$L^2$ boundedness of the Cauchy transform implies $L^2$ boundedness of all antisymmetric Calderón-Zygmund operators associated to odd kernels

Domènech, Xavier Tolsa

doi:10.5565/publmat_48204_09

Cited by 15 publications

(16 citation statements)

References 22 publications

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“…This decomposition of T µ is inspired in part by [Se]. See also [To1], [MT] and [To3] for some related techniques. Let us denote T (m) µ = j:j≤m T j µ.…”

Section: 2mentioning

confidence: 99%

Uniform rectifiability, Calderón-Zygmund operators with odd kernel, and quasiorthogonality

Tolsa

2008

Proceedings of the London Mathematical Society

104

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In this paper we study some questions in connection with uniform rectifiability and the L2 boundedness of Calderón–Zygmund operators (CZOs). We show that uniform rectifiability can be characterized in terms of some new adimensional coefficients that are related to the Jones’ β numbers. We also use these new coefficients to prove that n‐dimensional CZOs with odd kernel of type 𝒞2 are bounded in L2(μ), if μ is an n‐dimensional uniformly rectifiable measure.

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“…This decomposition of T µ is inspired in part by [Se]. See also [To1], [MT] and [To3] for some related techniques. Let us denote T (m) µ = j:j≤m T j µ.…”

Section: 2mentioning

confidence: 99%

Uniform rectifiability, Calderón-Zygmund operators with odd kernel, and quasiorthogonality

Tolsa

2008

Proceedings of the London Mathematical Society

104

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“…To prove the Main Lemma, we will closely follow the ideas by Tolsa in [To3], but we will use the dyadic lattice D associated to µ, which is introduced in Section 3, instead of the usual dyadic lattice of true cubes in R d . We apply Lemma D to obtain a Corona Decomposition for µ, and we decompose T µ in terms of that Corona Decomposition, since the terms that arise from that decomposition will be tractable.…”

Section: The Main Lemmamentioning

confidence: 99%

Geometric conditions for the L2-boundedness of singular integral operators with odd kernels with respect to measures with polynomial growth in ℝd

Girela-Sarrión

2019

JAMA

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Let µ be a finite Radon measure in R d with polynomial growth of degree n, although not necessarily n-AD regular. We prove that under some geometric conditions on µ that are closely related to rectifiability and involve the so-called β-numbers of Jones, David and Semmes, all singular integral operators with an odd and sufficiently smooth Calderón-Zygmund kernel are bounded in L 2 (µ). As a corollary, we obtain a lower bound for the Lipschitz harmonic capacity of a compact set in R d only in terms of its metric and geometric properties.

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“…If the CZO associated with the kernel k t , where (37) t ∈ (−∞; − √ 2) ∪ (0; ∞), is L 2 (µ)-bounded, then all 1-dimensional CZOs associated with odd and sufficiently smooth kernels are also L 2 (µ)-bounded. We refer the reader to [18,Sections 1 and 12] for the more precise description of what we mean by "sufficiently smooth kernels". Indeed, it follows from (18) and (22) with (n, N) = (1, 2) that for any t as in (37) and any cube Q ⊂ C, one has T κ 1 ,ε χ Q L 2 (µ⌊Q) C(t) T kt,ε χ Q L 2 (µ⌊Q) + µ(Q) , C(t) > 0,…”

Section: Additional Remarksmentioning

confidence: 99%

Singular integrals unsuitable for the curvature method whose L2-boundedness still implies rectifiability

Chunaev

Mateu

Tolsa

2019

JAMA

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The well-known curvature method initiated in works of Melnikov and Verdera is now commonly used to relate the L 2 (µ)-boundedness of certain singular integral operators to the geometric properties of the support of measure µ, e.g. rectifiability. It can be applied however only if Menger curvature-like permutations, directly associated with the kernel of the operator, are non-negative. We give an example of an operator in the plane whose corresponding permutations change sign but the L 2 (µ)-boundedness of the operator still implies that the support of µ is rectifiable. To the best of our knowledge, it is the first example of this type. We also obtain several related results with Ahlfors-David regularity conditions. 2010 Mathematics Subject Classification. 42B20 (primary); 28A75 (secondary).How to relate the L 2 (µ)-boundedness of a certain CZO to the geometric properties of the support of µ is an old problem in harmonic analysis. It stems from Calderón's paper [1] where it is proved that the Cauchy transform, i.e. the CZO with K(z) = 1/z, is L 2 (H 1 ⌊E)-bounded if E is a Lipschitz graphs with small slope. Later on, Coifman, McIntosh and Meyer [5] removed the small Lipschitz constant assumption. In [6] David fully characterized rectifiable curves Γ, for which the Cauchy transform is L 2 (H 1 ⌊Γ)-bounded. These results led to further development of tools for understanding the above-mentioned problem.A new quantitative characterization of rectifiability in terms of the so-called βnumbers introduced by Jones [12] and the concept of uniform rectifiability proposed by David and Semmes [7,8] are among these tools. Several related definitions for the plane are in order. (We refer the reader to [7,8] for definitions and results in the multidimensional case). A Radon measure µ on C is called (1-dimensional ) Ahlfors-David regular (or AD-regular, for short) if it satisfies the inequalities C −1 r µ(B(z, r)) Cr, where z ∈ spt µ, 0 < r < diam (spt µ)

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$L^2$ boundedness of the Cauchy transform implies $L^2$ boundedness of all antisymmetric Calderón-Zygmund operators associated to odd kernels

Cited by 15 publications

References 22 publications

Uniform rectifiability, Calderón-Zygmund operators with odd kernel, and quasiorthogonality

Uniform rectifiability, Calderón-Zygmund operators with odd kernel, and quasiorthogonality

Geometric conditions for the L2-boundedness of singular integral operators with odd kernels with respect to measures with polynomial growth in ℝd

Singular integrals unsuitable for the curvature method whose L2-boundedness still implies rectifiability

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