Synopses & Reviews
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.
Review
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)
Synopsis
This book presents a major accomplishment of modern complex analysis, the affirmative resolution of Vitushkin's conjecture. It also contains background material on removability, analytic capacity, Hausdorff measure, arclength measure and Garabedian duality.
Synopsis
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 employs Melnikov curvature. A brief postscript reports on a deep theorem of Tolsa and its relevance to going beyond Vitushkin's conjecture. 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.
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
Preface.- 1 Removable Sets and Analytic Capacity.- 1.1 Removable Sets.- 1.2 Analytic Capacity.- 2 Removable Sets and Hausdor Measure.- 2.1 Hausdor Measure and Dimension.- 2.2 Painlevé's Theorem.- 2.3 Frostman's Lemma.- 2.4 Conjecture & 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 H1(U).- 3.3 Interlude: Some F. & M. Riesz Theorems.- 3.4 Construction of the Boundary Garabedian Function.- 3.5 Construction of the Interior Garabedian Function.- 3.6 A Further Reduction.- 3.7 Interlude: Some Extension and Join Propositions.- 3.8 Analytically Extending the Ahlfors and Garabedian Functions.- 3.9 Interlude: Consequences of the Argument Principle.- 3.10 An Analytic Logarithm of the Garabedian Function.- 4 Melnikov and Verdera's Solution to the Denjoy Conjecture.- 4.1 Menger Curvature of Point Triples.- 4.2 Melnikov's Lower Capacity Estimate.- 4.3 Interlude: A Fourier Transform Review.- 4.4 Melnikov Curvature of Some Measures on Lipschitz Graphs.- 4.5 Arclength & Arclength Measure: Enough to Do the Job.- 4.6 The Denjoy Conjecture Resolved Affirmatively.- 4.7 Conjecture & Refutation: The Joyce-Mörters Set.- 5 Some Measure Theory.- 5.1 The Carathéodory Criterion and Metric Outer Measures.- 5.2 Arclength & Arclength Measure: The Rest of the Story.- 5.3 A Vitali Covering Lemma and Planar Lebesgue Measure.- 5.4 Regularity Properties of Hausdor Measures.- 5.5 The Besicovitch Covering Lemma and Lebesgue Points.- 6 A Solution to Vitushkin's Conjecture Modulo Two Difficult Results.- 6.1 Statement of the Conjecture and a Reduction.- 6.2 Cauchy Integral Representation.- 6.3 Estimates of Truncated Cauchy Integrals.- 6.4 Estimates of Truncated Suppressed Cauchy Integrals.- 6.5 Vitushkin's Conjecture Resolved Affirmatively Modulo Two Difficult Results.- 6.6 Postlude: The Original Vitushkin Conjecture.- 7 The T(b) Theorem of Nazarov, Treil, and Volberg.- 7.1 Restatement of the Result.- 7.2 Random Dyadic Lattice Construction.- 7.3 Lip(1)-Functions Attached to Random Dyadic Lattices.- 7.4 Construction of the Lip(1)-Function of the Theorem.- 7.5 The Standard Martingale Decomposition.- 7.6 Interlude: The Dyadic Carleson Imbedding Inequality.- 7.7 The Adapted Martingale Decomposition.- 7.8 Bad Squares and Their Rarity.- 7.9 The Good/Bad-Function Decomposition.- 7.10 Reduction to the Good Function Estimate.- 7.11 A Sticky Point, More Reductions, and Course Setting.- 7.12 Interlude: The Schur Test.- 7.13 G1: The Crudely Handled Terms.- 7.14 G2: The Distantly Interacting Terms.- 7.15 Splitting Up the G3 Terms.- 7.16 Gterm 3 : The Suppressed Kernel Terms.- 7.17 Gtran 3 : The Telescoping Terms.- 8 The Curvature Theorem of David and Léger.- 8.1 Restatement of the Result and an Initial Reduction.- 8.2 Two Lemmas Concerning High Density Balls.- 8.3 The Beta Numbers of Peter Jones.- 8.4 Domination of Beta Numbers by Local Curvature.- 8.5 Domination of Local Curvature by Global Curvature.- 8.6 Selection of Parameters for the Construction.- 8.7 Construction of a Baseline L0.- 8.8 De nition of a Stopping-Time Region S0.- 8.9 De nition of a Lipschitz Set K0 over the Base Line.- 8.10 Construction of Adapted Dyadic Intervals on the Base Line.- 8.11 Assigning Linear Functions to Adapted Dyadic Intervals.- 8.12 Construction of a Lipschitz Graph G Threaded through K0.- 8.13 Veri cation that the Graph is Indeed Lipschitz.- 8.14 A Partition of K n K0 into Three Sets: K1, K2, & K3.- 8.15 The Smallness of the Set K2.- 8.16 The Smallness of a Horrible Set H.- 8.17 Most of K Lies in the Vicinity of the Lipschitz Graph.- 8.18 The Smallness of the Set K1.- 8.19 Gamma Functions of the Lipschitz Graph.- 8.20 A Point Estimate on One of the Gamma Functions.- 8.21 A Global Estimate on the Other Gamma Function.- 8.22 Interlude: Calderon's Formula.- 8.23 A Decomposition of the Lipschitz Function.- 8.24 The Smallness of the Set K3.- Postscript.- Bibliography.- Symbol Glossary & List.- Index.