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© 2010

Vitushkin’s Conjecture for Removable Sets

Book

Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages i-xii
  2. James J. Dudziak
    Pages 1-17
  3. James J. Dudziak
    Pages 19-38
  4. James J. Dudziak
    Pages 39-68
  5. James J. Dudziak
    Pages 105-129
  6. James J. Dudziak
    Pages 159-220
  7. James J. Dudziak
    Pages 221-310
  8. Back Matter
    Pages 311-331

About this book

Introduction

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. Chapters 1-5 of this 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 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.

Keywords

Analytic capacity Arclength measure Argument principle Complex analysis Garabedian duality Hausdorff measure James Dudziak Melnikov curvature Melnikov's conjecture Removable sets for bounded analytic functions Vitushkin's conjecture differential equation gamma function logarithm measure

Authors and affiliations

  1. 1.Michigan State UniversityDepartment of MathematicsEast LansingUSA

About the authors

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.

Bibliographic information

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)