# 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} }

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…

## 16 Citations

### Higher dimensional Jordan curves.

- Mathematics
- 2020

We address the question of what is the correct higher dimensional analogue of Jordan curves from the point of view of quantitative rectifiability. More precisely, we show that 'topologically stable'…

### 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…

### A $d$-dimensional Analyst's Travelling Salesman Theorem for general sets in $\mathbb{R}^n$.

- Mathematics
- 2020

In his 1990 paper, Jones proved the following: given $E \subseteq \mathbb{R}^2$, there exists a curve $\Gamma$ such that $E \subseteq \Gamma$ and \[ \mathscr{H}^1(\Gamma) \sim \text{diam}\, E +…

### An Analyst's Travelling Salesman Theorem for general sets in $\mathbb{R}^n$

- Mathematics
- 2020

In his 1990 paper, Jones proved the following: given $E \subseteq \mathbb{R}^2$, there exists a curve $\Gamma$ such that $E \subseteq \Gamma$ and \[ \mathscr{H}^1(\Gamma) \sim \text{diam}\, E +…

### Almost sharp descriptions of traces of Sobolev $W_{p}^{1}(\mathbb{R}^{n})$-spaces to arbitrary compact subsets of $\mathbb{R}^{n}$. The case $p \in (1,n]$

- Mathematics
- 2021

Let S ⊂ R be an arbitrary nonempty compact set such that the d-Hausdorff content H ∞(S) > 0 for some d ∈ (0, n]. For each p ∈ (max{1, n−d}, n] an almost sharp intrinsic description of the trace space…

### A $d$-dimensional Analyst's Travelling Salesman Theorem for subsets of Hilbert space

- Mathematics
- 2021

We are interested in quantitative rectifiability results for subsets of infinite dimensional Hilbert space H. We prove a version of Azzam and Schul’s d-dimensional Analyst’s Travelling Salesman…

### Some porosity-type properties of sets related to the $d$-Hausdorff content

- Mathematics
- 2021

Let S ⊂ R n be a nonempty set. Given d ∈ [0 , n ) and a cube Q ⊂ R n with l = l ( Q ) ∈ (0 , 1] , we show that if the d -Hausdorﬀ content H d ∞ ( Q ∩ S ) < λl d for some λ ∈ (0 , 1) , then the set Q…

### Sets with topology, the Analyst's TST, and applications

- Mathematics
- 2019

This paper was motivated by three questions. First: in a recent paper, Azzam and Schul asked what sort of sets could play the role of curves in the context of the higher dimensional analyst's…

### Subsets of rectifiable curves in Banach spaces I: sharp exponents in traveling salesman 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 and…

### Analytic capacity and dimension of sets with plenty of big projections

- Mathematics
- 2022

. Our main result marks progress on an old conjecture of Vitushkin. We show that a compact set in the plane with plenty of big projections (PBP) has positive analytic capacity, along with a…

## References

SHOWING 1-10 OF 31 REFERENCES

### 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…

### 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

- Mathematics
- 2006

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

- Mathematics
- 1990

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

- MathematicsRevista 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…