Displaying similar documents to “On the analytic capacity and curvature of some Cantor sets with non-σ-finite length.”

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....

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...

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

Hany M. Farag (1999)

Publicacions Matemàtiques

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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...

Prevalence of "nowhere analyticity"

Françoise Bastin, Céline Esser, Samuel Nicolay (2012)

Studia Mathematica

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This note brings a complement to the study of genericity of functions which are nowhere analytic mainly in a measure-theoretic sense. We extend this study to Gevrey classes of functions.

Volume and multiplicities of real analytic sets

Guillaume Valette (2005)

Annales Polonici Mathematici

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We give criteria of finite determinacy for the volume and multiplicities. Given an analytic set described by {v = 0}, we prove that the log-analytic expansion of the volume of the intersection of the set and a "little ball" is determined by that of the set defined by the Taylor expansion of v up to a certain order if the mapping v has an isolated singularity at the origin. We also compare the cardinalities of finite fibers of projections restricted to such a set.