International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 746946, 19 pages
Research Article

Generalized Moisil-Théodoresco Systems and Cauchy Integral Decompositions

Ricardo Abreu Blaya,1 Juan Bory Reyes,2 Richard Delanghe,3 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

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.


Let 0,m+1(s) be the space of s-vectors (0sm+1) in the Clifford algebra 0,m+1 constructed over the quadratic vector space 0,m+1, let r,p,q with 0rm+1,0pq, and r+2qm+1, and let 0,m+1(r,p,q)=j=pq⨁ 0,m+1(r+2j). Then, an 0,m+1(r,p,q)-valued smooth function W defined in an open subset Ωm+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 m+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 0,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.