# Quantitative comparisons of multiscale geometric properties

@article{Azzam2021QuantitativeCO,
title={Quantitative comparisons of multiscale geometric properties},
author={Jonas Azzam and Michele Villa},
journal={Analysis \& PDE},
year={2021}
}
• Published 30 April 2019
• Mathematics
• Analysis & PDE
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 and Semmes showed that, given an Ahlfors $d$-regular set $E$, if we consider the set $\mathscr{B}$ of surface cubes (in the sense of Christ and David) near which $E$ does not look approximately like a union of planes, then $E$ is UR if and only if $\mathscr{B}$ satisfies a Carleson packing condition…
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## References

SHOWING 1-10 OF 31 REFERENCES

### Harmonic Measure and the Analyst's Traveling Salesman Theorem

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

### 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

### Subsets of rectifiable curves in Hilbert space-the analyst’s TSP

We study one dimensional sets (Hausdorff dimension) lying in a Hilbert space. The aim is to classify subsets of Hilbert spaces that are contained in a connected set of finite Hausdorff length. We do

### Rectifiable sets and the Traveling Salesman Problem

Let K c C be a bounded set. In this paper we shall give a simple necessary and sufficient condit ion for K to lie in a rectifiable curve. We say that a set is a rectifiable curve if it is the image

### 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 quantities

### A sharp necessary condition for rectifiable curves in metric spaces

• Mathematics
Revista Matemática Iberoamericana
• 2020
In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane, using a multiscale sum of what is now known as Jones $\beta$-numbers, numbers measuring flatness in a

### 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

### On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1

• Mathematics
• 2012
We prove that if μ is a d-dimensional Ahlfors-David regular measure in $${\mathbb{R}^{d+1}}$$Rd+1 , then the boundedness of the d-dimensional Riesz transform in L2(μ) implies that the non-BAUP

### Reifenberg Parameterizations for Sets with Holes

• Mathematics
• 2009
We extend the proof of Reifenberg's Topological Disk Theorem to allow the case of sets with holes, and give sufficient conditions on a set $E$ for the existence of a bi-Lipschitz parameterization of