Ebook: Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral

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.

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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 Painlev? problem.

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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 Painlev? problem.

Content:
1. Some geometric measure theory....Pages 1-15
2. P. Jones’ traveling salesman theorem....Pages 17-27
3. Menger curvature....Pages 29-54
4. The Cauchy singular integral operator on Ahlfors regular sets....Pages 55-65
5. Analytic capacity and the Painlev? problem....Pages 67-79
6. The Denjoy and Vitushkin conjectures....Pages 81-103
7. The capacity $gamma_{ + }$ and the Painlev? Problem....Pages 105-114
Bibliography....Pages 115-118
Index....Pages 119-119

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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 Painlev? problem.

Content:
1. Some geometric measure theory....Pages 1-15
2. P. Jones’ traveling salesman theorem....Pages 17-27
3. Menger curvature....Pages 29-54
4. The Cauchy singular integral operator on Ahlfors regular sets....Pages 55-65
5. Analytic capacity and the Painlev? problem....Pages 67-79
6. The Denjoy and Vitushkin conjectures....Pages 81-103
7. The capacity $gamma_{ + }$ and the Painlev? Problem....Pages 105-114
Bibliography....Pages 115-118
Index....Pages 119-119
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