BMO solvability and absolute continuity of harmonic measure

Hofmann, Steve; Le, Phi

doi:10.48550/arxiv.1607.00418

Cited by 2 publications

(2 citation statements)

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“…In Hofmann and Le [HL,eq. (4.12)] the estimate (1.5) (at least for n ≥ 2) is asserted for corkscrew domains with uniformly n-rectifiable boundaries and attributed to a forthcoming paper by Hofmann, Martell and Mayboroda.…”

Section: Introductionmentioning

confidence: 99%

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

Garnett¹,

Mourgoglou²,

Tolsa³

2018

Duke Math. J.

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Let Ω ⊂ R n+1 , n ≥ 1, be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that ∂Ω is uniformly n-rectifiable if every bounded harmonic function on Ω is ε-approximable or if every bounded harmonic function on Ω satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when Ω = R n+1 \ E and E is Ahlfors-David regular. Our results solve a conjecture posed by Hofmann, Martell, and Mayboroda in a recent work where they proved the converse statements. Here we also obtain two additional criteria for uniform rectifiability. One is given in terms of the so-called "S < N " estimates, and another in terms of a suitable corona decomposition involving harmonic measure.

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Section: Introductionmentioning

confidence: 99%

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

Garnett¹,

Mourgoglou²,

Tolsa³

2018

Duke Math. J.

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“…Seeking out this form of absolute continuity has applications for PDEs: in [HL16], for example, Hofmann and Le showed that BMO solvability of the Dirichlet problem for the Laplacian is implied by the A ∞ -property (in fact, it is implied by the weak A ∞ property).…”

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confidence: 99%

Semi-Uniform Domains and the A∞ Property for Harmonic Measure

Azzam

2019

International Mathematics Research Notices

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We study the properties of harmonic measure in semi-uniform domains. Aikawa and Hirata showed in [AH08] that, for John domains satisfying the capacity density condition (CDC), the doubling property for harmonic measure is equivalent to the domain being semi-uniform. Our first result removes the John condition by showing that any domain satisfying the CDC whose harmonic measure is doubling is semiuniform. Next, we develop a substitute for some classical estimates on harmonic measure in nontangentially accessible domains that works in semi-uniform domains.We also show that semi-uniform domains with uniformly rectifiable boundary have big pieces of chord-arc subdomains. We cannot hope for big pieces of Lipschitz subdomains (as was shown for chord-arc domains by David and Jerison [DJ90]) due to an example of Hrycak, which we review in the appendix.Finally, we combine these tools to study the A ∞ -property of harmonic measure. For a domain with Ahlfors-David regular boundary, it was shown by Hofmann and Martell that the A ∞ property of harmonic measure implies uniform rectifiability of the boundary [HM15, HLMN17] . Since A ∞ -weights are doubling, this also implies the domain is semi-uniform. Our final result shows that these two properties, semi-uniformity and uniformly rectifiable boundary, also imply the A ∞ property for harmonic measure, thus classifying geometrically all domains for which this holds.

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BMO solvability and absolute continuity of harmonic measure

Cited by 2 publications

References 20 publications

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

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

Semi-Uniform Domains and the A∞ Property for Harmonic Measure

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