# Two sufficient conditions for rectifiable measures

@article{Badger2014TwoSC, title={Two sufficient conditions for rectifiable measures}, author={Matthew Badger and Raanan Schul}, journal={arXiv: Classical Analysis and ODEs}, year={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 a prior result of Pajot, is a sufficient test in terms of $L^p$ affine approximability for a locally finite Borel measure $\mu$ on $\mathbb{R}^n$ satisfying the global regularity hypothesis $$\limsup_{r\downarrow 0} \mu(B(x,r))/r^m <\infty\quad \text{at $\mu$-a.e. $x\in\mathbb{R}^n$}$$ to be $m…

## 29 Citations

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

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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…

Characterization of n-rectifiability in terms of Jones’ square function: Part II

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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…

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

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- 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…

Characterization of n-rectifiability in terms of Jones’ square function: part I

- Mathematics
- 2015

In this paper it is shown that if $$\mu $$μ is a finite Radon measure in $${\mathbb R}^d$$Rd which is n-rectifiable and $$1\le p\le 2$$1≤p≤2, then $$\begin{aligned} \displaystyle \int _0^\infty \beta…

Cones, rectifiability, and singular integral operators

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- 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…

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}…

$\Omega$-symmetric measures and related singular integrals

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- 2019

Let $\mathbb{S} \subset \mathbb{C}$ be the circle in the plane, and let $\Omega: \mathbb{S} \to \mathbb{S}$ be an odd bi-Lipschitz map with constant $1+\delta_\Omega$, where $\delta_\Omega>0$ is…

Necessary Condition for Rectifiability Involving Wasserstein Distance W2

- Mathematics
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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…

Characterization of rectifiable measures in terms of 𝛼-numbers

- Mathematics
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We characterize Radon measures $\mu$ in $\mathbb{R}^{n}$ that are $d$-rectifiable in the sense that their supports are covered up to $\mu$-measure zero by countably many $d$-dimensional Lipschitz…

Geometry of Measures in Real Dimensions via Hölder Parameterizations

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We investigate the influence that s-dimensional lower and upper Hausdorff densities have on the geometry of a Radon measure in $$\mathbb {R}^n$$Rn when s is a real number between 0 and n. This topic…

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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…

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

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- 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…

Characterization of n-rectifiability in terms of Jones’ square function: part I

- Mathematics
- 2015

In this paper it is shown that if $$\mu $$μ is a finite Radon measure in $${\mathbb R}^d$$Rd which is n-rectifiable and $$1\le p\le 2$$1≤p≤2, then $$\begin{aligned} \displaystyle \int _0^\infty \beta…

Wasserstein distance and the rectifiability of doubling measures: part I

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