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Vitushkin's Conjecture for Removable Sets : Universitext - James J. Dudziak

Vitushkin's Conjecture for Removable Sets



Published: 23rd September 2010
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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 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 meausre, 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)

Preface: Painlevé's Problemp. ix
Removable Sets and Analytic Capacityp. 1
Removable Setsp. 1
Analytic Capacityp. 9
Removable Sets and Hausdorff Measurep. 19
Hausdorff Measure and Dimensionp. 19
Painlevé's Theoremp. 24
Frostman's Lemmap. 26
Conjecture and Refutation: The Planar Cantor Quarter Setp. 30
Garabedian Duality for Hole-Punch Domainsp. 39
Statement of the Result and an Initial Reductionp. 39
Interlude: Boundary Correspondence for H∞(U)p. 42
Interlude: An F. & M. Riesz Theoremp. 47
Construction of the Boundary Garabedian Functionp. 50
Construction of the Interior Garabedian Functionp. 51
A Further Reductionp. 52
Interlude: Some Extension and Join Propositionsp. 53
Analytically Extending the Ahlfors and Garabedian Functionsp. 59
Interlude: Consequences of the Argument Principlep. 62
An Analytic Logarithm of the Garabedian Functionp. 66
Melnikov and Verdera's Solution to the Denjoy Conjecturep. 69
Menger Curvature of Point Triplesp. 69
Melnikov's Lower Capacity Estimatep. 71
Interlude: A Fourier Transform Reviewp. 78
Melnikov Curvature of Some Measures on Lipschitz Graphsp. 82
Arclength and Arclength Measure: Enough to Do the Jobp. 86
The Denjoy Conjecture Resolved Affirmativelyp. 92
Conjecture and Refutation: The Joyce-Mörters Setp. 95
Some Measure Theoryp. 105
The Carathéodory Criterion and Metric Outer Measuresp. 105
Arclength and Arclength Measure: The Rest of the Storyp. 109
Vitali's Covering Lemma and Planar Lebesgue Measurep. 113
Regularity Properties of Hausdorff Measuresp. 120
Besicovitch's Covering Lemma and Lebesgue Pointsp. 124
A Solution to Vitushkin's Conjecture Modulo Two Difficult Resultsp. 131
Statement of the Conjecture and a Reductionp. 131
Cauchy Integral Representationp. 137
Estimates of Truncated Cauchy Integralsp. 140
Estimates of Truncated Suppressed Cauchy Integralsp. 143
Vitushkin's Conjecture Resolved Affirmatively Modulo Two Difficult Resultsp. 146
Postlude: Vitushkin's Original Conjecturep. 154
The T(b) Theorem of Nazarov, Treil, and Volbergp. 159
Restatement of the Resultp. 159
Random Dyadic Lattice Constructionp. 160
Lip(1)-Functions Attached to Random Dyadic Latticesp. 161
Construction of the Lip(1)-Function of the Theoremp. 162
The Standard Martingale Decompositionp. 164
Interlude: The Dyadic Carleson Imbedding Inequalityp. 169
The Adapted Martingale Decompositionp. 172
Bad Squares and Their Rarityp. 181
The Good/Bad-Function Decompositionp. 185
Reduction to the Good Function Estimatep. 186
A Sticky Point, More Reductions, and Course Settingp. 190
Interlude: The Schur Testp. 195
G1: The Crudely Handled Termsp. 197
G2: The Distantly Interacting Termsp. 202
Splitting Up the G3 Termsp. 208
G3term: The Suppressed Kernel Termsp. 209
G3tran: The Telescoping Termsp. 213
The Curvature Theorem of David and Légerp. 221
Restatement of the Result and an Initial Reductionp. 221
Two Lemmas Concerning High-Density Ballsp. 224
The Beta Numbers of Peter Jonesp. 228
Domination of Beta Numbers by Local Curvaturep. 232
Domination of Local Curvature by Global Curvaturep. 236
Selection of Parameters for the Constructionp. 238
Construction of a Baseline L0p. 241
Definition of a Stopping-Time Region S0p. 241
Definition of a Lipschitz Set K0 over L0p. 244
Construction of Adapted Dyadic Intervals {In}p. 248
Assigning a Good Linear Function ln to Each Inp. 250
Construction of a Function l Whose Graph Contains K0p. 253
Verification That l is Lipschitzp. 256
A Partition of KK0 into Three Sets: K1, K2, and K3p. 262
The Smallness of K2p. 263
The Smallness of a Horrible Set Hp. 264
Most of K Lies in the Vicinity of p. 267
The Smallness of K1p. 271
Gamma Functions Associated with lp. 273
A Point Estimate on One of the Gamma Functionsp. 275
A Global Estimate on the Other Gamma Functionp. 285
Interlude: Calderón's Formulap. 287
A Decomposition of lp. 295
The Smallness of K3p. 302
Postscript: Tolsa's Theoremp. 311
Bibliographyp. 317
Symbol Glossary and Listp. 321
Indexp. 327
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9781441967084
ISBN-10: 1441967087
Series: Universitext
Audience: Professional
Format: Paperback
Language: English
Number Of Pages: 332
Published: 23rd September 2010
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.5 x 15.5  x 1.83
Weight (kg): 1.07