Preorders on Subharmonic Functions and Measures with Applications to the Distribution of Zeros of Holomorphic Functions

TL;DR

This paper develops a dual framework for preorders on subharmonic functions using balayage and measures, with applications to understanding zero distributions of holomorphic functions under growth constraints.

Contribution

It introduces dual equivalent forms of a preorder on subharmonic functions and applies these to characterize zero distributions of holomorphic functions with growth restrictions.

Findings

Dual forms of the preorder are established via balayage processes.

Necessary and sufficient conditions for zero distributions are derived for specific domains.

Results connect subharmonic function theory with zero set distribution of holomorphic functions.

Abstract

Let $X$ be a class of extended numerical functions on a domain $D$ of $d$-dimensional Euclidean space $\mathbb R^d$, $H\subset X$. Given $u,M\in X$, we write $u\prec_H M$ if there is a function $h\in H$ such that $u+h\leq M$ on $D$. We consider this special preorder $\prec_H$ for a pair of subharmonic unctions $u, M$ on $D$ in cases where $H$ is the space of all harmonic functions on $D$ or $H$ is the convex cone of all subharmonic functions $h \not\equiv -\infty$ on $D$. Main results are dual equivalent forms for this preorder $\prec_H$ in terms of balayage processes for Riesz measures of subharmonic functions $u$ and $M$, for Jensen and Arens-Singer (representing) measures, for potentials of these measures, and for special test functions generated by subharmonic functions on complements $D\setminus S$ of non-empty precompact subsets $S\Subset D$. Applications to holomorphic functions $f$ on a domain $D\subset \mathbb C^n$ relate to the distribution of zero sets of functions $f$ under upper restrictions $|f|\leq \exp M$ on $D$. If a domain $D\subset \mathbb C$ is a finitely connected domain with non-empty exterior or a simply connected domain with two different points on the boundary of $D$, then our conditions for the distribution of zeros of $f\neq 0$ with $|f|\leq \exp M$ on $D$ are both necessary and sufficient.