Projection Theorems for Harmonic Measure in NTA Domains

Suppose D is an NTA domain, E D is any closed set, and P ^x ₀(E) is the projection with respect to a point x ₀D of the set E onto the boundary of D. The projection P ^x ₀ satisfies certain geometric properties so that it is a generalization of the notion of radial projection with respect to a point x ₀ onto a boundary of a domain. It is shown that the harmonic measure of E with respect to the domain DE evaluated at the point x ₀ is bounded below by a constant times the harmonic measure of the set P ^x ₀(E) with respect to the domain D evaluated at the point x ₀. The constant is independent of the set E but it may depend upon x ₀.