论文标题
电荷分布和统一功能的balayage在条带上
Balayage of charge distributions and subharmonic functions onto a strip
论文作者
论文摘要
我们考虑在复杂平面上的两个balayage构造，$ \ mathbb c $，带有Real Axis $ \ Mathbb r $，价格为$ 0 \ leq B \ in \ Mathbb r $。令$ \ e equiv -equiv - \ infty $是$ \ mathbb c $ of dord $$ $$ \ operatorname {orp} [or} [us} [u]：= \ limsup_ {z \ to \ infty} \ frac {\ frac { $ u = u-v $是subharmonic函数$ u $和$ v \ equiv-equiv-equiv-equiv-equiv-equiv- \ infty $ on $ \ mathbb c $带有$ \ propatatorName {ordaTorname {ord} [v] \ leq 1 $，即，即，$δ$-Δ$-Δ$-Δ$-Δ$ -UD-subharmonic在$ \ Mathbb c $ of co $ fore of co $ fore $ \ feperAte $ \ eperate u \ operateNAme usanemearn usort usearne useorn useorn。然后有一个$δ$ -Subharmonic函数$ v \ equiv \ equiv \ pm \ pm \ in $ \ mathbb c $ of Order $ \ operatatorName {orpataTorname {ord} [v] \ leq 1 $，以便$ v $是$ \ bigl \ big big \ big \ in \ mathbb c \ mathbb c \ mathbb c \ bigm | bigm | bigm | | \ re z |> b \ bigr \} $和$ u（z）\ equiv v（z）$ for in \ bigl \ {z \ in \ mathbb c \ bigm | | \ re z | \ leq b \ bigr \} \ setMinus e $其中$ e \ subset \ mathbb c $是极性的。 如果$ u $是根据订单$ 1 $（即$$ \ limsup_ {z \ to \ infty} \ frac {u（z）} {| z | | |} <+\ iftty，$ $ unmarnomon函数$ _ _ _ $ _ n under undifiention o_ _ $ u__ $ u__ $ u__ $ u__ $ u__ $ u__ $ u__在$ \ mathbb c \ setminus \ mathbb r $和$ \ bigl \ {z \ in \ mathbb c \ bigm | | \ re z |> b \ bigr \} $使得$ \ begin {case} u（z）\ equiv u _ {\ mathbb r} \ mathbb r}（z）（z）+u_b（z）\ text {用于{\ mathb r} | \ re z | \ leq b \ bigr \} $}，\\ u（z）\ leq uq u _ {\ mathbb r}（z） + u_b（z） + u_b（z）\ text {对于每个$ z \ in \ mathbb c $。亚谐波和$δ$ -Subharmonic功能。
We consider two balayage constructions on the complex plane $\mathbb C$ with real axis $\mathbb R$ for $0\leq b\in \mathbb R$. Let $u\not\equiv -\infty$ be a subharmonic function on $\mathbb C$ of order $$\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.
