Hartogs companions and holomorphic extensions in arbitrary dimension

Abstract

We show that every holomorphic map $f\in\mathcal{H}(\Omega\setminus K)$ ($K\subset\Omega\subset\mathbb{C}^n$, with $K$ compact, $\Omega$ open, and $n\ge2$), has a unique "\emph{Hartogs companion}" $\tilde f\in\mathcal{H}(\Omega)$ matching $f$ on an open subset $C_{K,\Omega}\subset\Omega\setminus K$. Furthermore, $\tilde f$ extends $f$, \emph{if and only if} $\mathbb{C}^n\setminus K$ is a connected set; this equivalence proves the converse implication from the Hartogs Kugelsatz. The existence of vector-valued Hartogs companions in any dimension yields a Hartogs-type extension theorem for G\^ateaux holomorphic maps $f\in\mathcal{H}_\mathrm{G}(\Omega\setminus K,Y)$ on finitely open sets in arbitrary complex vector spaces. The equivalence is very similar to that for $K\subset\Omega\subset\mathbb{C}^n$ and leads to a corresponding Hartogs Kugelsatz in arbitrary dimension and to extension theorems for five types of holomorphy (G\^ateaux, Mackey/Silva, hypoanalytic, Fr\'echet, locally bounded). We also show that the range $\tilde f(\Omega)$ of a vector-valued Hartogs companion cannot leave a domain of holomorphy containing $f(\Omega\setminus K)$. We establish a boundary principle for maps $f\in\mathcal{H}_\mathrm{G}(\Omega,Y)\cap\mathcal{C}(\bar\Omega,Y)$ on finitely bounded open sets. For $Y=\mathbb{C}$, the principle states that $f\big(\bar\Omega\big)=f(\partial\Omega)$ (hence $\sup_{x\in\Omega}|f(x)|=\sup_{x\in\partial\Omega}|f(x)|$). Several results require a new identity theorem, which yields a maximum norm principle and a "max-min" seminorm principle.