Integration near the Royden boundary of a Riemannian manifold

In studying the properties of the space of energy-finite solutions of the equation A u = Pu(P ~0) on an oriented Riemannian manifold R attention has been focused on the integral of P over neighborhoods of the Royden boundary points of R. In particular in [1] it was shown that the points of the Royden harmonic boundary that possess a neighborhood over which the integral of P is convergent completely determine this space. Recently Nakai [3] introduced the descriptive term P-nondensity point for points with this property. Off hand one would expect to be able to find a P that grows so rapidly that the P-nondensity points of the Royden boundary of R form a small set from the topological viewpoint. The purpose of this paper is to prove the rather startling fact (Theorem 2) that for any nonnegative locally integrable n-form P on a Riemannian n-manifold R the P-nondensity points on the Royden boundary of R form a dense subset. The key to this result is the compactification introduced by Nakai-Sario [4] and in Theorem 1 we give some relations between the Royden boundary and the boundary of this compactification. For a detailed account of the terminology and auxiliary results useA here see Chapter III of [51, [2] and [4]. We begin by fixing our notations. Let R be a Riemannian n-manifold, M its Royden algebra and M~ the potential subalgebra. The maximal ideal space R* of M with the weak* topology is the Royden compactification of of R. We denote by F = R*\R the Royden boundary and by d its harmonic part, i.e., A = {pER*lf(P)=O for every feM~}. Let P denote an arbitrary nonnegative locally integrable n-form on R and denote by E the subalgebra of M given by E = l f ~ M{~ f2p < + ~}. Under