In this paper, we introduce a pair of multiplication-like operators, \(L_0\) 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 \( \mathscr {D}^{(2)}f=g \) is solvable for any g satisfying \(\mathscr {D}_1^{(2)}g=0\) on a domain \(\Omega \subset \mathbb {R}^4\) if and only if \(H^3(\Omega , \mathbb {R}) = 0\) , or equivalently, if and only if every real-valued harmonic function on \(\Omega \) can be represented as the real part of a quaternionic regular function.
