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

Summary

We investigate local properties of the Green function of the complement of a compact set Ein R ^d, d>2. We give a Wiener type characterization for the Hölder 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 Hölder 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.