Vitushkin's Conjecture for Removable Sets

by James Dudziak

Overview

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.

Product Details

ISBN-13:	9781441967084
Publisher:	Springer New York
Publication date:	09/23/2010
Series:	Universitext
Edition description:	2010
Pages:	332
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

About the Author

James J. Dudziak received his Ph.D from Indiana University and is currently a visiting associate professor at Michigan State University at Lyman Briggs College. He published six excellent papers in good journals from 1984 to 1990 when he received tenure at Bucknell University.

Table of Contents

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.

Editorial Reviews

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)

From the Publisher