Vitushkin's Conjecture for Removable Sets

Description

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 employ.;Preface: Painlevé's Problem; 1 Removable Sets and Analytic Capacity; 1.1 Removable Sets; 1.2 Analytic Capacity; 2 Removable Sets and Hausdorff Measure; 2.1 Hausdorff Measure and Dimension; 2.2 Painlevé's Theorem; 2.3 Frostman's Lemma; 2.4 Conjecture and 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 H(U); 3.3 Interlude: An F. & M. Riesz Theorem; 3.4 Construction of the Boundary Garabedian Function. Preface: Painlevé's Problem 1 Removable Sets and Analytic Capacity 1.1 Removable Sets 1.2 Analytic Capacity 2 Removable Sets and Hausdorff Measure 2.1 Hausdorff Measure and Dimension 2.2 Painlevé's Theorem 2.3 Frostman's Lemma 2.4 Conjecture and 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 H(U) 3.3 Interlude: An F. & M. Riesz Theorem 3.4 Construction of the Boundary Garabedian Function. 3.5 Construction of the Interior Garabedian Function3.6 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 and Arclength Measure: Enough to Do the Job4.6 The Denjoy Conjecture Resolved Affirmatively 4.7 Conjecture and Refutation: The Joyce -- Mörters Set 5 Some Measure Theory 5.1 The Carathéodory Criterion and Metric Outer Measures 5.2 Arclength and Arclength Measure: The Rest of the Story 5.3 Vitali's Covering Lemma and Planar Lebesgue Measure 5.4 Regularity Properties of Hausdorff Measures 5.5 Besicovitch's 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 Representation6.3 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: Vitushkin's Original 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 Inequality7.7 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 Gterm3: The Suppressed Kernel Terms 7.17 Gtran3: The Telescoping Terms 8 The Curvature Theorem of David and Léger.