Corpus ID: 162168448

# Harmonic Measure and the Analyst's Traveling Salesman Theorem

@article{Azzam2019HarmonicMA,
title={Harmonic Measure and the Analyst's Traveling Salesman Theorem},
author={Jonas Azzam},
journal={arXiv: Classical Analysis and ODEs},
year={2019}
}
• Jonas Azzam
• Published 22 May 2019
• Mathematics
• arXiv: Classical Analysis and ODEs
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 boundaries are lower $d$-content regular admit Corona decompositions for harmonic measure if and only if the square sum $\beta_{\partial\Omega}$ of the generalized Jones $\beta$-numbers is finite. Secondly, for semi-uniform domains with Ahlfors regular boundaries, it is known that uniform rectifiability… Expand
4 Citations

#### Figures from this paper

Quantitative comparisons of multiscale geometric properties
• Mathematics
• Analysis & PDE
• 2021
We generalize some characterizations of uniformly rectifiable (UR) sets to sets whose Hausdorff content is lower regular (and in particular, do not need to be Ahlfors regular). For example, David andExpand
Subsets of rectifiable curves in Banach spaces: sharp exponents in Schul-type theorems
• Mathematics
• 2020
The Analyst's Traveling Salesman Problem is to find a characterization of subsets of rectifiable curves in a metric space. This problem was introduced and solved in the plane by Jones in 1990 andExpand
Cone and paraboloid points of arbitrary subsets of Euclidean space
• Mathematics
• 2021
In this paper we characterise cone points of arbitrary subsets of Euclidean space. Given E ⊂ R, x ∈ E is a cone point of E if and only if
SUBSETS OF RECTIFIABLE CURVES IN BANACH SPACES: SHARP EXPONENTS IN TRAVELING SALESMAN THEOREMS
The Analyst’s Traveling Salesman Problem is to find a characterization of subsets of rectifiable curves in a metric space. This problem was introduced and solved in the plane by Jones in 1990 andExpand

#### References

SHOWING 1-10 OF 67 REFERENCES
Uniform rectifiability from Carleson measure estimates and ε-approximability of bounded harmonic functions
• Mathematics
• Duke Mathematical Journal
• 2018
Let $\Omega\subset\mathbb R^{n+1}$, $n\geq1$, be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that $\partial\Omega$ is uniformly $n$-rectifiable if every boundedExpand
Absolute continuity of harmonic measure for domains with lower regular boundaries
• 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 $d$-AhlforsExpand
Characterization of n-rectifiability in terms of Jones’ square function: Part II
• Mathematics
• 2015
We show that a Radon measure $${\mu}$$μ in $${\mathbb{R}^d}$$Rd which is absolutely continuous with respect to the n-dimensional Hausdorff measure $${\mathcal{H}^n}$$Hn is n-rectifiable if the soExpand
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
An analyst’s traveling salesman theorem for sets of dimension larger than one
• Mathematics
• 2016
In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of $$\beta$$β-numbers. These $$\beta$$β-numbers are geometric quantitiesExpand
Quantitative comparisons of multiscale geometric properties
• Mathematics
• Analysis & PDE
• 2021
We generalize some characterizations of uniformly rectifiable (UR) sets to sets whose Hausdorff content is lower regular (and in particular, do not need to be Ahlfors regular). For example, David andExpand
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
Semi-Uniform Domains and the A∞ Property for Harmonic Measure
• Jonas Azzam
• 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 propertyExpand
Uniform Rectifiability and harmonic measure IV: Ahlfors regularity plus Poisson kernels in $L^p$ implies uniform rectifiability
• Mathematics
• 2015
Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $\Omega:=Expand Tangent measures and absolute continuity of harmonic measure • Mathematics • 2015 We show that for uniform domains$\Omega\subseteq \mathbb{R}^{d+1}\$ whose boundaries satisfy a certain nondegeneracy condition that harmonic measure cannot be mutually absolutely continuous withExpand