A New Family of Singular Integral Operators Whose $$L^2$$ L 2 -Boundedness Implies Rectifiability

Chunaev, Petr

doi:10.1007/s12220-017-9780-9

Cited by 6 publications

(10 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, taking into account this property and using a curvature-like method, the following Theorem B type result is proved in [4].…”

Section: Introductionmentioning

confidence: 99%

“…Clearly, one obtains a kernel of the form (5) from (8) when n = N (and t = −1) or t = 0. It turns out that this slight modification of the kernel leads to a diverse behaviour of the corresponding CZO depending on the parameter t. For example, it is shown in [4] that if t belongs to the set…”

Section: Introductionmentioning

confidence: 99%

“…From the other side, it is shown in [4] that there exist triples (z 1 , z 2 , z 3 ) such that p Kt (z 1 , z 2 , z 3 ) change sign if t belongs to the interval (11) ω(n, N) := (−N/n; 0)…”

Section: Introductionmentioning

confidence: 99%

“…For the details see also [4,Remark 2]. Thus we come to the question of what happens when t ∈ ω(n, N), i.e.…”

Section: Introductionmentioning

confidence: 99%

“…Thus we come to the question of what happens when t ∈ ω(n, N), i.e. the permutations p Kt (z 1 , z 2 , z 3 ) change sign, and curvature-like methods as in [2,4,13,14] do not work. In this paper a partial answer is given in the case of kernels (12).…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

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

Chunaev

Mateu

Tolsa

2019

JAMA

Self Cite

View full text Add to dashboard Cite

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)

show abstract

“…Moreover, taking into account this property and using a curvature-like method, the following Theorem B type result is proved in [4].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…From the other side, it is shown in [4] that there exist triples (z 1 , z 2 , z 3 ) such that p Kt (z 1 , z 2 , z 3 ) change sign if t belongs to the interval (11) ω(n, N) := (−N/n; 0)…”

Section: Introductionmentioning

confidence: 99%

“…For the details see also [4,Remark 2]. Thus we come to the question of what happens when t ∈ ω(n, N), i.e.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

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

Chunaev

Mateu

Tolsa

2019

JAMA

Self Cite

View full text Add to dashboard Cite

show abstract

A family of singular integral operators which control the Cauchy transform

2019

Self Cite

View full text Add to dashboard Cite

We study the behaviour of singular integral operators T kt of convolution type on C associated with the parametric kernelsIt is shown that for any positive locally finite Borel measure with linear growth the corresponding L 2 -norm of T k0 controls the L 2 -norm of T k∞ and thus of the Cauchy transform. As a corollary, we prove that the L 2 (H 1 E)-boundedness of T kt with a fixed t ∈ (−t 0 , 0), where t 0 > 0 is an absolute constant, implies that E is rectifiable. This is so in spite of the fact that the usual curvature method fails to be applicable in this case. Moreover, as a corollary of our techniques, we provide an alternative and simpler proof of the bi-Lipschitz invariance of the L 2 -boundedness of the Cauchy transform, which is the key ingredient for the bilipschitz invariance of analytic capacity.2010 Mathematics Subject Classification. 42B20 (primary); 28A75 (secondary).The authors of [CMPT1] used a curvature type method. It allowed them to modify the required parts of the proof from [L] to obtain their result. Extending this technique, Chunaev [Ch] proved that the same is true for a quite large range of the parameter t, additionally to t = 0 and t = ∞:It is also shown in [Ch] that a direct curvature type method cannot be applied for t ∈ (−2, 0). Moreover, it is known that for some of these t there exist counterexamples to the above-mentioned implication due to results of Huovinen [H] and Jaye and Nazarov [JN]: t = −1 or t = − 3 4 ⇒ ∃ purely unrectifiable E : T kt L 2 (H 1 E) < ∞.Then, integrating on x ∈ J i , y ∈ J j , and z ∈ J k with respect to σ, we get

show abstract

Symmetrization of a Family of Cauchy-Like Kernels: Global Instability

Lanzani

Pramanik

2022

La Matematica

View full text Add to dashboard Cite

The fundamental role of the Cauchy transform in harmonic and complex analysis has led to many different proofs of its L 2 boundedness. In particular, a famous proof of Melnikov-Verdera (A geometric proof of the L 2 boundedness of the Cauchy integral on Lipschitz graphs. Int Math Res Not 7: [325][326][327][328][329][330][331] 1995) relies upon an iconic symmetrization identity of Melnikov (Analytic capacity: a discrete approach and the curvature of measures. Mat Sb 186(6):57-76, 1995) linking the universal Cauchy kernel K 0 to Menger curvature. Analogous identities hold for the real and the imaginary parts of K 0 as well. Such connections have been immensely productive in the study of singular integral operators and in geometric measure theory. In this article, given any function h : C → R, we consider an inhomogeneous variant K h of K 0 which is inspired by complex function theory. While an operator with integration kernel K h is easily seen to be L 2 -bounded for all h, the symmetrization identities for each of the real and imaginary parts of K h show a striking lack of robustness in terms of boundedness and positivity, two properties that were critical in Melnikov and Verdera (A geometric proof of the L 2 boundedness of the Cauchy integral on Lipschitz graphs. Int Math Res Not 7: [325][326][327][328][329][330][331] 1995) and in subsequent works by many authors. Indeed here we show that for any continuous h on C, the only member of {K h } h whose symmetrization has the right properties is K 0 ! This global instability complements our previous investigation (Lanzani and Pramanik, Symmetrization of a Cauchy-like kernel on curves. J Funct Anal (Preprint). arXiv:2001.09375) of symmetrization identities in the restricted setting of a curve, where a sub-family of {K h } h displays very different behaviour than its global counterparts considered here. Our methods of proof have some overlap with B Malabika Pramanik

show abstract

A New Family of Singular Integral Operators Whose $$L^2$$ L 2 -Boundedness Implies Rectifiability

Cited by 6 publications

References 22 publications

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

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

A family of singular integral operators which control the Cauchy transform

Symmetrization of a Family of Cauchy-Like Kernels: Global Instability

Contact Info

Product

Resources

About