A property of holomorphic functions related to plurisubharmonic functions

Abstract

Let $\phi$ be a negative plurisubharmonic function in a pseudoconvex domain $\Omega$ in ${\mathbb{C}}^n$ and $f$ be a bounded holomorphic function belonging to $L^2(\Omega ,\phi )$. For all negative plurisubharmonic functions $\psi$ satisfying that $\psi$ is close to $\phi$ in the $L^1(\Omega )$ norm. Using the recent results about $L^2-$estimate for $\overline{\partial }-$equation and strong openness conjecture, we prove that there exists a holomorphic function $g$ belonging to $L^2(\Omega ,\psi )$ such that $g$ is close to $f$ in the $L^2$ norm on some relatively compact open subset ${\Omega }^{{}^{\prime }}$ of $\Omega .$

11 pages