# Analytic capacity, Calderón-Zygmund operators, and rectifiability

• Volume: 43, Issue: 1, page 3-25
• ISSN: 0214-1493

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## Abstract

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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) &gt; 0 when K has Hausdorff dimension larger than 1, and γ(K) = 0 when dim(K) &lt; 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 H1(K) is finite was recently settled. In this case, γ(K) = 0 if and only if K is unrectifiable (or Besicovitch irregular), i.e., if H1(K ∩ Γ) = 0 for all C1-curves Γ, as was conjectured by Vitushkin. In the present text, we try to explain the structure of the proof of this result, and present the necessary techniques. These include the introduction to Menger curvature in this context (by M. Melnikov and co-authors), and the important use of geometric measure theory (results on quantitative rectifiability), but we insist most on the role of Calderón-Zygmund operators and T(b)-Theorems.

## How to cite

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David, Guy. "Analytic capacity, Calderón-Zygmund operators, and rectifiability." Publicacions Matemàtiques 43.1 (1999): 3-25. <http://eudml.org/doc/41370>.

@article{David1999,
abstract = {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) &gt; 0 when K has Hausdorff dimension larger than 1, and γ(K) = 0 when dim(K) &lt; 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 H1(K) is finite was recently settled. In this case, γ(K) = 0 if and only if K is unrectifiable (or Besicovitch irregular), i.e., if H1(K ∩ Γ) = 0 for all C1-curves Γ, as was conjectured by Vitushkin. In the present text, we try to explain the structure of the proof of this result, and present the necessary techniques. These include the introduction to Menger curvature in this context (by M. Melnikov and co-authors), and the important use of geometric measure theory (results on quantitative rectifiability), but we insist most on the role of Calderón-Zygmund operators and T(b)-Theorems.},
author = {David, Guy},
journal = {Publicacions Matemàtiques},
keywords = {Funciones analíticas; Dimensión de Hausdorff; Teoría de la medida; Curvatura; Operadores integrales},
language = {eng},
number = {1},
pages = {3-25},
title = {Analytic capacity, Calderón-Zygmund operators, and rectifiability},
url = {http://eudml.org/doc/41370},
volume = {43},
year = {1999},
}

TY - JOUR
AU - David, Guy
TI - Analytic capacity, Calderón-Zygmund operators, and rectifiability
JO - Publicacions Matemàtiques
PY - 1999
VL - 43
IS - 1
SP - 3
EP - 25
AB - 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) &gt; 0 when K has Hausdorff dimension larger than 1, and γ(K) = 0 when dim(K) &lt; 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 H1(K) is finite was recently settled. In this case, γ(K) = 0 if and only if K is unrectifiable (or Besicovitch irregular), i.e., if H1(K ∩ Γ) = 0 for all C1-curves Γ, as was conjectured by Vitushkin. In the present text, we try to explain the structure of the proof of this result, and present the necessary techniques. These include the introduction to Menger curvature in this context (by M. Melnikov and co-authors), and the important use of geometric measure theory (results on quantitative rectifiability), but we insist most on the role of Calderón-Zygmund operators and T(b)-Theorems.
LA - eng
KW - Funciones analíticas; Dimensión de Hausdorff; Teoría de la medida; Curvatura; Operadores integrales
UR - http://eudml.org/doc/41370
ER -

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