Unrectifiable 1-sets have vanishing analytic capacity
- Autores: Guy David
- Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 14, Nº 2, 1998, págs. 369-479
- Idioma: inglés
- DOI: 10.4171/rmi/242
- Títulos paralelos:
- Los conjuntos unidimensionales no rectificables tienen la capacidad analítica nula.
- Enlaces
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Resumen
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.
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