Vitushkin's Conjecture for Removable Sets

Cover13; -- Contents -- Preface: Painlev233;'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 Painlev233;'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 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 and Arclength Measure: Enough to Do the Job -- 4.6 The Denjoy Conjecture Resolved Affirmatively -- 4.7 Conjecture and Refutation: The Joyce -- M246;rters Set -- 5 Some Measure Theory -- 5.1 The Carath233;odory 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 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: 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 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 Gterm3: The Suppressed Kernel Terms -- 7.17 Gtran3: The Telescoping Terms -- 8 The Curvature Theorem of David and L233;ger -- 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 Definition of a Stopping-Time Region S0 -- 8.9 Definition of a Lipschitz Set K0.