# 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}, journal={Advances in Mathematics}, year={2019} }

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

#### 14 Citations

A two-phase free boundary problem for harmonic measure and uniform rectifiability

- Mathematics
- 2017

We assume that $\Omega_1, \Omega_2 \subset \mathbb{R}^{n+1}$, $n \geq 1$ are two disjoint domains whose complements satisfy the capacity density condition and the intersection of their boundaries $F$… Expand

Harmonic measure and quantitative connectivity: geometric characterization of the $$L^p$$-solvability of the Dirichlet problem

- Mathematics
- Inventiones mathematicae
- 2020

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,… Expand

Harmonic Measure and the Analyst's Traveling Salesman Theorem

- Mathematics
- 2019

We study how generalized Jones $\beta$-numbers relate to harmonic measure. Firstly, we generalize a result of Garnett, Mourgoglou and Tolsa by showing that domains in $\mathbb{R}^{d+1}$ whose… Expand

Rectifiability of Pointwise Doubling Measures in Hilbert Space

- Mathematics
- 2020

In geometric measure theory, there is interest in studying the interaction of measures with rectifiable sets. Here, we extend a theorem of Badger and Schul in Euclidean space to characterize… Expand

A geometric characterization of the weak-$A_\infty$ condition for harmonic measure

- Mathematics
- 2018

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,… Expand

Tangent measures of elliptic measure and applications

- Mathematics
- Analysis & PDE
- 2019

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… Expand

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

- Mathematics
- International Mathematics Research Notices
- 2019

We study the properties of harmonic measure in semi-uniform domains. Aikawa and Hirata showed in [ 5] that, for John domains satisfying the capacity density condition (CDC), the doubling property… Expand

Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries

- Mathematics
- Potential Analysis
- 2019

We show that given a domain $\Omega\subseteq \mathbb{R}^{d+1}$ with uniformly non-flat Ahlfors $s$-regular boundary and $s\geq d$, the dimension of its harmonic measure is strictly less than $s$.

Boundary rectifiability and elliptic operators with W1,1 coefficients

- Mathematics
- 2017

Abstract We consider second-order divergence form elliptic operators with W1,1{W^{1,1}} coefficients, in a uniform domain Ω with Ahlfors regular boundary. We show that the A∞{A_{\infty}} property of… Expand

Hausdorff dimension of caloric measure

- Mathematics
- 2021

We examine caloric measures ω on general domains in R = R × R (space × time) from the perspective of geometric measure theory. On one hand, we give a direct proof of a consequence of a theorem of… Expand

#### References

SHOWING 1-10 OF 58 REFERENCES

On the absolute continuity of p-harmonic measure and surface measure in Reifenberg flat domains

- Mathematics
- 2015

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

- Mathematics
- 2015

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 high… Expand

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 that… Expand

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 $H^n$ ($n$-dimensional… Expand

Rectifiability of harmonic measure

- Mathematics
- 2015

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 with… Expand

Approximate tangents, harmonic measure, and domains with rectifiable boundaries

- Mathematics
- 2016

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 \leq… Expand

Sets of Absolute Continuity for Harmonic Measure in NTA Domains

- Mathematics
- 2014

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 quantitatively… Expand

Tangents, rectifiability, and corkscrew domains

- Mathematics
- 2015

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 higher… Expand

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 harmonic… Expand