On the Structure of Harmonic Multi-Vector Functions

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