Martin boundaries and thin sets for \(\Delta u = Pu\) on Riemann surfaces (Q1196283)

Let \(R\) be a hyperbolic Riemann surface and \(P\) a density on \(R\), that is, a non-negative Hölder continuous function on \(R\) which depends on the local parameter \(z=x+iy\) in such a way that the partial differential equation \[ L_ P u\equiv- \Delta u+Pu=0, \qquad \Delta=\partial^ 2/\partial x^ 2+ \partial^ 2/ \partial y^ 2 \tag{1} \] is invariantly defined on \(R\). A real valued function \(u\) is said to be a \(P\)-harmonic function (or a \(P\)-solution) on an open set \(U\) of \(R\), if \(u\) has continuous partial derivatives up to the order 2 and satisfies the equation (1) on \(U\). We consider two Martin compactifications of a hyperbolic Riemann surface \(R\), the first \(R^*_ P\) with respect to a hyperbolic density \(P\) on \(R\), the second \(R^*\) with respect to harmonic functions. Let \(K^ P(z,a)\), \(K(z,b)\) be Martin kernels on the compactifications \(R^*_ P\), \(R^*\) respectively. Let \(G(z,w)\) be the harmonic Green's function of \(R\). For a minimal boundary point \(a\) of \(R^*_ P\) such that \[ \int_ R P(w)G(w,z_ 1)K^ P(w,a)du dv<+\infty \tag{2} \] for some point \(z_ 1\) in \(R\), there exists a unique minimal boundary point of \(R^*\). Then we may define a mapping with the domain consisting points \(a\) which satisfy the condition (2) into the set of minimal boundary points of \(R^*\). The purpose of this paper is to show that a closed set \(E\) in \(R\) is thin at a point \(a\) with the condition (2) if and only if \(E\) is thin at the minimal boundary point of \(R^*\) assigned to the point \(a\) by the mapping.