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

Azzam, Jonas; Tolsa, Xavier

doi:10.1007/s00039-015-0334-7

Cited by 69 publications

(107 citation statements)

References 24 publications

(25 reference statements)

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“…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%

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Wasserstein distance and the rectifiability of doubling measures: part I

2015

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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.

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Section: Statement Of Resultsmentioning

confidence: 99%

“…2 Let μ and ν be measures on R n , whose restrictions to B := B(0, 1) are probability measures. We set W 1 (μ, ν) := sup ψ ˆψ dμ −ˆψdν , (1.…”

Section: Statement Of Resultsmentioning

confidence: 99%

Wasserstein distance and the rectifiability of doubling measures: part I

2015

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“…see [13] and [14]). Much work has been done in connecting beta numbers and recti ability (highlights include [50], [32], [33], [24], [34], [15], [5], [2], [6], [46]), but we single out two recent papers, [58] and [4], and state one theorem, which is a combination of their main results. …”

Section: P Jones Beta Numbers and Recti Abilitymentioning

confidence: 99%

Multiscale Analysis of 1-rectifiable Measures II: Characterizations

Badger

Schul

2017

Analysis and Geometry in Metric Spaces

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A measure is 1-recti able if there is a countable union of nite length curves whose complement has zero measure. We characterize 1-recti able Radon measures µ in n-dimensional Euclidean space for all n ≥ in terms of positivity of the lower density and niteness of a geometric square function, which loosely speaking, records in an L gauge the extent to which µ admits approximate tangent lines, or has rapidly growing density ratios, along its support. In contrast with the classical theorems of Besicovitch, Morse and Randolph, and Moore, we do not assume an a priori relationship between µ and 1-dimensional Hausdor measure H . We also characterize purely 1-unrecti able Radon measures, i.e. locally nite measures that give measure zero to every nite length curve. Characterizations of this form were originally conjectured to exist by P. Jones. Along the way, we develop an L variant of P. Jones' traveling salesman construction, which is of independent interest.

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“…In Section 8 we prove Theorem 1.6 by combining the estimates from the previous section with a characterization of rectifiable measures by Azzam and Tolsa [6].…”

Section: Plan Of the Articlementioning

confidence: 95%

“…We refer to the survey article [38] for related problems and various generalizations. Also, see [6,19,23] for more on the Jones's -numbers in the context of rectifiability and bi-Lipschitz parametrizations of sets in the euclidean space.…”

Section: Mean Flatnessmentioning

confidence: 99%

On the Singular Set of Free Interface in an Optimal Partition Problem

Alper

2019

Comm Pure Appl Math

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We study the singular set of free interface in an optimal partition problem for the Dirichlet eigenvalues. We prove that its upper (n − 2)‐dimensional Minkowski content, and consequently its (n − 2)‐dimensional Hausdorff measure, are locally finite. We also show that the singular set is countably (n − 2)‐rectifiable; namely, it can be covered by countably many C1‐manifolds of dimension (n − 2), up to a set of (n − 2)‐dimensional Hausdorff measure zero. Our results hold for optimal partitions on Riemannian manifolds and harmonic maps into homogeneous trees as well. © 2019 Wiley Periodicals, Inc.

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

Cited by 69 publications

References 24 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 II: Characterizations

On the Singular Set of Free Interface in an Optimal Partition Problem

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