Some properties of the Cauchy-type integral for the Laplace vector fields theory

Abstract

We study the analog of the Cauchy-type integral for the Laplace vector fields theory in case of a piece-wise Liapunov surface of integration and we prove the Sokhotski-Plemelj theorem for it as well as the necessary and sufficient condition for the possibility to extend a given Holder function from such a surface up to a Laplace vector field. Formula for the square of the singular Cauchy-type integral is given. The proofs of all these facts are based on intimate relations between Laplace vector field and some versions of quaternionic analysis.