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

@article{Azzam2015CharacterizationON, title={Characterization of n-rectifiability in terms of Jones’ square function: Part II}, author={Jonas Azzam and Xavier Tolsa}, journal={Geometric and Functional Analysis}, year={2015}, volume={25}, pages={1371-1412} }

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 called Jones’ square function is finite $${\mu}$$μ-almost everywhere. The converse of this result is proven in a companion paper by the second author, and hence these two results give a classification of all n-rectifiable measures which are absolutely continuous with respect to $${\mathcal{H}^{n}}$$Hn…

## 80 Citations

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

- Mathematics
- 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…

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…

Characterization of rectifiable measures in terms of 𝛼-numbers

- Mathematics
- 2018

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…

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}…

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…

Geometric criteria for $C^{1,\alpha}$ rectifiability

- Mathematics
- 2019

We prove criteria for $\mathcal{H}^k$-rectifiability of subsets of $\mathbb{R}^n$ with $C^{1,\alpha}$ maps, $0<\alpha\leq 1$, in terms of suitable approximate tangent paraboloids. We also provide a…

Sufficient conditions for $C^{1,\alpha}$ parametrization and rectifiability

- Mathematics
- 2017

We say a measure is $C^{1,\alpha}$ $d$-rectifiable if there is a countable union of $C^{1,\alpha}$ $d$-surfaces whose complement has measure zero. We provide sufficient conditions for a Radon measure…

Geometric conditions for the $L^2$-boundedness of singular integral operators with odd kernels with respect to measures with polynomial growth in $\mathbb{R}^d$

- Mathematics
- 2015

Let $\mu$ be a finite Radon measure in $\mathbb{R}^d$ with polynomial growth of degree $n$, although not necessarily $n$-AD-regular. We prove that under some geometric conditions on $\mu$ that are…

Wild examples of rectifiable sets

- Mathematics
- 2019

We study the geometry of sets based on the behavior of the Jones function, $J_{E}(x) = \int_{0}^{1} \beta_{E;2}^{1}(x,r)^{2} \frac{dr}{r}$. We construct two examples of countably $1$-rectifiable sets…

Menger curvatures and $$\varvec{C^{1,\alpha }}$$ rectifiability of measures

- Mathematics
- 2019

We further develop the relationship between $$\beta $$-numbers and discrete curvatures to provide a new proof that under weak density assumptions, finiteness of the pointwise discrete curvature…

## References

SHOWING 1-10 OF 51 REFERENCES

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…

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…

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…

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 $E$…

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{…

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…

Wasserstein distance and the rectifiability of doubling measures: part II

- Mathematics
- 2014

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…

Principal values for Riesz transforms and rectifiability

- Mathematics
- 2007

Let $E\subset R^d$ with $H^n(E) \ve} \frac{x-y}{|x-y|^{n+1}} dH^n(y)$$ exists H^n-almost everywhere in E. To prove this result we obtain precise estimates from above and from below for the $L^2$ norm…

Menger curvature and rectifiability

- Mathematics
- 1999

where 'HI is the 1-dimensional Hausdorff measure in Rn, c(x, y, z) is the inverse of the radius of the circumcircle of the triangle (x, y, z), that is, following the terminology of [6], the Menger…

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…