Subharmonic test functions and the distribution of zero sets of holomorphic functions

Abstract

Let $m,n\geq 1$ are integers and $D$ be a domain in the $$ $\mathbb C^n$ or in the $m$-dimensional real space $\mathbb R^m$. We build positive subharmonic functions on $D$ vanishing on the boundary $\partial D$ of $D$. We use such (test) functions to study the distribution of zero sets of holomorphic functions $f$ on $D\subset \mathbb C^n$ with restrictions on the growth of $f$ near the boundary $\partial D$.