Minimal Thinness and the Beurling Minimum Principle

Abstract

Every domain with the Green function has the Martin boundary and every positive harmonic function on the domain is represented as the integral over the Martin boundary. The classical Fatou theorem concerning nontangential limits of harmonic functions is extended to this general context by using the notion of the minimal thinnes. The identification of the Martin boundaries for specific domains is an interesting problem and has attracted many mathematicians. From smooth domains to nonsmooth domains the study of the Martin boundary has been expanded. It is now known that the Martin boundary of a uniform domain is homeomorphic to the Euclidean boundary. On the other hand, the study of the minimal thinnes has been exploited comparatively a little. The minimal thinnes of an NTA domain was studied by the author with the aid of the quasiadditivity of capacity, the Hardy inequality, and the Beurling minimum principle. Maz’ya was one of the first mathematicians who recognized the significance of the Beurling minimum principle. This article illustrates the backgrounds of the characterization of the minimal thinness and gives a characterization of the minimal thinness of a uniform domain.