Uniform rectifiability and harmonic measure II: Poisson kernels in LpL^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 $\partial\Omega$

A note on failure of energy reversal for classical fractional singular integrals

For alpha in [0,n) we demonstrate the failure of energy reversal for the vector of alpha-fractional Riesz transforms, and more generally for any vector of alpha-fractional convolution singular integrals having a kernel with vanishing integral on every great circle of the sphere.Comment: 24 pages. This version references a correct form of the T1 theorem, corrects typos and uses Bochner's theorem to complete the proof for the missing range of alpha, and also points out an easy extension to higher dimensions. arXiv admin note: text overlap with arXiv:1305.510