A characterization of the unit disk and the harmonic measure doubling condition

Abstract:

Suppose $\Omega \subset \mathbb {C}$ is a bounded Jordan domain. Let $\Omega ^*=\overline {\mathbb {C}}\setminus \overline {\Omega }$ denote its complementary domain in the extended plane. A well-known theorem by Jerison and Kenig states that $\partial \Omega$ is a quasicircle if and only if both $\Omega$ and $\Omega ^*$ are doubling domains with respect to the harmonic measure. This theorem fails if we only assume that $\Omega$ is a doubling domain. We show that if $\Omega$ is a doubling domain with constant $c=1$, then it must be a disk.