Unrectifiable 1-sets have vanishing analytic capacity.

Guy David

Revista Matemática Iberoamericana (1998)

  • Volume: 14, Issue: 2, page 369-479
  • ISSN: 0213-2230

Abstract

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We complete the proof of a conjecture of Vitushkin that says that if E is a compact set in the complex plane with finite 1-dimensional Hausdorff measure, then E has vanishing analytic capacity (i.e., all bounded anlytic functions on the complement of E are constant) if and only if E is purely unrectifiable (i.e., the intersection of E with any curve of finite length has zero 1-dimensional Hausdorff measure). As in a previous paper with P. Mattila, the proof relies on a rectifiability criterion using Menger curvature, and an extension of a construction of M. Christ. The main new part is a generalization of the T(b)-theorem to some spaces that are non necessarily of homogeneous type.

How to cite

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David, Guy. "Unrectifiable 1-sets have vanishing analytic capacity.." Revista Matemática Iberoamericana 14.2 (1998): 369-479. <http://eudml.org/doc/39553>.

@article{David1998,
abstract = {We complete the proof of a conjecture of Vitushkin that says that if E is a compact set in the complex plane with finite 1-dimensional Hausdorff measure, then E has vanishing analytic capacity (i.e., all bounded anlytic functions on the complement of E are constant) if and only if E is purely unrectifiable (i.e., the intersection of E with any curve of finite length has zero 1-dimensional Hausdorff measure). As in a previous paper with P. Mattila, the proof relies on a rectifiability criterion using Menger curvature, and an extension of a construction of M. Christ. The main new part is a generalization of the T(b)-theorem to some spaces that are non necessarily of homogeneous type.},
author = {David, Guy},
journal = {Revista Matemática Iberoamericana},
keywords = {Conjuntos medibles; Espacio de Hausdorff; Unidimensional; Funciones analíticas; analytic capacity; Hausdorff measure},
language = {eng},
number = {2},
pages = {369-479},
title = {Unrectifiable 1-sets have vanishing analytic capacity.},
url = {http://eudml.org/doc/39553},
volume = {14},
year = {1998},
}

TY - JOUR
AU - David, Guy
TI - Unrectifiable 1-sets have vanishing analytic capacity.
JO - Revista Matemática Iberoamericana
PY - 1998
VL - 14
IS - 2
SP - 369
EP - 479
AB - We complete the proof of a conjecture of Vitushkin that says that if E is a compact set in the complex plane with finite 1-dimensional Hausdorff measure, then E has vanishing analytic capacity (i.e., all bounded anlytic functions on the complement of E are constant) if and only if E is purely unrectifiable (i.e., the intersection of E with any curve of finite length has zero 1-dimensional Hausdorff measure). As in a previous paper with P. Mattila, the proof relies on a rectifiability criterion using Menger curvature, and an extension of a construction of M. Christ. The main new part is a generalization of the T(b)-theorem to some spaces that are non necessarily of homogeneous type.
LA - eng
KW - Conjuntos medibles; Espacio de Hausdorff; Unidimensional; Funciones analíticas; analytic capacity; Hausdorff measure
UR - http://eudml.org/doc/39553
ER -

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