On quasi Dirichlet bounded harmonic functions
Zenjiro Kuramochi · Hokkaido Mathematical Journal · 1979
In the present paper we denote by $P$ , $B$ , $D,H$, $SPH$ and SBHy posi- tive, bounded, Dirichlet bounded, harmonic, superharmonic and subharmonic respectively.Let $R$ be a Riemann surface $ ot\in O_{g}$ and let $\{R_{n}\}$ : $n=0,1$ , 2, $\cdots$ , be an exhaustion with compact analytic relative boundary $\partial R_{n}$ .We call a domain $G$ a subdomain, if $\partial G$ consists of at most an enumerably infinite number of analytic curves clustering nowhere in $R$ .In this note we use simply a domain)|^{2}d\theta<\infty$ , $\zeta=re^{i\theta}$ and $U(\zeta)$ is representable by Poisson's integral.This is equivalent to $U(z)=U_{1}(z)-U_{2}(z)$ , where $U_{1}(z)$ and $U_{2}(z)$ are positive quasibounded harmonic function (abbreviated by $QHB)$ .The purpose of this paper is to extend the above theorem.Let $G$ be a domain, if $G$ is compact, we denote by $H_{g}^{G}$ the solution of the Dirichlet problem with respect to the boundary valuean$HP$ is the limit of increasing sequence of $HB$ functions, $U(z)$ is called quasibounded harmonic function (QHB).In this note we denote min $(M, U(z))$ by $U^{M}(z)$ .Let $U(z)$ be a function (harmonic function).If