shapiro, M. (2007). The cauchy transform for the hodge/de rham system and some of its properties. Georgian Mathematical Journal, 1-20. https://doi.org/10.1515/GMJ.2007.1