On the Structure of Harmonic Multi-Vector Functions

Let F_r (0 < r < m + 1) be a smooth r-vector valued function in a suitable open domain of satisfying in Ω, where ∂ is the Dirac operator in . Then it is proved that there exists H _r , an r-vector valued harmonic function in Ω, such that F _r = . Two proofs of this structure theorem are given, one based on properties of harmonic differential forms and one relying upon primitivation of monogenic functions.