Conjuntos evitables, capacidad analítica y medidas de Hausdorff

Abstract

The main problem we consider in this work is the characterization of removable sets. A compact set E in the complex plane is removable if there exists an open set Ω containing E such that every bounded and analytic function in Ω \ E has an analytic extension to the whole set Ω. For this, we will study some concepts such as analytic capacity, Hausdorff measure and Hausdorff dimension of sets. The analytic capacity of E, γ(E), constitutes a form to measure “how removable is a compact set E” and its definition deals with the behaviour of holomorphic functions on C\E and not with the geometric properties of E. Hausdorff measure and dimension are geometric concepts related to the size of sets. We will see that a compact set E is removable if and only if γ(E)=0. With this result we can study the relationship between removability and Hausdorff dimensions. In particular, Painleve’s Theorem says that any set with length equal to zero is removable. The most important connection with Hausdorff dimension is that the length of any set is identical to the Hausdorff measure in one dimension. Moreover, we will provide an example due to J. Garnett to show that the inverse implication of this result is not true. In the opposite direction, we will prove that if a set E has Hausdorff dimension strictly largest than one then it is not removable. For this, we will need to prove the Frostman’s Lemma and introduce one of the most important tool in this area: the Cauchy transform.