An extension theorem of holomorphic functions on hyperconvex domains

Abstract

Let n ≥ 3 n \geq 3 and let Ω \Omega be a bounded domain in C n \mathbb {C}^n with a smooth negative plurisubharmonic exhaustion function φ \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 ∂ ∂ ¯ φ ) n − 2 (i\partial \bar \partial \varphi )^{n-2} in Ω \Omega can be extended to the whole domain Ω \Omega . To prove it, we combine an L 2 L^2 version of Serre duality and Donnelly-Fefferman type estimates on ( n , n − 1 ) (n,n-1) - and ( n , n ) (n,n) -forms.