A Pair of Multiplication-Type Operators in Quaternionic Analysis and the 2-Cauchy-Fueter Equation

In this paper, we introduce a pair of multiplication-like operations, $L_0$ and $L_1$, which derive $k$-regular functions from $(k+1)$-regular functions. The investigation of the inverse problem naturally leads to a deeper study of the 2-Cauchy-Fueter equation. In doing so, we provide a new acyclic resolution for the sheaf of $2$-regular functions ${\mathcal{R}}^{(2)}$. Furthermore, a complete topological characterization for the solvability of the $2$-Cauchy-Fueter equation is established. Specifically, we prove that the $2$-Cauchy-Fueter equation $${\mathcal{D}}^{(2)}f=g$$ is solvable for any $g$ satisfying ${\mathcal{D}}_1^{(2)}g=0$ on a domain $\mathrm{\Omega }\subset {\mathbb{R}}^4$ if and only if $H^3(\mathrm{\Omega },\mathbb{R})=0$, or equivalently, if and only if every real-valued harmonic function on $\mathrm{\Omega }$ can be represented as the real part of a quaternionic regular function.