A T(b) theorem with remarks on analytic capacity and the Cauchy integral

Christ, Michael

doi:10.4064/cm-60-61-2-601-628

Cited by 658 publications

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“…boundedness of a singular integral that was proved originally by M. Christ in the setting of homogeneous spaces [CH2]. We state below a very particular version of Christ's result, which is adapted to the principal value Cauchy integral and to a measure µ supported on ∂E M .…”

Section: The Local T (B)-theorem the Local T (B)-theorem Is A Critermentioning

confidence: 93%

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The planar Cantor sets of zero analytic capacity and the local 𝑇(𝑏)-Theorem

Mateu¹,

Tolsa

Verdera³

2002

J. Amer. Math. Soc.

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Section: The Local T (B)-theorem the Local T (B)-theorem Is A Critermentioning

confidence: 93%

“…The main tool used in our proof of Theorem 2 is the local T (b)-Theorem of M. Christ [CH2], a particular version of which will be discussed and stated in Section 2. Section 2 also contains some basic facts on the Cauchy transform and the Plemelj formulae.…”

Section: Theorem 1 the Cantor Set E(λ) Has Zero Analytic Capacity Ifmentioning

confidence: 99%

The planar Cantor sets of zero analytic capacity and the local 𝑇(𝑏)-Theorem

Mateu¹,

Tolsa

Verdera³

2002

J. Amer. Math. Soc.

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“…We make use of this by employing ideas of Okikiolu, as well as ideas similar to ones of G. David and M. Christ (see e.g. [Chr90], and [Dav91] page 93 for a simple version). Note however that one main difference with the case of Christ and David is the lack of homogeneity assumption on Γ!…”

Section: Outlinementioning

confidence: 99%

“…[Chr90], and [Dav91] page 93 for a simple version). One should note that condition (2) replaces a doubling condition.…”

Section: } (The Intersection Is An Empty Set a Single Point Or Twomentioning

confidence: 99%

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Subsets of rectifiable curves in Hilbert space-the analyst’s TSP

Schul¹

2007

J Anal Math

104

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We study one dimensional sets (Hausdorff dimension) lying in a Hilbert space. The aim is to classify subsets of Hilbert spaces that are contained in a connected set of finite Hausdorff length. We do so by extending and improving results of Peter Jones and Kate Okikiolu for sets in R d . Their results formed the basis of quantitative rectifiability in R d . We prove a quantitative version of the following statement: a connected set of finite Hausdorff length (or a subset of one), is characterized by the fact that inside balls at most scales around most points of the set, the set lies close to a straight line segment (which depends on the ball). This is done via a quantity, similar to the one introduced in [Jon90], which is a geometric analog of the Square function. This allows us to conclude that for a given set K, the ℓ 2 norm of this quantity (which is a function of K) has size comparable to a shortest (Hausdorff length) connected set containing K. In particular, our results imply that, with a correct reformulation of the theorems, the estimates in [Jon90,Oki92] are independent of the ambient dimension. Mathematics Subject Classification (2000): 28A75

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