Balayage of charge distributions and subharmonic functions onto a strip

TL;DR

This paper develops methods for balayage (sweeping) of charge distributions and subharmonic functions onto a strip in the complex plane, establishing conditions for harmonic approximation and analyzing their logarithmic characteristics.

Contribution

It introduces new balayage constructions for subharmonic and δ-subharmonic functions onto a strip, with harmonic approximation outside the strip and detailed analysis of their charge distributions.

Findings

Existence of harmonic functions matching given subharmonic functions outside a strip.

Decomposition of subharmonic functions into harmonic parts on different regions.

Analysis of logarithmic characteristics of charge distributions.

Abstract

We consider two balayage constructions on the complex plane $\mathbb C$ with real axis $\mathbb R$ for $0\leq b\in \mathbb R$. Let $u\not\equiv -\infty$ be a subharmonic function on $\mathbb C$ of order $$\operatorname{ord}[u]:=\limsup_{z\to \infty} \frac{\ln \max\{1,u(z)\}}{\ln |z|}\leq 1,$$ $U=u-v$ be the difference of subharmonic functions $u$ and $v\not\equiv -\infty$ on $\mathbb C$ with $\operatorname{ord}[v]\leq 1$, i.e., $\delta$-subharmonic function on $\mathbb C$ of order $\operatorname{ord}[U]\leq 1$. Then there is a $\delta$-subharmonic function $V\not\equiv \pm\infty$ on $\mathbb C$ of order $\operatorname{ord}[V]\leq 1$ such that $V$ is harmonic on $\bigl\{ z \in \mathbb C\bigm| |\Re z|> b\bigr\}$ and $U(z)\equiv V(z)$ for all $z\in \bigl\{ z \in \mathbb C\bigm| |\Re z|\leq b\bigr\}\setminus E$ where $E\subset \mathbb C$ is polar. If $u$ is a subharmonic function of finite type under order $1$, i.e., $$\limsup_{z\to \infty} \frac{u(z)}{|z|}<+\infty,$$ then there exist subharmonic functions $u_{\mathbb R}$ and $u_b$ of finite type under order $1$ that are harmonic respectively on $\mathbb C\setminus \mathbb R$ and $\bigl\{ z \in \mathbb C\bigm| |\Re z|> b\bigr\}$ such that $$\begin{cases} u(z)\equiv u_{\mathbb R}(z)+u_b(z) \text{ for all $z\in {\mathbb R}\bigcup \bigl\{ z \in \mathbb C\bigm| |\Re z|\leq b\bigr\}$},\\ u(z)\leq u_{\mathbb R}(z) + u_b(z) \text{ for each $z\in \mathbb C$.}\end{cases}$$ At the same time, we trace special relationships between the various logarithmic characteristics of the Riesz mass and charge distributions of subharmonic and $\delta$-subharmonic functions.