Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderon-Zygmund Theory

Xavier Tolsa

396 pages, parution le 22/08/2016

Ajouter à une liste

Indisponible

Résumé

This book studies some of the groundbreaking advances that have been made regarding analytic capacity and its relationship to rectifiability in the decade 1995-2005.This book studies some of the groundbreaking advances that have been made regarding analytic capacity and its relationship to rectifiability in the decade 1995-2005. The Cauchy transform plays a fundamental role in this area and is accordingly one of the main subjects covered. Another important topic, which may be of independent interest for many analysts, is the so-called non-homogeneous Calderon-Zygmund theory, the development of which has been largely motivated by the problems arising in connection with analytic capacity. The Painleve problem, which was first posed around 1900, consists in finding a description of the removable singularities for bounded analytic functions in metric and geometric terms. Analytic capacity is a key tool in the study of this problem. In the 1960s Vitushkin conjectured that the removable sets which have finite length coincide with those which are purely unrectifiable. Moreover, because of the applications to the theory of uniform rational approximation, he posed the question as to whether analytic capacity is semiadditive. This work presents full proofs of Vitushkin's conjecture and of the semiadditivity of analytic capacity, both of which remained open problems until very recently. Other related questions are also discussed, such as the relationship between rectifiability and the existence of principal values for the Cauchy transforms and other singular integrals. The book is largely self-contained and should be accessible for graduate students in analysis, as well as a valuable resource for researchers.Introduction.- Basic notation.- Chapter 1. Analytic capacity.- Chapter 2. Basic Calderon-Zygmund theory with non doubling measures.- Chapter 3. The Cauchy transform and Menger curvature.- Chapter 4. The capacity +.- Chapter 5. A Tb theorem of Nazarov, Treil and Volberg.- Chapter 6. The comparability between and +, and the semiadditivity of analytic capacity.- Chapter 7. Curvature and rectifiability.- Chapter 8. Principal values for the Cauchy transform and rectifiability.- Chapter 9. RBMO( ) and H1 atb( ).- Bibliography.- Index.Xavier Tolsa is Research Professor of Mathematics from ICREA - Universitat Autonoma de Barcelona. He is the author of many research papers in connection with the topics discussed in this book. The present monograph was awarded the 2013 Ferran Sunyer i Balaguer Prize.

Caractéristiques techniques

	PAPIER
Éditeur(s)	Springer
Auteur(s)	Xavier Tolsa
Parution	22/08/2016
Nb. de pages	396
EAN13	9783319345444

Avantages Eyrolles.com

Livraison à partir de 0,01 € en France métropolitaine

Paiement en ligne SÉCURISÉ

Livraison dans le monde

Retour sous 15 jours

+ d'un million et demi de livres disponibles

Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderon-Zygmund Theory

Résumé

Caractéristiques techniques

Consultez aussi