Unrectictifiable 1-sets have vanishing analytic capacity

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 analytic 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 not necessarily of homogeneous type.