Monogenic hull for the n-Cauchy-Fueter operator and twistor theory

Abstract

This is the first part in a series of three articles in which are studied the domains of monogenicity for the $n$-Cauchy-Fueter operator. Using the twistor theory, we will in this article show that for a given open subset $U$ of $\mathbb{Q}^n$, there is an open subset $\mathcal{H}(U)$, called the monogenic hull of $U$, of $M_{2n\times 2}^{\mathbb{C}}=\mathbb{Q}^n\otimes\mathbb{C}$ such that each monogenic function in $U$ extends to a unique pair of holomorphic functions on $\mathcal{H}(U)$. In the second part of the series we will exploit the twistor theory furthermore to prove that any pseudoconvex domain in $\mathbb{Q}^n$ is a domain of monogenicity. In the third part of the series, we show the other implication and provide a geometric characterization of the domains of monogenicity.