関数核の正則性とポテンシァルの無限遠点の近傍での挙動について

Abstract

Let G be a symmetric and continuous function-kernels on a locally compact Hausdorff space X and δ be the point at infinity. In this paper, first we define the several notions of thinness of a closed set at infinity δ and investigate the mutual relations among them. For a non-negative G-superharmonic function u, we denote by R^<X, δ>_G(u) the G-reduced function of u to δ. A kernel G satisfying the domination principle is said to be regular when we have R^<X, δ>_G(Gμ)(x)=0 G-nearly everywhere on X for the potential G_μ of any positive mesure μ with compact support. The regularity of kernels plays an important role in the theory of Hunt kernels. The purpose of this paper is to characterize the regularity of function-kernels by the behavior of potentials in a neighborhood of infinity δ. We shall prove that a continuous function-kernel G is regular if and only if, for every mesure p with finite G-energy, the potential G_μ is equal to 0 G-quasi-everywhere at infinity δ under the assumption that G satisfies the complete maximum principle.