Uniform rectifiability and harmonic measure II: Poisson kernels in $L^p$ imply uniform rectifiability

We present the converse to a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz. More precisely, for $n\geq 2$, for an ADR domain $Ω\subset \re^{n+1}$ which satisfies the Harnack Chain condition plus an interior (but not exterior) Corkscrew condition, we show that absolute continuity of harmonic measure with respect to surface measure on $\partialΩ$, with scale invariant higher integrability of the Poisson kernel, is sufficient to imply uniformly rectifiable of $\partialΩ$.

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