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 $\Omega\subset \re^{n+1}$ 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 $\partial\Omega$, with scale invariant higher integrability of the Poisson kernel, is sufficient to imply uniformly rectifiable of $\partial\Omega$. 
Steve Hofmann, Jos\'e Mar\'ia Martell, Ignacio Uriarte-Tuero
math.CA math.AP