Octonionic and Monogenic Completions of Harmonic Functions

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a domain, with $n\ge 2$. We characterize the real-valued harmonic functions that can occur as prescribed real components of monogenic functions. For any fixed nonzero real Clifford module $M$ and any fixed unit vector $s\in M$, a smooth real-valued function $h$ can be written as $h=⟨F,s⟩$ with $F$ monogenic if and only if the $(n-1)$-form $\ast dh$ is exact. Consequently, every real-valued harmonic function on $\mathrm{\Omega }$ admits such a representation if and only if $H_{\mathrm{d}\mathrm{R}}^{n-1}(\mathrm{\Omega })=0$. In dimension eight, the same exactness criterion characterizes the real parts of octonionic regular functions and yields the corresponding equivalence with $H_{\mathrm{d}\mathrm{R}}^7(\mathrm{\Omega })=0$. The proofs use global solvability of the Poisson equation and elementary identities for differential forms.