Mass Transport and Uniform Rectifiability

Tolsa, Xavier

doi:10.1007/s00039-012-0160-0

Cited by 18 publications

(20 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…More precisely we explore quantitative conditions which imply that a doubling measure in Euclidean space is rectifiable. Recently this question has been addressed by several authors in the context of Ahlfors regular measures (1.5); see [5,27,32,33] in connection with the L 2 -boundedness of Riesz transforms (also see [7,18,24,25], and earlier papers for singular integrals in noninteger dimensions), and a long line of papers between [29] and [17] in connection with harmonic measure. The reader may also be interested in [22] (for rectifiability with even less structure) and the more recent preprints [2,[34][35][36].…”

Section: Statement Of Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Wasserstein distance and the rectifiability of doubling measures: part I

2015

View full text Add to dashboard Cite

Let μ be a doubling measure in R n . We investigate quantitative relations between the rectifiability of μ and its distance to flat measures. More precisely, for x in the support of μ and r > 0, we introduce a number α(x, r ) ∈ (0, 1] that measures, in terms of a variant of the L 1 -Wasserstein distance, the minimal distance between the restriction of μ to B(x, r ) and a multiple of the Lebesgue measure on an affine subspace that meets B(x, r/2). We show that the set of points of wheré 1 0 α(x, r ) dr r < ∞ can be decomposed into rectifiable pieces of various dimensions. We obtain additional control on the pieces and the size of μ when we assume that some Carleson measure estimates hold. Résumé en FrançaisSoit μ une mesure doublante dans R n . On étudie des relations quantifiées entre la rectifiabilité de μ et la distance entre μ et les mesures plates. Plus précisément, on utilise une variante de la L 1 -distance de Wasserstein pour définir, pour x dans le support de μ et r > 0, un nombre α(x, r ) qui mesure la distance minimale entre la restriction de μ à B(x, r ) et une mesure de Lebesgue sur un sousespace affine passant par B(x, r/2). On décompose l'ensemble des points x ∈ tels que´1 0 α(x, r ) dr r < ∞ en parties rectifiables de dimensions diverses, et on obtient un meilleur contrôle de ces parties et de la taille de μ quand les α(x, r ) vérifient certaines conditions de Carleson.

show abstract

Section: Statement Of Resultsmentioning

confidence: 99%

“…We could also have used a slightly smoother version of W 1 ; see Section 5 of [1]. See [38] for additional information about the Wasserstein distance, and [33] for its relationship to W 1 and uniform rectifiability.…”

Section: Statement Of Resultsmentioning

confidence: 99%

Wasserstein distance and the rectifiability of doubling measures: part I

2015

View full text Add to dashboard Cite

show abstract

“…To end the paper, in §4, we discuss some connections between Theorem A and Corollary B, and prior work of Léger [Lég99] (Menger curvature), Lerman [Ler03] (curve learning) and Tolsa [Tol12] (mass transport).…”

Section: The Ordinary Lmentioning

confidence: 99%

“…Finally, we wish to mention a recent paper by Tolsa [Tol12], which introduced the use of tools from the theory of mass transportation to the theory of quantitative rectifiability. In particular, Tolsa established a new characterization of uniformly rectifiable measures, expressed in terms of the L 2 Wasserstein distance W 2 (·, ·) between probability measures.…”

Section: Related Workmentioning

confidence: 99%

Multiscale analysis of 1-rectifiable measures: necessary conditions

Badger

Schul

2014

Math. Ann.

View full text Add to dashboard Cite

Abstract. We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in R n , n ≥ 2. To each locally finite Borel measure µ, we associate a function J 2 (µ, x) which uses a weighted sum to record how closely the mass of µ is concentrated near a line in the triples of dyadic cubes containing x. We show that J 2 (µ, ·) < ∞ µ-a.e. is a necessary condition for µ to give full mass to a countable family of rectifiable curves. This confirms a conjecture of Peter Jones from 2000. A novelty of this result is that no assumption is made on the upper Hausdorff density of the measure.Thus we are able to analyze general 1-rectifiable measures, including measures which are singular with respect to 1-dimensional Hausdorff measure.

show abstract

“…This question, which was my initial motivation for the results of § 1, was asked to me by Xavier Tolsa, who needed such a result for his paper [Tolsa, 2012] on characterizing uniform rectifiability in terms of mass transport. Actually Xavier managed to devise a proof of his own [Tolsa, 2012, Theorem 1.1], but it was quite long (about thirty pages) and involved arguments of multi-scale analysis.…”

Section: Introductionmentioning

confidence: 99%

Comparison between W₂ distance and Ḣ⁻¹ norm, and Localization of Wasserstein distance

Peyre

2018

ESAIM: COCV

View full text Add to dashboard Cite

It is well known that the quadratic Wasserstein distance W 2 (·, ·) is formally equivalent, for infinitesimally small perturbations, to some weighted H −1 homogeneous Sobolev norm. In this article I show that this equivalence can be integrated to get non-asymptotic comparison results between these distances. Then I give an application of these results to prove that the W 2 distance exhibits some localisation phenomenon: if µ and ν are measures on R n and ϕ : R n → R + is some bump function with compact support, then under mild hypotheses, you can bound above the Wasserstein distance between ϕ · µ and ϕ · ν by an explicit multiple of W 2 (µ, ν). ForewordThis article is divided into two sections, each of which having its own introduction.§ 1 deals with general results of comparison between Wasserstein distance and homogeneous Sobolev norm, while § 2 handles an application to localisation of W 2 distance.

show abstract

Mass Transport and Uniform Rectifiability

Cited by 18 publications

References 19 publications

Wasserstein distance and the rectifiability of doubling measures: part I

Wasserstein distance and the rectifiability of doubling measures: part I

Multiscale analysis of 1-rectifiable measures: necessary conditions

Comparison between W₂ distance and Ḣ⁻¹ norm, and Localization of Wasserstein distance

Contact Info

Product

Resources

About

Mass Transport and Uniform Rectifiability

Cited by 18 publications

References 19 publications

Wasserstein distance and the rectifiability of doubling measures: part I

Wasserstein distance and the rectifiability of doubling measures: part I

Multiscale analysis of 1-rectifiable measures: necessary conditions

Comparison between W2 distance and Ḣ−1 norm, and Localization of Wasserstein distance

Contact Info

Product

Resources

About

Comparison between W₂ distance and Ḣ⁻¹ norm, and Localization of Wasserstein distance