Sharp Estimates for Harmonic Measure in Convex Domains

In this note we will prove estimates for harmonic measure in convex and convex C 1 domains. It is not hard to show that in a convex domain, surface measure belongs to the Muckenhoupt class A 1 with respect to harmonic measure (Lemma 3). If the boundary of the domain is also of class C 1, then it follows from [JK1] that the constant in the A 1 condition tends to 1 as the radius of the ball tends to 0 (Lemma 7′). Our main estimates (Theorems A and B) are of the same type. The novelty is that they are not calculated with respect to balls, but rather with respect to “slices” formed by the intersection of the boundary with an arbitrary half-space. In addition to proving Theorems A and B we will also explain how these estimates are related to an approach to regularity for the Monge-Ampere equation due to L. Caffarelli and to a problem of prescribing harmonic measure as a function of the unit normal.