Let be a harmonic function in the half-plane , . We define a family of functionals , that are analogs of the family of local times associated to the process where is Brownian motion in . We show that is bounded in if and only if belongs to , an equivalence already proved by Barlow and Yor for the supremum of the local times. Our proof relies on the theory of singular integrals due to Caldéron and Zygmund, rather than the stochastic calculus.
It is shown that the methods developed in an earlier paper of the author about a Dirichlet problem for the Silov boundary [Annales Inst. Fourier, 11 (1961)] lead in a new and natural way to the most important results about the convergence of positive linear operators on spaces of continuous functions defined on a compact space. Choquet’s notion of an adapted space of continuous functions in connection with results of Mokobodzki-Sibony opens the possibility of extending these results to the case...
Let be a domain of type in a Brelot potential theory. A compact in is a in iff has at most countably many components. If is a relatively closed locally polar subset of , any in is a in . If is a domain in , all Borel subsets of are Baire even if is not metrizable. The known results concerning equivalences between weak thinness, thinness, and strong thinness of a set at a point are extended from the case where is a to the cases in which meets only countably...