An extension theorem of holomorphic functions on hyperconvex domains

Abstract:

Let $n \geq 3$ and let $\Omega$ be a bounded domain in $\mathbb {C}^n$ with a smooth negative plurisubharmonic exhaustion function $\varphi$. As a generalization of Y. Tiba’s result, we prove that any holomorphic function on a connected open neighborhood of the support of $(i\partial \bar \partial \varphi )^{n-2}$ in $\Omega$ can be extended to the whole domain $\Omega$. To prove it, we combine an $L^2$ version of Serre duality and Donnelly-Fefferman type estimates on $(n,n-1)$- and $(n,n)$-forms.