Copyright © 2008 Ricardo Abreu Blaya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let ℝ0,m+1(s) be the space of s-vectors (0≤s≤m+1) in the Clifford algebra ℝ0,m+1 constructed over the quadratic vector space ℝ0,m+1, let r,p,q∈ℕ with 0≤r≤m+1, 0≤p≤q, and r+2q≤m+1, and let ℝ0,m+1(r,p,q)=∑j=pq⨁ ℝ0,m+1(r+2j). Then, an ℝ0,m+1(r,p,q)-valued smooth function W defined in an open subset Ω⊂ℝm+1 is said to satisfy the generalized Moisil-Théodoresco system of type (r,p,q) if ∂xW=0 in Ω, where ∂x is the Dirac operator in ℝm+1. A structure theorem is proved for such functions, based on the construction of conjugate harmonic pairs. Furthermore, if Ω is bounded with boundary Γ, where Γ is an Ahlfors-David regular surface, and if W is a ℝ0,m+1(r,p,q)-valued Hölder continuous function on Γ, then necessary and sufficient conditions are given under which W admits on Γ a Cauchy integral decomposition W=W++W−.