A property of holomorphic functions related to plurisubharmonic functions

Abstract

Let $\varphi $ be a negative plurisubharmonic function in a pseudoconvex domain $\Omega$ in $\mathbb{C}^{n}$ and $f$ be a bounded holomorphic function belonging to $L^{2}(\Omega, \varphi)$. For all negative plurisubharmonic functions $\psi$ satisfying that $\psi$ is close to $\varphi$ 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 $\Omega^{'}$ of $\Omega.$