Displaying similar documents to “Singular integrals and rectifiability.”

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

Joan Verdera (2002)

Publicacions Matemàtiques

Similarity:

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...

Analytic capacity, Calderón-Zygmund operators, and rectifiability

Guy David (1999)

Publicacions Matemàtiques

Similarity:

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....

Henkin measures, Riesz products and singular sets

Evgueni Doubtsov (1998)

Annales de l'institut Fourier

Similarity:

The mutual singularity problem for measures with restrictions on the spectrum is studied. The d -pluriharmonic Riesz product construction on the complex sphere is introduced. Singular pluriharmonic measures supported by sets of maximal Hausdorff dimension are obtained.

The Riesz kernels do not give rise to higher dimensional analogues of the Menger-Melnikov curvature.

Hany M. Farag (1999)

Publicacions Matemàtiques

Similarity:

Ever since the discovery of the connection between the Menger-Melnikov curvature and the Cauchy kernel in the L norm, and its impressive utility in the analytic capacity problem, higher dimensional analogues have been coveted. The lesson from 1-sets was that any such (nontrivial, nonnegative) expression, using the Riesz kernels for m-sets in R, even in any L norm (k ∈ N), would probably carry nontrivial information on whether the boundedness of these kernels in the appropriate norm implies...

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

Pertti Mattila (1996)

Publicacions Matemàtiques

Similarity:

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.