Analytic Capacity, rectifiability, Menger Curvature and the Cauchy Integral

Hervé Pajot

128 pages, parution le 31/12/2002

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Résumé

Based on a graduate course given by the author at Yale University this book deals with complex analysis (analytic capacity), geometric measure theory (rectifiable and uniformly rectifiable sets) and harmonic analysis (boundedness of singular integral operators on Ahlfors-regular sets). In particular, these notes contain a description of Peter Jones' geometric traveling salesman theorem, the proof of the equivalence between uniform rectifiability and boundedness of the Cauchy operator on Ahlfors-regular sets, the complete proofs of the Denjoy conjecture and the Vitushkin conjecture (for the latter, only the Ahlfors-regular case) and a discussion of X. Tolsa's solution of the Painlevé problem.

Contents

Chapter 1. Some geometric measure theory

1. Carleson measures
2. Lipschitz maps
3. Hausdorff dimension and Hausdorff measures
4. Density properties of Hausdorff measures
5. Rectifiable and purely unrectifiable sets

Chapter 2. P. Jones' traveling salesman theorem

1. The β numbers
2. Characterization of subsets of rectifiable curves
3. Uniformly rectifiable sets

Chapter 3. Menger curvature

1. Definition and basic properties
2. Menger curvature and Lipschitz graphs
3. Menger curvature and β numbers
4. Menger curvature and Cantor type sets
5. P. Jones' construction of "good" measures supported on continua

Chapter 4. The Cauchy singular integral operator on Ahlfors regular sets

1. The Hilbert transform
2. Singular integral operators
3. The Hardy-Littlewood maximal operator
4. The Calderon-Zygmund theory
5. The T1 and the Tb theorems
6. L2 boundedness of the Cauchy singular operator on Lipschitz graphs
7. Cauchy singular operator and rectifiability

Chapter 5. Analytic capacity and the Painlevé problem

1. Removable singularities
2. The Painlevé Problem
3. Some examples
4. Analytic capacity and metric size of sets
5. Garnett-Ivanov's counterexample
6. Who was Painlevé ?

Chapter 6. The Denjoy and Vitushkin conjectures

1. The statements
2. The standard duality argument
3. Proof of the Denjoy conjecture
4. Proof of the Vitushkin conjecture
5. The Vitushkin conjecture for sets with infinite length

Chapter 7. The capacity + and the Painlevé Problem

1. Melnikov's inequality
2. Tolsa's solution of the Painlevé problem
3. Concluding remarks and open problems

L'auteur - Hervé Pajot

University of Cergy-Pontoise, France

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Caractéristiques techniques

	PAPIER
Éditeur(s)	Springer
Auteur(s)	Hervé Pajot
Parution	31/12/2002
Nb. de pages	128
Format	15,5 x 23,5
Couverture	Broché
Poids	218g
Intérieur	Noir et Blanc
EAN13	9783540000013