On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1

Nazarov, Fëdor; Volberg, Alexander; Tolsa, Xavier

doi:10.1007/s11511-014-0120-7

Cited by 115 publications

(120 citation statements)

References 13 publications

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“…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%

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%

Wasserstein distance and the rectifiability of doubling measures: part I

2015

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“…The latter allows us to invoke the recent resolution of the David-Semmes conjecture in co-dimension 1 ( [NToV1], [NToV2], see also [HMM] in the context of uniform domains), establishing that boundedness of the Riesz transforms implies rectifiability. We note that in Theorem 1.1 connectivity is just a cosmetic assumption needed to make sense of harmonic measure at a given pole.…”

Section: (B)mentioning

confidence: 89%

“…Then from the results of Nazarov, Tolsa and Volberg in [NToV1] and [NToV2], it follows that ω p | G 0 is n-rectifiable. This suffices to prove the full n-rectifiability of ω p | E .…”

Section: Lemma 37 Let R ∈ D and Let Q ⊂ R Be A Cube Such That All Tmentioning

confidence: 89%

Rectifiability of harmonic measure

et al. 2016

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“…At the geometrical level, the Nazarov, Tolsa, Volberg recent main result in [44] mentioned earlier may be rephrased, in light of (1.7), as follows: under the background assumption that Σ is an Ahlfors regular subset of R n , one has Σ uniformly rectifiable set ⇐⇒ R j (1) ∈ BMO(Σ) for each j ∈ {1, . .…”

Section: (12)mentioning

confidence: 99%

Characterizing regularity of domains via the Riesz transforms on their boundaries

Mitrea

Verdera

2016

Anal. PDE

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Abstract. Under mild geometric measure theoretic assumptions on an open subset Ω of R n , we show that the Riesz transforms on its boundary are continuous mappings on the Hölder space C α (∂Ω) if and only if Ω is a Lyapunov domain of order α (i.e., a domain of class C 1+α ). In the category of Lyapunov domains we also establish the boundedness on Hölder spaces of singular integral operators with kernels of the form P (x − y)/|x − y| n−1+l , where P is any odd homogeneous polynomial of degree l in R n . This family of singular integral operators, which may be thought of as generalized Riesz transforms, includes the boundary layer potentials associated with basic PDE's of mathematical physics, such as the Laplacian, the Lamé system, and the Stokes system. We also consider the limiting case α = 0 (with VMO(∂Ω) as the natural replacement of C α (∂Ω)), and discuss an extension to the scale of Besov spaces.

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On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1

Cited by 115 publications

References 13 publications

Wasserstein distance and the rectifiability of doubling measures: part I

Wasserstein distance and the rectifiability of doubling measures: part I

Rectifiability of harmonic measure

Characterizing regularity of domains via the Riesz transforms on their boundaries

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