Harmonic measure and approximation of uniformly rectifiable sets

Bortz, Simon; Hofmann, Steve

doi:10.4171/rmi/940

Cited by 10 publications

(10 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An alternate proof of the result of J. Azzam and R. Schul [AS] in the case d = n − 1, based on corona-type constructions, was given by the first and third author in [BH1]. While Theorem 1.1 applies in far more general settings beyond the setting of UR sets in Euclidean spaces, a consequence of Theorem 1.1, and the characterization of UR sets by coronizations with respect to Lipschitz graphs (see [DS1]), is that we here provide a 'corona analysis' type of proof of the result of J. Azzam and R. Schul [AS] for d < n. However, it should be noted that in their work [AS] J. Azzam and R. Schul also establish several other results beyond the fact that UR sets are BP 2 (LG).…”

Section: Introductionmentioning

confidence: 99%

Coronizations and big pieces in metric spaces

Bortz¹,

Hoffman²,

Hofmann³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

We prove that coronizations with respect to arbitrary d-regular sets (not necessarily graphs) imply big pieces squared of these (approximating) sets. This is known (and due to David and Semmes in the case of sufficiently large co-dimension, and to Azzam and Schul in general) in the (classical) setting of Euclidean spaces with Hausdorff measure of integer dimension, where the approximating sets are Lipschitz graphs. Our result is a far reaching generalization of these results and we prove that coronizations imply big pieces squared is a generic property. In particular, our result applies, when suitably interpreted, in metric spaces having a fixed positive (perhaps non-integer) dimension, equipped with a Borel regular measure and with arbitrary approximating sets. As a novel application we highlight how to utilize this general setting in the context of parabolic uniform rectifiability.

show abstract

Section: Introductionmentioning

confidence: 99%

Coronizations and big pieces in metric spaces

Bortz¹,

Hoffman²,

Hofmann³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…This corresponds to a quantitative version of the implication (c) implies (a) of our Theorem 1.2 in a setting without connectivity assumptions. The converse of this result, that is, that the complement of a Uniformly Rectifiable set has "interior big pieces of good harmonic measure estimates", has been recently proved by Bortz and the third author of this paper [9]. This can be seen as a quantitative version of (c) (or (e)) implies (a) in Theorem 1.2.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 68%

Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries

Akman¹,

Badger²,

Hofmann³

et al. 2017

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. Let Ω ⊂ R n+1 , n ≥ 2, be 1-sided NTA domain also known as uniform domain), i.e., a domain which satisfies interior Corkscrew and Harnack Chain conditions, and assume that ∂Ω is n-dimensional Ahlfors-David regular. We characterize the rectifiability of ∂Ω in terms of the absolute continuity of surface measure with respect to harmonic measure. We also show that these are equivalent to the fact that ∂Ω can be covered H n -a.e. by a countable union of portions of boundaries of bounded chord-arc subdomains of Ω and to the fact that ∂Ω possesses exterior corkscrew points in a qualitative way H n -a.e. Our methods apply to harmonic measure and also to elliptic measures associated with real symmetric second order divergence form elliptic operators with locally Lipschitz coefficients whose derivatives satisfy a natural qualitative Carleson condition.

show abstract

“…We then use Theorem II and the maximum principle to prove A ∞ for our original measure. The work of Bortz and Hofmann [BH17] comes close to what we need by building a union of (possibly disjoint) chord-arc domains, and in essence what we do is show that these chord arc domains can be connected into one single CAD, although our construction in the end is quite different and uses some additional techniques in order to prove semi-uniformity.…”

mentioning

confidence: 67%

“…The rest of this section is dedicated to the proof of 6.4. The arguments below take inspiration not just from [BH17] and [HM14], but also from [DS91, Chapter 16]. We understand that there are many constructions of chord-arc subdomains for uniform domains and NTA domains, and though we are working in semi-uniform domains, the details are similar and in some cases identical to steps in these other cases (see for example [HM14]).…”

Section: Chord-arc Subdomains Of Semi-uniform Domains With Ur Boundarymentioning

confidence: 99%

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

Azzam

2019

International Mathematics Research Notices

View full text Add to dashboard Cite

We study the properties of harmonic measure in semi-uniform domains. Aikawa and Hirata showed in [AH08] 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 first result removes the John condition by showing that any domain satisfying the CDC whose harmonic measure is doubling is semiuniform. Next, we develop a substitute for some classical estimates on harmonic measure in nontangentially accessible domains that works in semi-uniform domains.We also show that semi-uniform domains with uniformly rectifiable boundary have big pieces of chord-arc subdomains. We cannot hope for big pieces of Lipschitz subdomains (as was shown for chord-arc domains by David and Jerison [DJ90]) due to an example of Hrycak, which we review in the appendix.Finally, we combine these tools to study the A ∞ -property of harmonic measure. For a domain with Ahlfors-David regular boundary, it was shown by Hofmann and Martell that the A ∞ property of harmonic measure implies uniform rectifiability of the boundary [HM15, HLMN17] . Since A ∞ -weights are doubling, this also implies the domain is semi-uniform. Our final result shows that these two properties, semi-uniformity and uniformly rectifiable boundary, also imply the A ∞ property for harmonic measure, thus classifying geometrically all domains for which this holds.

show abstract

Harmonic measure and approximation of uniformly rectifiable sets

Cited by 10 publications

References 22 publications

Coronizations and big pieces in metric spaces

Coronizations and big pieces in metric spaces

Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries

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

Contact Info

Product

Resources

About