For uniformly elliptic divergence-form operators with DKP coefficients on uniform Ahlfors regular domains, A∞ absolute continuity of elliptic measure is equivalent to uniform rectifiability of the boundary and to being a chord-arc domain.
Uniform Rectifiability and harmonic measure IV: Ahlfors regularity plus Poisson kernels in $L^p$ implies uniform rectifiability
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abstract
Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $\Omega:= \mathbb{R}^{n+1}\setminus E$, implies uniform rectifiability of $E$.
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Uniform rectifiability and elliptic operators satisfying a Carleson measure condition. Part II: The large constant case
For uniformly elliptic divergence-form operators with DKP coefficients on uniform Ahlfors regular domains, A∞ absolute continuity of elliptic measure is equivalent to uniform rectifiability of the boundary and to being a chord-arc domain.