Mutual Absolute Continuity of Interior and Exterior Harmonic Measure Implies Rectifiability

Azzam, Jonas; Mourgoglou, Mihalis; Tolsa, Xavier

doi:10.1002/cpa.21687

“…Among these we would like to highlight [HM1] and [HMU], from which it follows that for a bounded uniform domain Ω ⊂ R n+1 (so that ∂Ω is n-AD-regular), the harmonic measure ω p in Ω is an A ∞ weight with respect to the surface measure if and only if ∂Ω is uniformly n-rectifiable. On the other hand, the connection between harmonic measure and the Riesz transforms, in combination with the rectifiability criteria from [NToV1] and [NToV2], has been successfully exploited in other recent works such as [AHM 3 TV], [MT], and [AMT2].…”

Section: Introductionmentioning

confidence: 99%

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

Garnett¹,

Mourgoglou²,

Tolsa³

2018

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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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“…By Lemma 4.13, (4.27) holds. In particular, 1 2 B ∩ supp ω ∞ is a smooth real analytic variety, and arguing as in [AMTV16], for example, one deduces that…”

Section: 3mentioning

confidence: 85%

“…The main blow-up lemma. We now introduce the main tool of this paper, which is a variant of previous blow-up arguments, first introduced by Kenig and Toro for NTA domains [KT06], then extended to CDC domains in [AMT16]. Both these cases applied to harmonic measure, but it can be extended to elliptic measures with a VMO condition on the coefficients.…”

Section: 4mentioning

confidence: 99%

“…Proof. The proof of this lemma can be found in [AMT16] for harmonic measure for the case that K = {ξ} (i.e. so that (1.4) holds).…”

Section: 4mentioning

confidence: 99%

“…Thus, there is B ⊂ Ω j for all large j (after passing to a subsequence). Proof of (d): Arguing as in [AMT16], there exists u + ∞ which is continuous in R n+1 and vanishes on (Ω + ∞ ) c such that (after passing to a subsequence) u + j → u + ∞ uniformly on compact sets of R n+1 . Moreover, it is not hard to see that u + j ∈ W 1.2 (B) for large j.…”

Section: Proof Of (B)mentioning

confidence: 99%

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Tangent measures of elliptic measure and applications

Azzam

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Mourgoglou

²

2019

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Tangent measure and blow-up methods, are powerful tools for understanding the relationship between the infinitesimal structure of the boundary of a domain and the behavior of its harmonic measure. We introduce a method for studying tangent measures of elliptic measures in arbitrary domains associated with (possibly non-symmetric) elliptic operators in divergence form whose coefficients have vanishing mean oscillation at the boundary. In this setting, we show the following for domains Ω ⊂ R n+1 :(1) We extend the results of Kenig, Preiss, and Toro [KPT09] by showing mutual absolute continuity of interior and exterior elliptic measures for any domains implies the tangent measures are a.e. flat and the elliptic measures have dimension n. (2) We generalize the work of Kenig and Toro [KT06] and show that VMO equivalence of doubling interior and exterior elliptic measures for general domains implies the tangent measures are always elliptic polynomials. (3) In a uniform domain that satisfies the capacity density condition and whose boundary is locally finite and has a.e. positive lower n-Hausdorff density, we show that if the elliptic measure is absolutely continuous with respect to n-Hausdorff measure then the boundary is rectifiable. This generalizes the work of Akman, Badger, Hofmann, and Martell [ABHM17]. Finally, we generalize one of the main results of [Bad11] by showing that if ω is a Radon measure for which all tangent measures at a point are harmonic polynomials vanishing at the origin, then they are all homogeneous harmonic polynomials. 2010 Mathematics Subject Classification. 31A15,28A75,28A78,28A33.

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L2-Boundedness of Gradients of Single Layer Potentials for Elliptic Operators with Coefficients of Dini Mean Oscillation-Type

Molero

¹

,

Mourgoglou

²

,

Puliatti

³

et al. 2023

Arch Rational Mech Anal

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We consider a uniformly elliptic operator $$L_A$$ L A in divergence form associated with an $$(n+1)\times (n+1)$$ ( n + 1 ) × ( n + 1 ) -matrix A with real, merely bounded, and possibly non-symmetric coefficients. If "Equation missing"then, under suitable Dini-type assumptions on $$\omega _A$$ ω A , we prove the following: if $$\mu $$ μ is a compactly supported Radon measure in $$\mathbb {R}^{n+1}$$ R n + 1 , $$n \ge 2$$ n ≥ 2 , and $$ T_\mu f(x)=\int \nabla _x\Gamma _A (x,y)f(y)\, \textrm{d}\mu (y) $$ T μ f ( x ) = ∫ ∇ x Γ A ( x , y ) f ( y ) d μ ( y ) denotes the gradient of the single layer potential associated with $$L_A$$ L A , then $$\begin{aligned} 1+ \Vert T_\mu \Vert _{L^2(\mu )\rightarrow L^2(\mu )}\approx 1+ \Vert {\mathcal {R}}_\mu \Vert _{L^2(\mu )\rightarrow L^2(\mu )}, \end{aligned}$$ 1 + ‖ T μ ‖ L 2 ( μ ) → L 2 ( μ ) ≈ 1 + ‖ R μ ‖ L 2 ( μ ) → L 2 ( μ ) , where $${\mathcal {R}}_\mu $$ R μ indicates the n-dimensional Riesz transform. This allows us to provide a direct generalization of some deep geometric results, initially obtained for $${\mathcal {R}}_\mu $$ R μ , which were recently extended to $$T_\mu $$ T μ associated with $$L_A$$ L A with Hölder continuous coefficients. In particular, we show the following: If $$\mu $$ μ is an n-Ahlfors-David-regular measure on $$\mathbb {R}^{n+1}$$ R n + 1 with compact support, then $$T_\mu $$ T μ is bounded on $$L^2(\mu )$$ L 2 ( μ ) if and only if $$\mu $$ μ is uniformly n-rectifiable. Let $$E\subset \mathbb {R}^{n+1}$$ E ⊂ R n + 1 be compact and $${\mathcal {H}}^n(E)<\infty $$ H n ( E ) < ∞ . If $$T_{{\mathcal {H}}^n|_E}$$ T H n | E is bounded on $$L^2({\mathcal {H}}^n|_E)$$ L 2 ( H n | E ) , then E is n-rectifiable. If $$\mu $$ μ is a non-zero measure on $$\mathbb {R}^{n+1}$$ R n + 1 such that $$\limsup _{r\rightarrow 0}\tfrac{\mu (B(x,r))}{(2r)^n}$$ lim sup r → 0 μ ( B ( x , r ) ) ( 2 r ) n is positive and finite for $$\mu $$ μ -a.e. $$x\in \mathbb {R}^{n+1}$$ x ∈ R n + 1 and $$\liminf _{r\rightarrow 0}\tfrac{\mu (B(x,r))}{(2r)^n}$$ lim inf r → 0 μ ( B ( x , r ) ) ( 2 r ) n vanishes for $$\mu $$ μ -a.e. $$x\in \mathbb {R}^{n+1}$$ x ∈ R n + 1 , then the operator $$T_\mu $$ T μ is not bounded on $$L^2(\mu )$$ L 2 ( μ ) . Finally, we prove that if $$\mu $$ μ is a Radon measure on $${\mathbb {R}}^{n+1}$$ R n + 1 with compact support which satisfies a proper set of local conditions at the level of a ball $$B=B(x,r)\subset {\mathbb {R}}^{n+1}$$ B = B ( x , r ) ⊂ R n + 1 such that $$\mu (B)\approx r^n$$ μ ( B ) ≈ r n and r is small enough, then a significant portion of the support of $$\mu |_B$$ μ | B can be covered by a uniformly n-rectifiable set. These assumptions include a flatness condition, the $$L^2(\mu )$$ L 2 ( μ ) -boundedness of $$T_\mu $$ T μ on a large enough dilation of B, and the smallness of the mean oscillation of $$T_\mu $$ T μ at the level of B.

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Mutual Absolute Continuity of Interior and Exterior Harmonic Measure Implies Rectifiability

Cited by 25 publications

References 38 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

Tangent measures of elliptic measure and applications

L2-Boundedness of Gradients of Single Layer Potentials for Elliptic Operators with Coefficients of Dini Mean Oscillation-Type

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