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

TL;DR

The paper develops positive subharmonic test functions to analyze the distribution of zero sets of holomorphic functions in complex domains, considering growth restrictions near the boundary.

Contribution

It introduces a method to construct subharmonic test functions for studying zero set distributions of holomorphic functions with boundary growth constraints.

Findings

Constructed positive subharmonic functions vanishing on boundary

Applied these functions to analyze zero set distributions

Provided insights into growth restrictions near boundary

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 $\partial D$.