Abstract: This is a joint work with Julia Cufi, Juan Jesus Donaire and Joan Verdera. We show that on some standard Cantor sets in the plane with Hausdorff dimension 1 and non-sigma-finite one-dimensional Hausdorff measure, the principal values of the Cauchy transform with respect to the natural probability measure exist on a set of Hausdorff dimension one, although they often fail to exist almost everywhere. I shall also discuss questions related to more general settings.
