带上电荷分布与下调和函数的平衡
Balayage of charge distributions and subharmonic functions onto a strip
摘要：
我们考虑复平面$\mathbb C$上的两个平衡结构，其中实轴$\mathbb R$为$0\leq b\in \mathbb R$。令$u\not\equiv -\infty$是$\mathbb C$上的阶为$$\operatorname{ord}[u]:=\limsup_{z\to \infty} \frac{\ln \max\{1,u(z)\}}{\ln |z|}\leq 1,$$的次调和函数，$U=u-v$是$\mathbb C$上的下调和函数$u$和$v\not\equiv -\infty$与$\operatorname{ord}[v]\leq 1$的差，即，$δ$ -阶数为$\operatorname{ord}[U]\leq 1$的$\mathbb C$上的次调和函数。然后有一个$δ$ -次调和函数$V\not\equiv \pm\infty$在$\mathbb C$的顺序$\operatorname{ord}[V]\leq 1$，使得$V$是调和的$\bigl\{ z \in \mathbb C\bigm| |\Re z|> b\bigr\}$和$U(z)\equiv V(z)$的所有$z\in \bigl\{ z \in \mathbb C\bigm| |\Re z|\leq b\bigr\}\setminus E$，其中$E\subset \mathbb C$是极。如果$u$是$1$阶下的有限型次调和函数，即，$$\limsup_{z\to \infty} \frac{u(z)}{|z|}<+\infty,$$则存在$1$阶下的有限型下调和函数$u_{\mathbb R}$和$u_b$，它们分别在$\mathbb C\setminus \mathbb R$和$\bigl\{ z \in \mathbb C\bigm| |\Re z|> b\bigr\}$上调和，使得$$\begin{cases} u(z)\equiv u_{\mathbb R}(z)+u_b(z) \text{ for all $z\in {\mathbb R}\bigcup \bigl\{ z \in \mathbb C\bigm| |\Re z|\leq b\bigr\}$},\\ u(z)\leq u_{\mathbb R}(z) + u_b(z) \text{ for each $z\in \mathbb C$.}\end{cases}$$同时，我们追踪了次调和函数和$δ$ -下调和函数的Riesz质量和电荷分布的各种对数特征之间的特殊关系。
英文摘要：
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., $δ$-subharmonic function on $\mathbb C$ of order $\operatorname{ord}[U]\leq 1$. Then there is a $δ$-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 $δ$-subharmonic functions.
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