Analytic Capacity, Rectifiability, Menger Curvature and Cauchy Integral

1. Front Matter

   Pages i-xii

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2. 1. Some geometric measure theory

   • Hervé Pajot

   Pages 1-15

3. 2. P. Jones’ traveling salesman theorem

   • Hervé Pajot

   Pages 17-27

4. 3. Menger curvature

   • Hervé Pajot

   Pages 29-54

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

   • Hervé Pajot

   Pages 55-65

6. 5. Analytic capacity and the Painlevé problem

   • Hervé Pajot

   Pages 67-79

7. 6. The Denjoy and Vitushkin conjectures

   • Hervé Pajot

   Pages 81-103

8. 7. The capacity \(\gamma_{ + }\) and the Painlevé Problem

   • Hervé Pajot

   Pages 105-114

9. Bibliography

   • Hervé Pajot

   Pages 115-118

10. Index

    • Hervé Pajot

    Pages 119-119

11. Back Matter

    Pages 119-125

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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 Painlevé problem.

• Cauchy integral
• Complex analysis
• Singular integral
• analysis
• analytic capacity
• harmonic analysis
• menger curvature
• rectifiability
• singular integral operators

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