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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 arc length 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.
Chapters 1-5 of this book provide important background material on removability, analytic capacity, Hausdorff measure, arc length 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 employs Melnikov curvature. A brief postscript reports on a deep theorem of Tolsa and its relevance to going beyond Vitushkin's conjecture. Although standard notation is used throughout, there is a symbol glossary at the back of the book for the reader's convenience.
This text can be used for a topics course or seminar in complex analysis. To understand it, the reader should have a firm grasp of basic real and complex analysis.
Chapters 1-5 of this book provide important background material on removability, analytic capacity, Hausdorff measure, arc length 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 employs Melnikov curvature. A brief postscript reports on a deep theorem of Tolsa and its relevance to going beyond Vitushkin's conjecture. Although standard notation is used throughout, there is a symbol glossary at the back of the book for the reader's convenience.
This text can be used for a topics course or seminar in complex analysis. To understand it, the reader should have a firm grasp of basic real and complex analysis.
From the reviews:
"This is a very nice and well-written book that presents a complete proof of the so-called Vitushkin conjecture on removable sets for bounded analytic functions ... . it is accessible to both graduate and undergraduate students." (Xavier Tolsa, Mathematical Reviews, Issue 2011 i)
"The aim of the book is to present a complete proof of the recent affirmative solution to the Vitushkin conjecture, which was preceded by a proof of the Denjoy conjecture. ... The book is a guide for graduate students and a helpful survey for experts." (Dmitri V. Prokhorov, Zentralblatt MATH, Vol. 1205, 2011)
"This is a very nice and well-written book that presents a complete proof of the so-called Vitushkin conjecture on removable sets for bounded analytic functions ... . it is accessible to both graduate and undergraduate students." (Xavier Tolsa, Mathematical Reviews, Issue 2011 i)
"The aim of the book is to present a complete proof of the recent affirmative solution to the Vitushkin conjecture, which was preceded by a proof of the Denjoy conjecture. ... The book is a guide for graduate students and a helpful survey for experts." (Dmitri V. Prokhorov, Zentralblatt MATH, Vol. 1205, 2011)