# Absolute continuity of harmonic measure for domains with lower regular boundaries

@article{Akman2019AbsoluteCO,
title={Absolute continuity of harmonic measure for domains with lower regular boundaries},
author={Murat Akman and Jonas Azzam and Mihalis Mourgoglou},
year={2019}
}
• Published 24 May 2016
• Mathematics
We study absolute continuity of harmonic measure with respect to surface measure on domains $\Omega$ that have large complements. We show that if $\Gamma\subset \mathbb{R}^{d+1}$ is $d$-Ahlfors regular and splits $\mathbb{R}^{d+1}$ into two NTA domains then $\omega_{\Omega}\ll \mathscr{H}^{d}$ on $\Gamma\cap \partial\Omega$. This result is a natural generalisation of a result of Wu in [Wu86]. We also prove that almost every point in $\Gamma\cap\partial\Omega$ is a cone point if $\Gamma$ is a… Expand

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#### References

SHOWING 1-10 OF 58 REFERENCES
On the absolute continuity of p-harmonic measure and surface measure in Reifenberg flat domains
In this paper, we study the set of absolute continuity of p-harmonic measure, $\mu$, and $(n-1)-$dimensional Hausdorff measure, $\mathcal{H}^{n-1}$, on locally flat domains in $\mathbb{R}^{n}$,Expand
Uniform domains with rectifiable boundaries and harmonic measure
We assume that $\Omega \subset \mathbb{R}^{d+1}$, $d \geq 2$, is a uniform domain with lower $d$-Ahlfors-David regular and $d$-rectifiable boundary. We show that if $\mathcal{H}^d|_{\partial \Omega}$Expand
Singular sets for harmonic measure on locally flat domains with locally finite surface measure
• Mathematics
• 2015
A theorem of David and Jerison asserts that harmonic measure is absolutely continuous with respect to surface measure in NTA domains with Ahlfors regular boundaries. We prove that this fails in highExpand
Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries
• Mathematics
• 2015
Let $\Omega \subset \mathbb{R}^{n+1}$, $n\geq 2$, be 1-sided NTA domain (aka uniform domain), i.e. a domain which satisfies interior Corkscrew and Harnack Chain conditions, and assume thatExpand
Rectifiability, interior approximation and harmonic measure
• Mathematics
• Arkiv för Matematik
• 2019
We prove a structure theorem for any $n$-rectifiable set $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, satisfying a weak version of the lower ADR condition, and having locally finite $H^n$ ($n$-dimensionalExpand
Rectifiability of harmonic measure
In the present paper we prove that for any open connected set $${\Omega\subset\mathbb{R}^{n+1}}$$Ω⊂Rn+1, $${n\geq 1}$$n≥1, and any $${E\subset \partial \Omega}$$E⊂∂Ω withExpand
Approximate tangents, harmonic measure, and domains with rectifiable boundaries
We show that if $E \subset \mathbb R^d$, $d \geq 2$ is a closed and weakly lower Ahlfors-David $m$--regular set, then the set of points where there exists an approximate tangent $m$-plane, $m \leqExpand Sets of Absolute Continuity for Harmonic Measure in NTA Domains We show that if Ω is an NTA domain with harmonic measure ω and E⊆∂Ω is contained in an Ahlfors regular set, then ω|E≪ℋd|E$\omega |_{E}\ll \mathcal {H}^{d}|_{E}$. Moreover, this holds quantitativelyExpand Tangents, rectifiability, and corkscrew domains In a recent paper, Cs\"ornyei and Wilson prove that curves in Euclidean space of$\sigma$-finite length have tangents on a set of positive$\mathscr{H}^{1}\$-measure. They also show that a higherExpand
Boundary structure and size in terms of interior and exterior harmonic measures in higher dimensions
• Mathematics
• 2007
In this work we introduce the use of powerful tools from geometric measure theory (GMT) to study problems related to the size and structure of sets of mutual absolute continuity for the harmonicExpand