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arXiv · 1202.3860

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

Steve Hofmann·José María Martell·Ignacio Uriarte-Tuero
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Abstract

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 $\partial\Omega$.

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Steve Hofmann, José María Martell, Ignacio Uriarte-Tuero. 2012-02-17. Uniform rectifiability and harmonic measure II: Poisson kernels in $L^p$ imply uniform rectifiability. https://doi.org/10.1215/00127094-2713809

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