Bilipschitz maps, analytic capacity, and the Cauchy integral

Tolsa, Xavier

doi:10.4007/annals.2005.162.1243

Cited by 75 publications

(103 citation statements)

References 14 publications

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“…Although this result has a definite geometric flavor, it is not clear if this is a really good geometric characterization. Nevertheless, in [To7] it has been shown that the characterization is invariant under bilipschitz mappings, using a corona type decomposition for non doubling measures. Previously, Garnett and Verdera [GV] had proved an analogous result for some Cantor sets.…”

Section: Is a Countable (Or Finite) Family Of Compact Sets We Havementioning

confidence: 99%

“…Using the corona type decomposition for measures with finite curvature and linear growth obtained in [To7], it has been proved in [To8] that if µ is a measure without atoms such that the Cauchy transform is bounded on L 2 (µ), then any Calderón-Zygmund operator associated to an odd kernel sufficiently smooth is also bounded in L 2 (µ).…”

Section: α(∂ I E) = 0 Then Any Function Analytic Inmentioning

confidence: 99%

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Analytic capacity, rectifiability, and the Cauchy integral

Domènech¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract. A compact set E ⊂ C is said to be removable for bounded analytic functions if for any open set Ω containing E, every bounded function analytic on Ω\E has an analytic extension to Ω. Analytic capacity is a notion that, in a sense, measures the size of a set as a non removable singularity. In particular, a compact set is removable if and only if its analytic capacity vanishes. The so called Painlevé problem consists in characterizing removable sets in geometric terms. Recently many results in connection with this very old and challenging problem have been obtained. Moreover, it has also been proved that analytic capacity is semiadditive. We review these results and other related questions dealing with rectifiability, the Cauchy transform, and the Riesz transforms. Mathematics Subject Classification (2000). Primary 30C85; Secondary 42B20, 28A75.

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Section: Is a Countable (Or Finite) Family Of Compact Sets We Havementioning

confidence: 99%

Section: α(∂ I E) = 0 Then Any Function Analytic Inmentioning

confidence: 99%

Analytic capacity, rectifiability, and the Cauchy integral

Domènech¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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“…Let η ∈ (1,∞). It was proved in [17] that we obtain an equivalent norm of RBLO(μ) if (2.10) and (2.11) in Definition 2.13 are, respectively, replaced by that there exists a nonnegative constant C such that for any cube Q centered at some point of supp(μ), 12) and for any two doubling cubes Q ⊂ R,…”

Section: Definition 24mentioning

confidence: 99%

Uniform Boundedness for Approximations of the Identity with Nondoubling Measures

Yang

2007

J Inequal Appl

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Let μ be a nonnegative Radon measure on R d which satisfies the growth condition that there exist constants C 0 > 0 and n where B(x,r) is the open ball centered at x and having radius r. In this paper, the authors establish the uniform boundedness for approximations of the identity introduced by Tolsa in the Hardy space H 1 (μ) and the BLO-type space RBLO (μ). Moreover, the authors also introduce maximal operators . ᏹs (homogeneous) and ᏹ s (inhomogeneous) associated with a given approximation of the identity S, and prove that . ᏹs is bounded from H 1 (μ) to L 1 (μ) and ᏹ s is bounded from the local atomic Hardy space h 1,∞ atb (μ) to L 1 (μ). These results are proved to play key roles in establishing relations between H 1 (μ) and h 1,∞ atb (μ), BMO-type spaces RBMO (μ) and rbmo (μ) as well as RBLO (μ) and rblo (μ), and also in characterizing rbmo (μ) and rblo (μ).

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“…Thus for analytic removability, dimension 1 is the critical point both for L ∞ and BM O. However, the solution to the original Painlevé problem lies much deeper and was only recently achieved by Tolsa ([Tol03], [Tol05]) in terms of curvatures of measures. Under the assumption that H 1 (E) is finite, Painlevé's problem was earlier solved by G. David [Dav98], who showed that a set E with 0 < H 1 (E) < ∞ is removable for bounded analytic functions if it is purely unrectifiable.…”

Section: Introductionmentioning

confidence: 99%