Vol. 38, No. 2, 1971

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Sario potentials on Riemannian spaces

Hideo Imai

Vol. 38 (1971), No. 2, 441–455

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.

Mathematical Subject Classification 2000
Primary: 31C15
Received: 10 September 1970
Published: 1 August 1971
Hideo Imai