International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 746946, 19 pages
doi:10.1155/2008/746946

Generalized Moisil-Théodoresco systems and Cauchy integral decompositions

Ricardo Abreu Blaya1 , Juan Bory Reyes2 , Richard Delanghe3 and Frank Sommen3

1Facultad de Informática y Matemática, Universidad de Holguín, Holguín 80100, Cuba
 2Departamento de Matemática, Facultad de Matemática y Computación, Universidad de Oriente, Santiago de Cuba 90500, Cuba
 3Department of Mathematical Analysis, Ghent University, 9000 Ghent, Belgium

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−.