Ebook: Vitushkin's Conjecture for Removable Sets

Vitushkin's conjecture, a special case of Painleve's problem, states that a compact subset of the complex plane with finite linear Hausdorff measure is removable for bounded analytic functions if and only if it intersects every rectifiable curve in a set of zero arclength measure. Chapters 1-5 of the book provide important background material on removability, analytic capacity, Hausdorff measure, arclength measure, and Garabedian duality that will appeal to many analysts with interests independent of Vitushkin's conjecture. The fourth chapter contains a proof of Denjoy's conjecture that employ.;Preface: Painlevé's Problem; 1 Removable Sets and Analytic Capacity; 1.1 Removable Sets; 1.2 Analytic Capacity; 2 Removable Sets and Hausdorff Measure; 2.1 Hausdorff Measure and Dimension; 2.2 Painlevé's Theorem; 2.3 Frostman's Lemma; 2.4 Conjecture and Refutation: The Planar Cantor Quarter Set; 3 Garabedian Duality for Hole-Punch Domains; 3.1 Statement of the Result and an Initial Reduction; 3.2 Interlude: Boundary Correspondence for H(U); 3.3 Interlude: An F. & M. Riesz Theorem; 3.4 Construction of the Boundary Garabedian Function.