# BMO Solvability and Absolute Continuity of Harmonic Measure

@article{Hofmann2016BMOSA, title={BMO Solvability and Absolute Continuity of Harmonic Measure}, author={Steve Hofmann and Phi Long Le}, journal={The Journal of Geometric Analysis}, year={2016}, volume={28}, pages={3278-3299} }

We show that for a uniformly elliptic divergence form operator L, defined in an open set $$\Omega $$Ω with Ahlfors–David regular boundary, BMO solvability implies scale-invariant quantitative absolute continuity (the weak-$$A_\infty $$A∞ property) of elliptic-harmonic measure with respect to surface measure on $$\partial \Omega $$∂Ω. We do not impose any connectivity hypothesis, qualitative, or quantitative; in particular, we do not assume the Harnack Chain condition, even within individual…

## 20 Citations

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We consider the Dirichlet boundary value problem for divergence form elliptic operators with bounded measurable coefficients. We prove that for uniform domains with Ahlfors regular boundary, the BMO…

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We show that BMO-solvability implies scale invariant quantitative absolute continuity (specifically, the weak-
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Let $\Omega\subset\mathbb R^{n+1}$ be an open set with $n$-AD-regular boundary. In this paper we prove that if the harmonic measure for $\Omega$ satisfies the so-called weak-$A_\infty$ condition,…

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Let $\Omega\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an open set, not necessarily connected, with an $n$-dimensional uniformly rectifiable boundary. We show that harmonic measure for $\Omega$ is…

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It is a well-known folklore result that quantitative, scale invariant absolute continuity (more precisely, the weak-A∞ property) of harmonic measure with respect to surface measure, on the bound¬ary…

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Let $\Omega\subset\mathbb{R}^{n+1}$, $n\geq 2$, be an open set with Ahlfors-David regular boundary. We consider a uniformly elliptic operator $L$ in divergence form associated with a matrix $A$ with…

Uniform rectifiability from Carleson measure estimates and ε-approximability of bounded harmonic functions

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Let $\Omega\subset\mathbb R^{n+1}$, $n\geq1$, be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that $\partial\Omega$ is uniformly $n$-rectifiable if every bounded…

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