# Sufficient Condition for Rectifiability Involving Wasserstein Distance $$W_2$$

@article{Dbrowski2019SufficientCF, title={Sufficient Condition for Rectifiability Involving Wasserstein Distance \$\$W\_2\$\$}, author={Damian Dąbrowski}, journal={arXiv: Classical Analysis and ODEs}, year={2019} }

A Radon measure $\mu$ is $n$-rectifiable if it is absolutely continuous with respect to $\mathcal{H}^n$ and $\mu$-almost all of $\text{supp}\,\mu$ can be covered by Lipschitz images of $\mathbb{R}^n$. In this paper we give two sufficient conditions for rectifiability, both in terms of square functions of flatness-quantifying coefficients. The first condition involves the so-called $\alpha$ and $\beta_2$ numbers. The second one involves $\alpha_2$ numbers -- coefficients quantifying flatness via…

## 6 Citations

Cones, rectifiability, and singular integral operators

- Mathematics
- 2020

Let $\mu$ be a Radon measure on $\mathbb{R}^d$. We define and study conical energies $\mathcal{E}_{\mu,p}(x,V,\alpha)$, which quantify the portion of $\mu$ lying in the cone with vertex…

Rectifiability of Pointwise Doubling Measures in Hilbert Space

- Mathematics
- 2020

In geometric measure theory, there is interest in studying the interaction of measures with rectifiable sets. Here, we extend a theorem of Badger and Schul in Euclidean space to characterize…

Necessary Condition for Rectifiability Involving Wasserstein Distance W2

- Mathematics
- 2020

A Radon measure μ is n-rectifiable if μ ≪ H n and μ-almost all of supp μ can be covered by Lipschitz images of R. In this paper we give a necessary condition for rectifiability in terms of the…

Birth and life of the $L^{2}$ boundedness of the Cauchy Integral on Lipschitz graphs

- Mathematics
- 2021

We review various motives for considering the problem of estimating the Cauchy Singular Integral on Lipschitz graphs in the L2 norm. We follow the thread that led to the solution and then describe a…

Cones, rectifiability, and singular integral operators

- MathematicsRevista Matemática Iberoamericana
- 2021

Let μ be a Radon measure on R. We define and study conical energies Eμ,p(x, V, α), which quantify the portion of μ lying in the cone with vertex x ∈ R , direction V ∈ G(d, d−n), and aperture α ∈ (0,…

Radon measures and Lipschitz graphs

- Mathematics
- 2020

For all $1\leq m\leq n-1$, we investigate the interaction of locally finite measures in $\mathbb{R}^n$ with the family of $m$-dimensional Lipschitz graphs. For instance, we characterize Radon…

## References

SHOWING 1-10 OF 50 REFERENCES

Necessary Condition for Rectifiability Involving Wasserstein Distance W2

- Mathematics
- 2020

A Radon measure $\mu$ is $n$-rectifiable if $\mu\ll\mathcal{H}^n$ and $\mu$-almost all of $\text{supp}\,\mu$ can be covered by Lipschitz images of $\mathbb{R}^n$. In this paper we give a necessary…

Two sufficient conditions for rectifiable measures

- Mathematics
- 2014

We identify two sufficient conditions for locally finite Borel measures on $\mathbb{R}^n$ to give full mass to a countable family of Lipschitz images of $\mathbb{R}^m$. The first condition, extending…

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 so…

Wasserstein distance and the rectifiability of doubling measures: part I

- Mathematics
- 2016

Let $$\mu $$μ be a doubling measure in $${\mathbb {R}}^n$$Rn. We investigate quantitative relations between the rectifiability of $$\mu $$μ and its distance to flat measures. More precisely, for…

Multiscale analysis of 1-rectifiable measures: necessary conditions

- Mathematics
- 2015

We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in $$\mathbb {R}^n$$Rn, $$n\ge 2$$n≥2. To each locally finite Borel measure $$\mu…

Mass Transport and Uniform Rectifiability

- Mathematics
- 2011

In this paper we characterize the so called uniformly rectifiable sets of David and Semmes in terms of the Wasserstein distance W2 from optimal mass transport. To obtain this result, we first prove a…

Rectifiable measures, square functions involving densities, and the Cauchy transform

- Mathematics
- 2014

This paper is devoted to the proof of two related results. The first one asserts that if $\mu$ is a Radon measure in $\mathbb R^d$ satisfying $$\limsup_{r\to 0} \frac{\mu(B(x,r))}{r}>0\quad \text{…

A characterization of $1$-rectifiable doubling measures with connected supports

- Mathematics
- 2015

Garnett, Killip, and Schul have exhibited a doubling measure $\mu$ with support equal to $\mathbb{R}^{d}$ which is $1$-rectifiable, meaning there are countably many curves $\Gamma_{i}$ of finite…

Rectifiability via a square function and Preiss' theorem

- Mathematics
- 2014

Let $E$ be a set in $\mathbb R^d$ with finite $n$-dimensional Hausdorff measure $H^n$ such that $\liminf_{r\to0}r^{-n} H^n(B(x,r)\cap E)>0$ for $H^n$-a.e. $x\in E$. In this paper it is shown that $E$…

Boundedness of the density normalised Jones' square function does not imply $1$-rectifiability

- Mathematics
- 2016

Recently, M. Badger and R. Schul proved that for a $1$-rectifiable Radon measure $\mu$, the density weighted Jones' square function $$ J_{1}(x) = \mathop{\sum_{Q \in \mathcal{D}}}_{\ell(Q) \leq 1}…