Quaternionic and Clifford Analysis for Non-smooth Domains

Abstract

Clifford analysis, which covers a great part of the quaternionic analysis, is known as the theory of monogenic functions and represents a refinement of classical harmonic analysis and a generalization of complex analysis to higher R. Abreu-Blaya ( ) Faculty of Mathematics and Informatics, University of Holguin, Holguin, Cuba e-mail: rabreu@facinf.uho.edu.cu J. Bory-Reyes Department of Mathematics, Universidad de Oriente, Santiago de Cuba, Cuba e-mail: jbory@rect.uo.edu.cu © Springer Basel 2015 D. Alpay (ed.), Operator Theory, DOI 10.1007/978-3-0348-0667-1_31 1447 1448 R. Abreu-Blaya and J. Bory-Reyes dimensions. In this chapter a condensed account is given of a selection of studies connected to the Cauchy transform on the non-smooth boundary of a bounded domain in Euclidean space, in a Clifford and quaternionic analysis context. Particular emphasis will be placed on the study of the so-called jump problem (reconstruction problem) for monogenic functions, that is, the problem of reconstructing a monogenic function in both the interior and exterior of the domain vanishes at infinity and has a prescribed jump across the boundary.