Analytic capacity and singular integrals

Abstract

[en] In this project we introduce the notion of analytic capacity $(\gamma)$ as well as some of its essential properties. Using this concept we identify the family of removable compact subsets of $\mathbb{C}$, which are those such that, for any bounded holomorphic function defined on their complementary, they allow to extend analytically such function to the whole complex plane. From this point on, we discuss a possible geometric characterization for removable subsets, popularly known as the Painlevé problem. The previous task is done in terms of the Hausdorff dimension of these subsets, obtaining a full classification for values different than 1. This remaining case, usually referred to as the critical dimension associated to $\gamma$, has to be dealt with apart. It is at this point that we invoke the theory of singular integrals in order to study a particular family of these subsets: those contained in graphs of Lipschitz functions. We end our project by tackling this case, introduced by Arnaud Denjoy in the early 1900's, and providing a proof of a characterization theorem in this particular setting.