BMO Solvability and Absolute Continuity of Harmonic Measure

Hofmann, Steve; Le, Phi

doi:10.1007/s12220-017-9959-0

Cited by 31 publications

(29 citation statements)

References 14 publications

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“…Remark that a related result has been proved in [HoL,Propostion 4.5], however the assumptions in that proposition require an estimate for the left hand side of (7.1) for all x away from 4B, which is not suitable for our purposes.…”

Section: From the Regularity Problem To The Dirichlet Problemmentioning

confidence: 99%

The regularity problem for the Laplace equation in rough domains

Mourgoglou¹,

Tolsa²

2021

Preprint

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Let Ω ⊂ R n+1 , n ≥ 1, be a bounded open and connected set satisfying the corkscrew condition. Assume also that its boundary ∂Ω is uniformly n-rectifiable and its measure theoretic boundary agrees with its topological boundary up to a set of n-dimensional Hausdorff measure zero. In this paper we study the equivalence between the solvability of (D p ′ ), the Dirichlet problem for the Laplacian with boundary data in L p ′ (∂Ω), and (Rp) (resp. ( Rp)), the regularity problem for the Laplacian with boundary data in the Hajłasz Sobolev space W 1,p (∂Ω) (resp. W 1,p (∂Ω), the usual Sobolev space in terms of the tangential derivative), where p ∈ (1, 2 + ε) and 1/p + 1/p ′ = 1. In particular, we show that if (D p ′ ) is solvable then so is (Rp), while in the opposite direction, solvability of ( Rp) implies solvability of (Ds), for all s > p ′ . Under additional geometric assumptions (two-sided local John condition or weak Poincare inequality on the boundary), we show that (D p ′ ) ⇒ ( Rp) and (Rp) ⇒ (Ds), for all s > p ′ . In particular, our results show that in chord-arc domains (resp. two-sided chord-arc domains), there exists p0 ∈ (1, 2 + ε) so that (Rp 0 ) (resp. ( Rp 0 )) is solvable. We also provide a counterexample of a chord-arc domain Ω0 ⊂ R n+1 , n ≥ 3, so that ( Rp) is not solvable for any p ∈ [1, ∞).

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Section: From the Regularity Problem To The Dirichlet Problemmentioning

confidence: 99%

The regularity problem for the Laplace equation in rough domains

Mourgoglou¹,

Tolsa²

2021

Preprint

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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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“…For any non-negative locally integrable function w if log w has small norm BMO norm then, roughly speaking, w is 'almost' an 'optimal' A ∞ weight. The connection between the space of BMO (or V MO) and A ∞ (Muckenhoupt) weights is well documented [GCRdF85,Sar75] as is the connection between the A ∞ condition for the elliptic measure and the solvability of an L p -Dirichlet problem [HL18].…”

Section: Introductionmentioning

confidence: 99%

Elliptic measures for Dahlberg-Kenig-Pipher operators: Asymptotically optimal estimates

Bortz¹,

Toro²,

Zhao³

2021

Preprint

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We show that if a divergence form elliptic operator in the upper half space has coefficients satisfying a vanishing Carleson condition, then the elliptic measure is an asymptotically optimal A ∞ weight. In particular, for such operators the logarithm of the elliptic kernel is in the space of (locally) vanishing mean oscillation.To achieve this, we obtain local, quantitative estimates on a quantity that controls the A ∞ constant introduced by Feffereman, Kenig and Pipher by using estimates produced recently in a work of David, Li and Mayboroda. These quantitative estimates may offer a new framework to approach similar problems. Contents 1. Introduction 1 2. Preliminaries and Notation 3 3. Classical Estimates and the [DLM] energy estimates 9 4. A local quantitative estimate on the FKP Carleson measure for ω L 12 5. Proof of Theorem 1.2 15 6. Globalizing local DKP conditions and the works of Kenig and Pipher and Dindos, Petermichl and Pipher 20 6.1. Extending local DKP conditions and an alternative approach Kenig and Pipher's theorem 21 6.2. Remarks concerning the work of Dindos, Petermichl and Pipher 25 References 26

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BMO Solvability and Absolute Continuity of Harmonic Measure

Cited by 31 publications

References 14 publications

The regularity problem for the Laplace equation in rough domains

The regularity problem for the Laplace equation in rough domains

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

Elliptic measures for Dahlberg-Kenig-Pipher operators: Asymptotically optimal estimates

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