Uniform rectifiability and harmonic measure i: uniform rectifiability implies Poisson kernels in Lp

We present a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz [37]. More precisely, we establish scale invariant absolute continuity of harmonic measure with respect to surface measure, along with higher integrability of the Poisson kernel, for a domain Ω ⊂ Rn+1; n 2, with a uniformly rectiable boundary, which satises the Harnack chain condition plus an interior (but not exterior) Corkscrew condition. In a companion paper to this one [28], we also establish a converse, in which we deduce uniform rectifiability of the boundary, assuming scale invariant Lq bounds, with q > 1, on the Poisson kernel.The first author was supported by NSF grant DMS-0801079. The second author was supported by MINECO Grant MTM2010-16518 and ICMAT Severo Ochoa project SEV-2011-0087.Peer Reviewe