Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón–Zygmund Theory

Tolsa, Xavier

doi:10.1007/978-3-319-00596-6

Cited by 142 publications

(263 citation statements)

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“…A good way to access a more comprehensive survey about uniform recti ability, its connection to singular integrals, and its connection to analytic capacity, is to read David and Semmes [14], Pajot [51], and Tolsa [57]. We also make some remarks in [5, §4].…”

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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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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“…For the detailed proof in the case of the Cauchy transform, see [To2,Theorem 8.13]. The same arguments with very minor modifications work for the Riesz transform.…”

Section: Lemma 43 (Key Lemma)mentioning

confidence: 98%

Rectifiability of harmonic measure

et al. 2016

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“…We further apply the T b theorem for suppressed operators by Nazarov-TreilVolberg [31] (see also Corollary 5.33 in [33]) and it follows that R Φ,σ :…”

Section: Riesz Transform and Rectifiabilitymentioning

confidence: 99%