Compactifications of Martin type of harmonic spaces (Q1119782)

The author introduces and studies compactifications \((X^*,k(x,z),\Delta_ 1,\mu)\) of Martin type for \({\mathcal P}\)-harmonic spaces X (with countable base and superharmonic positive constants). By definition, such a compactification is metrizable and resolutive, and for every bounded harmonic function u on X there exists a resolutive function f on \(\Delta =X^*\setminus X\) such that \[ u(x)=H_ f(x)=\int k(x,z)f(z)d\mu (z)\quad (x\in X). \] Several examples are discussed. In particular, it is shown that nuclearity of the sheaf \({\mathcal H}\) implies the existence of a compactification of Martin type. The Dirichlet problem associated with fine filters is treated and the Fatou-Doob-Naim theorem for compactifications of Martin type is established. Finally, the relationship to arbitrary metrizable and resolutive compactifications are discussed.