The Lindelöf Condition for Charge Distribution and Balayage

Abstract

Let $ν$ be a charge distribution on the complex plane $\mathbb C$, i.e. the real Radon measure on $\mathbb C$ with total variation $|ν|$. The charge distribution $ν$ is of finite upper density under order of $1$ if $$ \limsup_{0<r\to+\infty} \frac{1}{t}|ν|\Bigl(\bigl\{z\in \mathbb C \bigm| |z|\leq r\bigr\}\Bigr)<+\infty. $$ The charge distribution $ν$ satisfies the Lindelöf condition of genus $1$ if $$ \limsup_{1\leq r\to +\infty}\biggl|\int_{1\leq |z|\leq r} \frac{1}{z}\operatorname{d}\!ν(z)\biggr|<+\infty. $$ 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 $ν$ from the half-plane. We show that balayage of genus $q=1$ of the charge distribution $ν$ from the half-plane preserves this pair of properties under some condition on $ν$.