Dudziak J.J. Vitushkin’s Conjecture for Removable Sets

Springer, 2010. — 338 p. — (Universitext). — ISBN 978-1-4419-6708-4.Гипотеза Vitushkin’s - частный случай проблемы Painlevé's, утверждает, что компактное подмножество комплексной плоскости с конечной линейной мерой Hausdorff устранимо для ограниченных аналитических функций тогда и только тогда, когда оно пересекает каждую спрямляемую кривую в множестве нулевой меры длины дуги.Vitushkin's conjecture, a special case of Painlevé'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 6-8 of this carefully written text present a major recent accomplishment of modern complex analysis, the affirmative resolution of this conjecture.
Four of the five mathematicians whose work solved Vitushkin's conjecture have won the prestigious Salem Prize in analysis.Preface: Painlevé’s Problem Removable Sets and Analytic Capacity
Removable Sets and Hausdorff Measure
Garabedian Duality for Hole-Punch Domains
Melnikov and Verdera’s Solution to the Denjoy Conjecture
Some Measure Theory
A Solution to Vitushkin’s Conjecture Modulo Two Difficult Results
The T(b) Theorem of Nazarov, Treil, and Volberg
The Curvature Theorem of David and Léger
Postscript: Tolsa’s Theorem