Displaying similar documents to “Rectifiability, analytic capacity, and singular integrals.”

Analytic capacity, Calderón-Zygmund operators, and rectifiability

Guy David (1999)

Publicacions Matemàtiques

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For K ⊂ C compact, we say that K has vanishing analytic capacity (or γ(K) = 0) when all bounded analytic functions on CK are constant. We would like to characterize γ(K) = 0 geometrically. Easily, γ(K) > 0 when K has Hausdorff dimension larger than 1, and γ(K) = 0 when dim(K) < 1. Thus only the case when dim(K) = 1 is interesting. So far there is no characterization of γ(K) = 0 in general, but the special case when the Hausdorff measure H(K) is finite was recently settled....

On the analytic capacity and curvature of some Cantor sets with non-σ-finite length.

Pertti Mattila (1996)

Publicacions Matemàtiques

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We show that if a Cantor set E as considered by Garnett in [G2] has positive Hausdorff h-measure for a non-decreasing function h satisfying ∫ r h(r) dr < ∞, then the analytic capacity of E is positive. Our tool will be the Menger three-point curvature and Melnikov’s identity relating it to the Cauchy kernel. We shall also prove some related more general results.

Remark on computing the analytic SVD

Janovská, Dáša, Janovský, Vladimír

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A new technique for computing analytic SVD is proposed. The idea is to follow branches for just one selected singular value and the corresponding left/right singular vector.

Singular integrals and rectifiability.

Pertti Mattila (2002)

Publicacions Matemàtiques

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We shall discuss singular integrals on lower dimensional subsets of Rn. A survey of this topic was given in [M4]. The first part of this paper gives a quick review of some results discussed in [M4] and a survey of some newer results and open problems. In the second part we prove some results on the Riesz kernels in Rn. As far as I know, they have not been explicitly stated and proved, but they are very closely related to some earlier results and...

The fall of the doubling condition in Calderón-Zygmund theory.

Joan Verdera (2002)

Publicacions Matemàtiques

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The most important results of standard Calderón-Zygmund theory have recently been extended to very general non-homogeneous contexts. In this survey paper we describe these extensions and their striking applications to removability problems for bounded analytic functions. We also discuss some of the techniques that allow us to dispense with the doubling condition in dealing with singular integrals. Special attention is paid to the Cauchy Integral. [Proceedings...