Analytic Capacity, Rectifiability, Menger Curvature and 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.

List of contents

Preface.- Notations and conventions.- Some geometric measures theory.- Jones' traveling salesman theorem.- Menger curvature.- The Cauchy singular integral operator on Ahlfors-regular sets.- Analytic capacity and the Painlevé Problem.- The Denjoy and Vitushkin conjectures.- The capacity $gamma (+)$ and the Painlevé Problem.- Bibliography.- Index.

Summary

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.

Product details

• Authors: H. M. Pajot, Hervé M Pajot, Hervé M. Pajot
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 03.12.2002
• EAN: 9783540000013
• ISBN: 978-3-540-00001-3
• No. of pages: 119
• Dimensions: 156 mm x 241 mm x 6 mm
• Weight: 240 g
• Illustrations: VIII, 119 p.
• Series: Lecture Notes in Mathematics; Lecture Notes in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Analysis; Analysis, B, Funktionalanalysis und Abwandlungen, Integralrechnung und -gleichungen, Komplexe Analysis, komplexe Variablen, Funktionentheorie, measure theory, geometry, Mathematics and Statistics, Complex analysis, complex variables, Integral calculus & equations, Analysis (Mathematics), Functional analysis & transforms, Mathematical analysis, Measure and Integration, Fourier Analysis, Functions of a Complex Variable, Functions of complex variables