Generalized Moisil-Théodoresco Systems and Cauchy Integral Decompositions

Abstract

Letℝ0,m+1(s)be the space ofs-vectors(0≤s≤m+1)in the Clifford algebraℝ0,m+1constructed over the quadratic vector spaceℝ0,m+1, letr,p,q∈ℕwith0≤r≤m+1, 0≤p≤q, andr+2q≤m+1, and letℝ0,m+1(r,p,q)=∑j=pq⨁ ℝ0,m+1(r+2j). Then, anℝ0,m+1(r,p,q)-valued smooth functionWdefined in an open subsetΩ⊂ℝm+1is said to satisfy the generalized Moisil-Théodoresco system of type(r,p,q)if∂xW=0inΩ, where∂xis the Dirac operator inℝm+1. A structure theorem is proved for such functions, based on the construction of conjugate harmonic pairs. Furthermore, ifΩis bounded with boundaryΓ, whereΓis an Ahlfors-David regular surface, and ifWis aℝ0,m+1(r,p,q)-valued Hölder continuous function onΓ, then necessary and sufficient conditions are given under whichWadmits onΓa Cauchy integral decompositionW=W++W−.