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

TL;DR

This paper introduces new multiplication-like operators in quaternionic analysis, explores the 2-Cauchy-Fueter equation, and characterizes its solvability through topological and cohomological conditions.

Contribution

It provides a new acyclic resolution for 2-regular functions and a topological criterion for solving the 2-Cauchy-Fueter equation.

Findings

Established a solvability condition: $H^3(\,\Omega,\mathbb{R})=0$.

Connected solutions to harmonic functions and quaternionic regular functions.

Developed a new framework for understanding quaternionic differential equations.

Abstract

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 $$\mathscr{D}^{(2)}f=g$$ 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.