Generalized Moisil-Théodoresco Systems and Cauchy Integral Decompositions

Let be the space of -vectors in the Clifford algebra constructed over the quadratic vector space , let with , and , and let . Then, an -valued smooth function defined in an open subset is said to satisfy the generalized Moisil-Théodoresco system of type if in , where is the Dirac operator in . A structure theorem is proved for such functions, based on the construction of conjugate harmonic pairs. Furthermore, if is bounded with boundary , where is an Ahlfors-David regular surface, and if is a -valued Hölder continuous function on , then necessary and sufficient conditions are given under which admits on a Cauchy integral decomposition .