Calderón–Zygmund kernels and rectifiability in the plane☆

  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},

Capacities Associated with Calderón-Zygmund Kernels

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

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

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

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

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

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


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

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

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



The Cauchy integral, analytic capacity, and uniform rectifiability

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

Menger curvature and rectifiability

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

Unrectictifiable 1-sets have vanishing analytic capacity

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

A geometric proof of the L2 boundedness of the Cauchy integral on Lipschitz graphs

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

Analysis of and on uniformly rectifiable sets

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

Cauchy Singular Integrals and Rectifiability of Measures in the Plane

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,

Painlevé's problem and the semiadditivity of analytic capacity

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

Rectifiable sets and the Traveling Salesman Problem

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

Cauchy integrals on Lipschitz curves and related operators.

  • 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.