Existence of singular harmonic functions (Q971976)

Let \(HP(R)\) be the vector subspace of the vector space \(H(R)\) of harmonic functions on a Riemann surface \(R\) consisting of the essentially positive harmonic functions on \(R\), where \(u\) is essentially positive if \(u=u_1-u_2\) with \(u_j\in H(R)^+:=\{v\in H(R):v\geq0\}\) (\(j=1,2\)). We denote by \(u\vee v\) (\(u\wedge v\), resp.) the least (greatest, resp.) harmonic majorant (minorant, resp.) of \(u\) and \(v\) on \(R\) for \(u\) and \(v\) in \(HP(R)\). Let \(HB(R)\subset HP(R)\) be the vector space of bounded harmonic functions on \(R\). A function \(u\in HP(R)\) is called quasibounded if \[ u=\lim_{s,t\in\mathbb R^+;s,t\uparrow\infty}(u\wedge s)\vee (-t) \] locally uniformly on \(R\), so that, in particular, every \(u\in HB(R)\) is quasibounded on \(R\). A function \(u\in HP(R)\) is said to be singular if \((u\wedge s)\vee(-t)=0\) identically on \(R\) for every \(s\) and \(t\) in \(\mathbb R^+\). We denote by \(HP_q(R)\) (\(HP_s(R)\), resp.) the vector subspace of \(HP(R)\) consisting of quasibounded (singular, resp.) harmonic functions on \(R\). Then we have the Parreau decomposition of \(HP(R)\): \(HP(R)=HP(R)_q\oplus HP(R)_s\). It can happen that \(HP(R)=HP(R)_q\), or equivalently, \(HP(R)_s=\{0\}\). We denote by \(\mathcal O_s\) the class of hyperbolic Riemann surfaces \(R\) with \(HP(R)_s=\{0\}\). Let \(\mathcal O_G\) be the family of parabolic Riemann surfaces \(R\). An afforested surface \(W:=\big\langle P,(T_n)_{n\in\mathbb N},(\sigma_n)_{n\in\mathbb N}\big\rangle\) is an open Riemann surfaces consisting of three ingredients: a hyperbolic Riemann surface \(P\notin\mathcal O_G\) called a plantation, a sequence \((T_n)_{n\in\mathbb N}\) of hyperbolic Riemann surfaces \(T_n\) each of which is called a tree, and a sequence \((\sigma_n)_{n\in\mathbb N}\) of slits \(\sigma_n\) called the roots of \(T_n\), which are contained both in \(P\) and \(T_n\) and which are mutually disjont and not accumulating in \(P\). Then the surface \(W\) is formed by foresting trees \(T_n\) on the plantation \(P\) at the roots, for all \(n\in\mathbb N\), or, more precisely, by pasting the surface \(T_n\) to \(P\) along the slit \(\sigma_n\), for all \(n\in\mathbb N\). One might think that any afforested surface \(W\) belongs to the family \(\mathcal O_s\) if its plantation \(P\) and all its trees \(T_n\) belong to \(\mathcal O_s\). The aim of this paper is to prove that this is not the case. \textbf{The Main Theorem.} There exists an afforested surface \(W:=\big\langle P,(T_n)_{n\in\mathbb N},(\sigma_n)_{n\in\mathbb N}\big\rangle\) such that \(P\) and all \(T_n\) (\(n\in\mathbb N\)) belong to the class \(\mathcal O_s\), whereas \(W\) does not belong to the class \(\mathcal O_s\).