# Unrectifiable 1-sets have vanishing analytic capacity.

Revista Matemática Iberoamericana (1998)

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

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topDavid, 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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