Sharp Estimates for Harmonic Measure in Convex Domains

In this note we will prove estimates for harmonic measure in convex and convex C1 domains. It is not hard to show that in a convex domain, surface measure belongs to the Muckenhoupt class A₁ with respect to harmonic measure (Lemma 3). If the boundary of the domain is also of class C1, then it follows from [JK1] that the constant in the A₁ 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-Ampère equation due to L. Caffarelli and to a problem of prescribing harmonic measure as a function of the unit normal.