Smoothness of Green’s functions and density of sets

The continuity of Green's functions at boundary points has been extensively studied for a long time. Our aim is to give conditions for the stronger Hölder continuity in terms of the geometry of the set. We consider both the planar and the higher dimensional case.

We investigate local properties of the Green function of the complement of a compact set $ E$. First we consider the case $ E\subset [0,1]$ in the extended complex plane. We extend results of V. Andrievskii, L. Carleson and V. Totik on the condition for optimal smoothness. A characterization is also given with the Markov inequality for compact sets $ E\subset\mathbb{C}$ whose Green function satisfies the Lipschitz (or Hölder-1) condition. Finally, we consider the case when $ E$ is a compact set in $ \mathbb{R}^d$, $ d>2$ and give a Wiener type characterization for the Hölder continuity of the Green function. This result solves a long standing open problem - raised by Maz'ja in the 1960's - under the simple cone condition.