FLATNESS OF DOMAINS AND DOUBLING PROPERTIES OF MEASURES SUPPORTED ON THEIR BOUNDARY, WITH APPLICATIONS TO HARMONIC MEASURE

The results discussed here are joint work with Tatiana Toro, contained in [14] and [15]. In this note, Omega is always taken to be an open c onnected unbounded domain in Rn+1 whose boundary separates Rn+1. Namel y Rn+1\partial derivative Omega has exactly two non-empty connected co mponents Omega and int Omega(c). The canonical example to keep in mind is the upper half space R-+(n+1). In the context of this article, a d omain is always taken to be of this topological type. Similar results to the ones stared below hold for bounded domains satisfying the appro priate separation and connectivity conditions. For the sake of exposit ion, we restrict our discussion to the unbounded setting.