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}
}
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27 Citations
Background Citations
7
Methods Citations
4
Results Citations
1

27 Citations

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 Cauchy-like kernel on curves

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…

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

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…

References

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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…

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  • 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.