# Quantitative comparisons of multiscale geometric properties

@article{Azzam2019QuantitativeCO, title={Quantitative comparisons of multiscale geometric properties}, author={Jonas Azzam and Michele Villa}, journal={Analysis \& PDE}, year={2019} }

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

### The weak lower density condition and uniform rectifiability

- Mathematics
- 2020

We show that an Ahlfors $d$-regular set $E$ in $\mathbb{R}^{n}$ is uniformly rectifiable if the set of pairs $(x,r)\in E\times (0,\infty)$ for which there exists $y \in B(x,r)$ and $0 0$.
To prove…

### 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…

### 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 and…

### 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…

### A d-dimensional analyst’s travelling salesman theorem for subsets of Hilbert space

- MathematicsMathematische Annalen
- 2022

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…

### 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…

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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…

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