Domains of existence of polymonogenic functions (Q833058)

An open set \(\Omega\subset \mathbb C^n\) is called a domain of holomorphy of a function \(f:\Omega\to\mathbb C\) if \(f\) is holomorphic in \(\Omega \) and for any point \(z\in \Omega\) it has no holomorphic extension from the ball \(B_z(r)\) with center \(z\) to the ball \(B_z(r')\) for some \(r' > r \doteq\text{dist}(z, \partial\Omega)\). A domain is called non-extensible to a boundary point \(a\), if there exists a neighbourhood \(U\) of \(a\), an open set \(V\subset U\cap\Omega\) and a holomorphic function \(g : V \to \mathbb C\) that has no holomorphic extension to the point \(a\). Any domain of holomorphy is non-extensible to any boundary point. The Levi problem (1910) asks the question whether the converse is true, that is: is any open set \(\Omega\subset \mathbb C^n\) a domain of holomorphy, if it is not extensible to any boundary point? The paper addresses the Levi problem for a system of \(n\) Fueter equations in a domain in the quaternionic space \(\mathbb H^n\). This problem relates to various conditions of convexity and pseudoconvexity of the boundary of the domain.