The Lindel\"of Condition for Charge Distribution and Balayage

TL;DR

This paper investigates the Lindel"of condition for charge distributions on the complex plane and demonstrates that balayage of genus 1 preserves key properties related to entire and subharmonic functions of finite type.

Contribution

It extends previous balayage techniques to genus 1 charge distributions, showing preservation of the Lindel"of condition and finite upper density under certain conditions.

Findings

Balayage of genus 1 preserves the Lindel"of condition.

Balayage maintains finite upper density under specified conditions.

Results applicable to entire and subharmonic functions of finite type.

Abstract

Let $\nu$ be a charge distribution on the complex plane $\mathbb C$, i.e. the real Radon measure on $\mathbb C$ with total variation $|\nu|$. The charge distribution $\nu$ is of finite upper density under order of $1$ if $$ \limsup_{0<r\to+\infty} \frac{1}{t}|\nu|\Bigl(\bigl\{z\in \mathbb C \bigm| |z|\leq r\bigr\}\Bigr)<+\infty. $$ The charge distribution $\nu$ satisfies the Lindel\"of condition of genus $1$ if $$ \limsup_{1\leq r\to +\infty}\biggl|\int_{1\leq |z|\leq r} \frac{1}{z}\operatorname{d}\!\nu(z)\biggr|<+\infty. $$ These concepts play a key role in the study of entire functions of exponential type and subharmonic functions of finite type under order of $1$, as well as in their applications. In our previous works, a technique was developed for balayage of finite genus $q=0,1,\dots$ of the charge distribution $\nu$ from the half-plane. We show that balayage of genus $q=1$ of the charge distribution $\nu$ from the half-plane preserves this pair of properties under some condition on $\nu$.