Wasserstein distance and the rectifiability of doubling measures: part II

Azzam, Jonas; David, Guy; Toro, Tatiana

doi:10.1007/s00209-016-1788-5

Cited by 7 publications

(12 citation statements)

References 8 publications

(22 reference statements)

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“…61) where α > 0 is the same very small constant as in the previous section, and α(Q) is defined by (6.10).Set G = B \B. We want to decompose G into stopping time regions with constant α.…”

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confidence: 99%

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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confidence: 99%

Wasserstein distance and the rectifiability of doubling measures: part I

2015

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“…Moreover, if either µ or ν is a doubling, one has α s,µ,ν (I) α µ,ν (I). These facts are contained in Proposition 5.4 (or see [2,Section 5]).…”

Section: Introductionmentioning

confidence: 95%

“…The difficulties arise from the non-stability of the numbers α µ,ν . See [2,Section 5], and in particular [2, Lemma 5.3], for related discussion.…”

Section: The Following Question Remains Openmentioning

confidence: 99%

“…Remark 6.8. Lemma 5.4 in [2] implies that 1/2 1/4 α µ,ν (B(0, t)) dt α s,µ,ν (B(0, 1)), whenever ν is doubling, and ν(B(0, 1/4)) > 0, µ(B(0, 1/4)) > 0. So, at the level of L 1averages over scales, the smooth and regular α-numbers are comparable.…”

Section: Parts (A) Of the Main Theoremsmentioning

confidence: 99%

“…Then W 1 (ν 1 , ν 2 ) = 0, but the alternative definition, sayW 1 , would giveW 1 (ν 1 , ν 2 ) = 1. The main reason for using W 1 instead ofW 1 in this paper is to comply with the definitions in [1,2].…”

Section: Introductionmentioning

confidence: 99%

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Absolute continuity and α-numbers on the real line

Orponen¹

2019

Analysis & PDE

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Let µ, ν be Radon measures on R, with µ non-atomic and ν doubling, and write µ = µa + µs for the Lebesgue decomposition of µ relative to ν. For an interval I ⊂ R, define αµ,ν (I) := W1(µI , νI ), the Wasserstein distance of normalised blow-ups of µ and ν restricted to I. Let Sν be the square function 2010 Mathematics Subject Classification. 42A99 (Primary). The research was supported by the grants 274512 and 309365 of the Finnish Academy. 18 TUOMAS ORPONEN Then, recall that µ(|ψ I |) ≤ µ(ϕ I ). Further, it follows from the doubling of ν that ν(ϕ J ) D,θ ν(ϕ I ). Finally, notice that ψ I = (ψ I • T −1 J ) • T J and ϕ I = (ϕ I • T −1 J ) • T J , where both ψ I • T −1 J and ϕ I • T −1 J are (|J|/|I|)-Lipschitz functions supported on T J (I) ⊂ [0, 1]. Consequently,

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

Tolsa

2015

Calc. Var.

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Abstract. In this paper it is shown that if µ is a finite Radon measure in R d which is n-rectifiable and 1 ≤ p ≤ 2, thenwhere, with the infimum taken over all the n-planes L ⊂ R d . The β n µ,p coefficients are the same as the ones considered by David and Semmes in the setting of the so called uniform n-rectifiability. An analogous necessary condition for n-rectifiability in terms of other coefficients involving some variant of the Wasserstein distance W 1 is also proved.

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Wasserstein distance and the rectifiability of doubling measures: part II

Cited by 7 publications

References 8 publications

Wasserstein distance and the rectifiability of doubling measures: part I

Wasserstein distance and the rectifiability of doubling measures: part I

Absolute continuity and α-numbers on the real line

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

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