Uniform rectifiability and harmonic measure, II: Poisson kernels in Lp imply uniform rectifiability

Abstract

We present the converse to a higher-dimensional, scale-invariant version of the classical F. and M. Riesz theorem, proved by the first two authors. More precisely, for $n\ge 2$, for an Ahlfors–David regular domain $\Omega \subset {\mathbb{R}}^{n+1}$ which satisfies the Harnack chain condition plus an interior (but not exterior) corkscrew condition, we show that absolute continuity of the harmonic measure with respect to the surface measure on $\partial \Omega$, with scale-invariant higher integrability of the Poisson kernel, is sufficient to imply quantitative rectifiability of $\partial \Omega$.