Uniform rectifiability and harmonic measure II: Poisson kernels in Lᵖ imply uniform rectifiability

الأصل · EN

We present the converse to a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz. More precisely, for n≥ 2, for an ADR domain Ω⊂ 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 ∂Ω, with scale invariant higher integrability of the Poisson kernel, is sufficient to imply uniformly rectifiable of ∂Ω.