# Calderón–Zygmund kernels and rectifiability in the plane☆

@article{Chousionis2011CaldernZygmundKA, title={Calder{\'o}n–Zygmund kernels and rectifiability in the plane☆}, author={Vasilis Chousionis and Joan Mateu and Laura Prat and Xavier Tolsa}, journal={Advances in Mathematics}, year={2011}, volume={231}, pages={535-568} }

## 27 Citations

### Capacities Associated with Calderón-Zygmund Kernels

- Mathematics
- 2013

Analytic capacity is associated with the Cauchy kernel 1/z and the L∞-norm. For n ∈ ℕ, one has likewise capacities related to the kernels $K_i(x)=x_i^{2n-1}/|x|^{2n}$, 1 ≤ i ≤ 2,…

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

- MathematicsJournal d'Analyse Mathématique
- 2019

The well-known curvature method initiated in works of Melnikov and Verdera is now commonly used to relate the L2(μ)-boundedness of certain singular integral operators to the geometric properties of…

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

- MathematicsLa Matematica
- 2022

The fundamental role of the Cauchy transform in harmonic and complex analysis has led to many different proofs of its L boundedness. In particular, a famous proof of Melnikov-Verdera [18] relies upon…

### Capacities associated with Calder\'on-Zygmund kernels

- Mathematics
- 2011

Analytic capacity is associated with the Cauchy kernel $1/z$ and the $L^\infty$-norm. For $n\in\mathbb{N}$, one has likewise capacities related to the kernels $K_i(x)=x_i^{2n-1}/|x|^{2n}$, $1\le i\le…

### Symmetrization estimates for Cauchy-like kernels, Part I: Global Results

- Mathematics
- 2020

We investigate the robustness of the symmetrization identities that link the Cauchy kernel $K_0$ and its real and imaginary parts with the Menger curvature. We show that certain properties of these…

### Small local action of singular integrals on spaces of non-homogeneous type

- MathematicsRevista Matemática Iberoamericana
- 2020

Fix $d\geq 2$ and $s\in (0,d)$. In this paper we introduce a notion called small local action associated to a singular integral operator, which is a necessary condition for the existence of principal…

### NON-EXISTENCE OF REFLECTIONLESS MEASURES FOR THE s-RIESZ TRANSFORM WHEN 0 < s < 1

- Mathematics
- 2015

A measure µ on R d is called reflectionless for the s-Riesz transform if the singular integral R s µ(x) = ´ y x |y x|s+1 dµ(y) is constant on the support of µ in some weak sense and, moreover, the…

### A note on weak convergence of singular integrals in metric spaces

- Mathematics
- 2013

We prove that in any metric space $(X,d)$ the singular integral operators {equation*} T^k_{\mu,\ve}(f)(x)=\int_{X\setminus B(x,\varepsilon)}k(x,y)f(y)d\mu (y).{equation*} converge weakly in some…

### Some Calderón–Zygmund kernels and their relations to Wolff capacities and rectifiability

- Mathematics
- 2014

We consider the Calderón–Zygmund kernels $$K_ {\alpha ,n}(x)=(x_i^{2n-1}/|x|^{2n-1+\alpha })_{i=1}^d$$Kα,n(x)=(xi2n-1/|x|2n-1+α)i=1d in $${\mathbb R}^d$$Rd for $$0<\alpha \le 1$$0<α≤1 and $$n\in…

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