Masanori Kishi

Maximum principles in the potential theory

Ninomiya, in his thesis [13] on the potential theory with respect to a positive symmetric continuous kernel G on a locally compact Hausdorff space i2, proves that G satisfies the balayage (resp. equilibrium) principle if and only if G satisfies the domination (resp. maximum) principle. He starts from the Gauss-Ninomiya variation and shows that for any given compact set K in Ω and any positive upper semi-continuous function u on K% there exists a positive measure μ on K such that its potential Gμ is < u on the support of μ and Gμ > u on K almost everywhere with respect to any positive measure with finite energy. His method can not be applied to non-symmetric kernels, because for those kernels the Gauss-Ninomiya variation is useless in its original form. In this paper we shall prove that the above existence theorem is valid for non-symmetric kernels under certain additional conditions—separability of a compact set K and the continuity principle for adjoint kernels. We first prove it in a reduced form on a compact space consisting of a finite number of points and then extend it to a kernel on a locally compact Hausdorff space. Using our existence theorem, we shall prove that if G and its adjoint G satisfy the continuity principle, then G satisfies the balayage (resp. equilibrium) principle when and only when G satisfies the domination (resp. maximum) principle. It will be also shown that G satisfies the balayage principle if and only if 6 does. Other maximum principles which are closely connected with the domination and maximum principles will be dealt with also. We shall give an answer to a question, raised by Deny, which concerns with the complete maximum principle. A summary of this paper was published in [11].

An Existence Theorem in Potential Theory

1. Concerning a positive lower semicontinuous kernel G on a locally compact Hausdorff space X the following existence theorem was obtained in [3].

Puissance fractionnaire d'un noyau positif dont le carré est encore un noyau

Le noyau de Riesz-Frostman dans l’espace euclidien a d (≥3) dimensions est a la puissance fractionnaire du noyau de Newton. On trouve la conformite dans la theorie des espaces de Dirichlet; on construit un noyau a la puissance fractionnaire d’un noyau associe a l’espace de Dirichlet special, qui determine a nouveau un espace de Dirichlet special [2]. En notant que le noyau de Dirichlet est positif et symetrique, et satisfait au principe complet du maximum, on s’interroge: est-il possible de construire de bons noyaux a la puissance fractionnaire pour des noyaux de genre plus large?

Capacitabilty of analytic sets

Let Ω be a locally compact separable metric space and let Ф be a positive symmetric kernel. Then the inner and outer capacities of subsets of Ω are defined by means of Ф -potentials of positive measures in the following manner. We define the capacity c(K) of a compact set K in a certain manner by means of Ф -potentials.