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

@article{Azzam2019SemiUniformDA, title={Semi-Uniform Domains and the A∞ Property for Harmonic Measure}, author={Jonas Azzam}, journal={International Mathematics Research Notices}, year={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 for harmonic measure is equivalent to the domain being semi-uniform. Our 1st result removes the John condition by showing that any domain satisfying the CDC whose harmonic measure is doubling is semi-uniform. Next, we develop a substitute for some classical estimates on harmonic measure in…

## 14 Citations

Square function estimates, the BMO Dirichlet problem, and absolute continuity of harmonic measure on lower-dimensional sets

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Let $\mathbb{H}$ be the first Heisenberg group, and let $S \subset \mathbb{H}$ be an $\mathbb{H}$-regular surface. Can almost all of $S$ be covered by countably many Lipschitz images of subsets of…

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Let $\mu$ be a Radon measure on $\mathbb{R}^d$. We define and study conical energies $\mathcal{E}_{\mu,p}(x,V,\alpha)$, which quantify the portion of $\mu$ lying in the cone with vertex…

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In the present article, we purpose a method to deal with Dahlberg-Kenig-Pipher (DPK) operators in boundary value problems on the upper half plane. We give a nice subclass of the weak DKP operators…

## References

SHOWING 1-10 OF 76 REFERENCES

Characterization of a uniform domain by the boundary Harnack principle

- Mathematics
- 2004

Ever since the pioneering works of Carleson [8] and Hunt-Wheeden [10, 11] for Lipschitz domains, a large number of works have been devoted to the study of potential theory for nonsmooth domains, such…

Uniform rectifiability and harmonic measure I: Uniform rectifiability implies Poisson kernels in $L^p$

- Mathematics
- 2012

We present a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz. More precisely, we establish scale invariant absolute continuity of harmonic measure with respect…

BMO Solvability and Absolute Continuity of Harmonic Measure

- Mathematics
- 2016

We show that for a uniformly elliptic divergence form operator L, defined in an open set $$\Omega $$Ω with Ahlfors–David regular boundary, BMO solvability implies scale-invariant quantitative…

Harmonic measure and approximation of uniformly rectifiable sets

- Mathematics
- 2015

Let $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, be a uniformly rectifiable set of dimension $n$. We show $E$ that has big pieces of boundaries of a class of domains which satisfy a 2-sided corkscrew…

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 $F$…

Absolute continuity of harmonic measure for domains with lower regular boundaries

- MathematicsAdvances in Mathematics
- 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…

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…

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…

The weak-A∞ property of harmonic and p-harmonic measures implies uniform rectifiability

- Mathematics
- 2017

Let $E\subset \ree$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set$\Omega:= \ree\setminus E$, implies u…

Analysis of and on uniformly rectifiable sets

- Mathematics
- 1993

The notion of uniform rectifiability of sets (in a Euclidean space), which emerged only recently, can be viewed in several different ways. It can be viewed as a quantitative and scale-invariant…