# 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}
}
• Published 6 October 2011
• Mathematics
• Advances in Mathematics
27 Citations
• 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,
• Mathematics
Journal 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
• Mathematics
La 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
• 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 $K_i(x)=x_i^{2n-1}/|x|^{2n}$, $1\le i\le • Mathematics • 2020 We investigate the robustness of the symmetrization identities that link the Cauchy kernel$K_0$and its real and imaginary parts with the Menger curvature. We show that certain properties of these • Mathematics Revista Matemática Iberoamericana • 2020 Fix$d\geq 2$and$s\in (0,d)$. In this paper we introduce a notion called small local action associated to a singular integral operator, which is a necessary condition for the existence of principal • Mathematics • 2015 A measure µ on R d is called reflectionless for the s-Riesz transform if the singular integral R s µ(x) = ´ y x |y x|s+1 dµ(y) is constant on the support of µ in some weak sense and, moreover, the • Mathematics • 2013 We prove that in any metric space$(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 • 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 ## References SHOWING 1-10 OF 24 REFERENCES • Mathematics • 1996 Several explanations concerning notation, terminology, and background are in order. First notation: by 7Hi we have denoted the one-dimensional Hausdorff measure (i.e. length), and A(z,r) stands for where 'HI is the 1-dimensional Hausdorff measure in Rn, c(x, y, z) is the inverse of the radius of the circumcircle of the triangle (x, y, z), that is, following the terminology of [6], the Menger We complete the proof of a conjecture of Vitushkin that says that if E is a compact set in the complex plane with finite 1-dimensional Hausdorff measure, then E has vanishing analytic capacity (i.e., Let μ be a finite nonzero Borel measure in Rn satisfying 0 < c−1rs ≤ μB(x, r) ≤ crs <∞ for all x ∈ sptμ and 0 < r ≤ 1 and some c > 0. If the Riesz s-transform Cs,μ(x) = ∫ y − x |y − x|s+1 dμy is • Mathematics • 1995 In this paper we give a new proof of the L2 boundedness of the Cauchy integral on Lipschitz graphs (and chord-arc curves). Our method consists in controlling the Cauchy integral by an appropiate • Mathematics • 1993 The notion of uniform rectifiability of sets (in a Euclidean space), which emerged only recently, can be viewed in several different ways. It can be viewed as a quantitative and scale-invariant Abstract Let μ be a finite non-negative Borel measure on the complex plane C . We shall prove the following result: If for μ almost all a ∈ C [formula]and the limit [formula] exists and is finite, Let$\gamma(E)$be the analytic capacity of a compact set$E$and let$\gamma_+(E)$be the capacity of$E\$ originated by Cauchy transforms of positive measures. In this paper we prove that
Let K c C be a bounded set. In this paper we shall give a simple necessary and sufficient condit ion for K to lie in a rectifiable curve. We say that a set is a rectifiable curve if it is the image
• A. Calderón
• Mathematics
Proceedings of the National Academy of Sciences of the United States of America
• 1977
Certain properties of the Cauchy integral on Lipschitz curves are established and the L(p)-boundedness of some related operators are proved and the recent results of R. Coifman and Y. Meyer are obtained.