International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 746946, 19 pages
doi:10.1155/2008/746946
Research Article
Generalized Moisil-Théodoresco Systems and Cauchy Integral Decompositions
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
Received 20 September 2007; Revised 13 January 2008; Accepted 17 February 2008
Academic Editor: Heinrich Begehr
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−.