In potential theory on Riemann
surfaces three kernels are considered: the Green’s kernel on hyperbolic Riemann
surfaces; the Evans kernel on parabolic Riemann surfaces; and the Sario kernel on
arbitrary Riemann surfaces. Since the Sario kernel has no restriction on the domain
surface, in contrast with the two other kernels, its potential theory enjoys the
advantage of full generality. From the point of view of Riemannian spaces
potential theory on Riemann surfaces is included in that on Riemannian
spaces.
The object of this note is to construct the Sario kernel and to develop the
corresponding theory of Sario kernel on Riemannian spaces of dimension n ≧ 3. The
Sario kernel, which is positive, symmetric and jointly continuous, posseses the
property of Riez type decomposition (Theorem 1). The continuity principle, unicity
principle, Frostman’s maximum principle, energy principle and capacity principle are
valid for potentials with respect to the Sario kernel. It is also shown that a
set of capacity zero with respect to the Sario kernel is, considered locally,
of Newtonian capacity zero (Theorem 7), and so the relation of capacity
function and the equilibrium Newtonian potential in Euclidean n-space is
obtained.
