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… 
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References

SHOWING 1-10 OF 36 REFERENCES
Multiscale analysis of 1-rectifiable measures: necessary conditions
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 $$\mu
Rectifiable measures, square functions involving densities, and the Cauchy transform
This paper is devoted to the proof of two related results. The first one asserts that if $\mu$ is a Radon measure in $\mathbb R^d$ satisfying $$\limsup_{r\to 0} \frac{\mu(B(x,r))}{r}>0\quad \text{
Rectifiability via a square function and Preiss' theorem
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 $E$
Characterization of n-rectifiability in terms of Jones’ square function: Part II
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
Mass Transport and Uniform Rectifiability
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
A characterization of $1$-rectifiable doubling measures with connected supports
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
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 $$\begin{aligned} \displaystyle \int _0^\infty \beta
Wasserstein distance and the rectifiability of doubling measures: part I
Let $$\mu $$μ be a doubling measure in $${\mathbb {R}}^n$$Rn. We investigate quantitative relations between the rectifiability of $$\mu $$μ and its distance to flat measures. More precisely, for
Quantitative conditions of rectifiability for varifolds
Our purpose is to state quantitative conditions ensuring the rectifiability of a $d$--varifold $V$ obtained as the limit of a sequence of $d$--varifolds $(V_i)_i$ which need not to be rectifiable.
Wasserstein distance and the rectifiability of doubling measures: part II
We study the structure of the support of a doubling measure by analyzing its self-similarity properties, which we estimate using a variant of the $$L^1$$L1 Wasserstein distance. We show that a
...
1
2
3
4
...