A Wiener-type condition for Hoelder continuity for Green's functions

We investigate local properties of the Green function of the complement of a compact set En R(d), d > 2. We give a Wiener type characterization for the Hoelder continuity of the Green function, thus extending a result of L. Carleson and V. Totik. The obtained density condition is necessary, and it is sufficient as well, provided Esatisfies the cone condition. It is also shown that the Hoelder condition for the Green function at a boundary point can be equivalently stated in terms of the equilibrium measure and the solution to the corresponding Dirichlet problem. The results solve a long standing open problem -- raised by Maz'ja in the 1960's -- under the simple cone condition.