On Evans' and Choquet's Theorems for Polar Sets

Hansen W, Netuka I. On Evans' and Choquet's Theorems for Polar Sets. Potential Analysis. 2021;2022(56):423–43.By classical results of G.C. Evans and G. Choquet on "good" kernels G in potential theory, for every polar K-sigma-set P, there exists a finite measure mu on P such that its potential G mu is infinite on P, and a set P admits a finite measure mu on P such that G mu is infinite exactly on P if and only if P is a polar G(delta)-set. A known application of Evans' theorem yields the solutions of the generalized Dirichlet problem for open sets by the Perron-Wiener-Brelot method using only harmonic upper and lower functions. It is shown that, by an elementary "metric sweeping" of measures and without using any potential theory, such results can be obtained for general kernels G satisfying a local triangle property, a property which amounts to G being locally equivalent to some negative power of some metric. The particular case, G(x,y) = vertical bar x-y vertical bar(alpha-d) on R-d, 2 < alpha < d, solves a long-standing open problem