Fine potential theory

Abstract

The title of the present survey is short for Potential theory based on the fine This topology (on R n , say) was introduced by H. Cartan in 1940 as the weakest topology making all subharmonic functions continuous. The fine topology is strictly stronger than the usual (EUclidean) topology (when n 2); and an open set in the fine topology need not have inner points in the Euclidean topology. In the sequel we outline some of the developments in fine potential theory during the past two decades. We do not consider the fine topology at the Martin boundary or other ideal boundaries, cf. Brelot [12]. The important probabilistic aspects of fine potential theory will barely be hinted at, cf. Doob [24]. For the non-linear fine potential theory we refer to the lecture by G.Wildenhain in these Proceedings. As applications we discuss the existence of asymptotic paths for subharmonic functions (§6), uniform approximation by or continuous functions (§7), approximation and duality in Dirichlet space (§8), and finely holomorphic functionsan extension of one-dimensional complex analysis in the spirit of Borel's monogenic functions (§§9,10). The classical theory of and subharmonic or functions, defined in open subsets of Rn, has to a large extent been carried over to various axiomatic frameworks such as spaces, more generally to balayage spaces [4] and still more generally to H-cones [7]. In the present lecture we shall mainly consider spaces. On such a space, local notions of harmonic and superharmonic functions are available. We refer the reader to the excellent survey article by Bauer [2]. One may even forget about any reference to axiomatic potential theory and confine oneself to the two basic examples I and II below, which together are fairly typical as spaces.