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Generalized Moisil-Théodoresco systems and cauchy integral decompositions

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 ∂x W = 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 isan 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-.