Martin boundary over an isolated singularity of rotation free density
Mitsuru Nakai · Journal of the Mathematical Society of Japan · 1974
Consider a density $P(z)dxdy$ on a Riemann surface $R,$ $i$ .$e$ .a 2-form $P(z)dxdy$ whose coefficients $P(z)$ are nonnegative locally H\"older continuous functions of local parameters $z=x+iy$ on $R$ .Let $\delta$ be an isolated parabolic ideal boundary component of $R$ .This means that there exists the base $\{\Omega^{*}\}$ of neighborhoods of the point $\delta$ in the Ker\'ekj\'art6-Stoi1ow compactification of $R$ such that each $\Omega=\Omega*\cap R$ is an end of $R,$ $i$ .$e$ .asubregion of $R$ with compact analytic relative boundary $\partial\Omega$ and a single ideal boundary component $\delta$ .The parabolicity of $\delta$ is characterized by the parabolicity of the double $\hat{\Omega}$ of $\Omega=\Omega*\cap R$ about $\partial\Omega$ for every $\Omega^{*}\in\{\Omega^{*}\}$ .To describe a potential theoretic behavior of $P$ at $\delta$ we introduce the $P$ -elliPtic dimension, $\dim_{p}\delta$ , of $\delta$ as follows: Let $\mathcal{F}_{P}(\Omega)$ be the half module of nonnegative solutions $u$ of the elliptic equation $\Delta u(z)=P(z)u(z)$ $(i.e. d^{*}du(z)=u(z)P(z)dxdy)$ on $\Omega$ with continuously vanishing boundary values on $\partial\Omega$ for $\Omega=\Omega^{*}\cap R$ with $\Omega*\in\{\Omega^{*}\}$ .Since $\mathcal{F}_{P}(\Omega)$ are isomorphic to each other as half modules for all $\Omega$ ( $Oz$ awa [15,16]), the common half module structure $\mathcal{F}_{P}(\delta)$ is deter- mined only by $\delta$ and $P$ .Then we define $\dim_{p}\delta$ to be the dimension of $\mathcal{F}_{P}(\delta)$ , $i$ .$e$ .the minimal cardinal number of sets of generators of $\mathcal{F}_{P}(\delta)$ .The simplest $\delta$ is the $\delta_{0}$ which is represented as the origin $z=0$ of the punctured disk $00$ .For the particular rotation free densities $P_{\lambda}(z)=|z|^{-\lambda}$ in $U$ with $\lambda\in[-\infty, \infty$ ) we will show that $\alpha(P_{\lambda})=0f$ or $\lambda\in[-\infty, 2]$ and $\alpha(P_{\lambda})>0f$ or $\lambda\in(2, \infty)$ .Therefore the main conclusion of this paper is that the range under $P\rightarrow diin_{P}\delta_{0}$ of the class of rotation free densities $P$ is the two element set $\{1, c\}$ .In particular the Picard principle is valid for rotation free densities $P$ if and only if their singularity indices $\alpha(P)=0$ .For comparison we will append a proof of the Riemann theorem: $\lim_{z-0}u(z)$ exists for every bounded solution $u$ of $\Delta u=Pu$ on $0<|z|\leqq 1$ with a rotation free density $P(z)$ on $0<|z|\leqq 1$ and in fact a bit more general density $P$ which we call almost rotation free.\S 1. Singularity indices.1.1.Consider a nonnegative locally H\"older continuous $f$ unction (density) $P(z)$ on the closed punctured ciisk $0<|z|\leqq 1$ which is rotation free in the sense that $P(z)=P(|z|)$ for every $z$ in $0<|z|\leqq 1$ .The function $P(z)$ may or may not be defined at $z=0$ and therefore $P(z)$ is supposed to have an isolated singularity at $z=0$, removable or genuine.To describe the degree