Unrectictifiable 1-sets have vanishing analytic capacity

  • Guy David

    Université Paris-Sud, Orsay, France


We complete the proof of a conjecture of Vitushkin that says that if EE is a compact set in the complex plane with fi nite 1-dimensional Hausdorff measure, then EE has vanishing analytic capacity (i.e., all bounded analytic functions on the complement of EE are constant) if and only if EE is purely unrectifi able (i.e., the intersection of EE with any curve of fi nite length has zero 1-dimensional Hausdorff measure). As in a previous paper with P. Mattila, the proof relies on a rectifi ability criterion using Menger curvature and an extension of a construction of M. Christ. The main new part is a generalization of the T(b)T(b)-Theorem to some spaces that are not necessarily of homogeneous type .

Cite this article

Guy David, Unrectictifiable 1-sets have vanishing analytic capacity. Rev. Mat. Iberoam. 14 (1998), no. 2, pp. 369–479

DOI 10.4171/RMI/242