The Cauchy Integral Analytic Capacity And Uniform Rectifiability
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Analytic Capacity, Rectifiability, Menger Curvature and Cauchy Integral
Author | : Hervé M. Pajot |
Publisher | : Springer |
Total Pages | : 133 |
Release | : 2002-01-01 |
Genre | : Mathematics |
ISBN | : 3540360743 |
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.
Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón–Zygmund Theory
Author | : Xavier Tolsa |
Publisher | : Springer Science & Business Media |
Total Pages | : 402 |
Release | : 2013-12-16 |
Genre | : Mathematics |
ISBN | : 3319005960 |
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 Calderón-Zygmund theory, the development of which has been largely motivated by the problems arising in connection with analytic capacity. The Painlevé 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.
Analytic Capacity, Rectifiability, Menger Curvature and Cauchy Integral
Author | : Hervé Pajot |
Publisher | : Springer Science & Business Media |
Total Pages | : 140 |
Release | : 2002-11-26 |
Genre | : Mathematics |
ISBN | : 9783540000013 |
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.
Harmonic Analysis and Boundary Value Problems
Author | : Luca Capogna |
Publisher | : American Mathematical Soc. |
Total Pages | : 170 |
Release | : 2001 |
Genre | : Mathematics |
ISBN | : 0821827456 |
This volume presents research and expository articles by the participants of the 25th Arkansas Spring Lecture Series on ``Recent Progress in the Study of Harmonic Measure from a Geometric and Analytic Point of View'' held at the University of Arkansas (Fayetteville). Papers in this volume provide clear and concise presentations of many problems that are at the forefront of harmonic analysis and partial differential equations. The following topics are featured: the solution of the Kato conjecture, the ``two bricks'' problem, new results on Cauchy integrals on non-smooth curves, the Neumann problem for sub-Laplacians, and a new general approach to both divergence and nondivergence second order parabolic equations based on growth theorems. The articles in this volume offer both students and researchers a comprehensive volume of current results in the field.
Harmonic Analysis at Mount Holyoke
Author | : William Beckner |
Publisher | : American Mathematical Soc. |
Total Pages | : 474 |
Release | : 2003 |
Genre | : Mathematics |
ISBN | : 0821829033 |
This volume contains the proceedings of the conference on harmonic analysis and related areas. The conference provided an opportunity for researchers and students to exchange ideas and report on progress in this large and central field of modern mathematics. The volume is suitable for graduate students and research mathematicians interested in harmonic analysis and related areas.
Rectifiability
Author | : Pertti Mattila |
Publisher | : Cambridge University Press |
Total Pages | : 182 |
Release | : 2023-01-12 |
Genre | : Mathematics |
ISBN | : 1009288091 |
Rectifiable sets, measures, currents and varifolds are foundational concepts in geometric measure theory. The last four decades have seen the emergence of a wealth of connections between rectifiability and other areas of analysis and geometry, including deep links with the calculus of variations and complex and harmonic analysis. This short book provides an easily digestible overview of this wide and active field, including discussions of historical background, the basic theory in Euclidean and non-Euclidean settings, and the appearance of rectifiability in analysis and geometry. The author avoids complicated technical arguments and long proofs, instead giving the reader a flavour of each of the topics in turn while providing full references to the wider literature in an extensive bibliography. It is a perfect introduction to the area for researchers and graduate students, who will find much inspiration for their own research inside.
Perspectives in Partial Differential Equations, Harmonic Analysis and Applications
Author | : Dorina Mitrea |
Publisher | : American Mathematical Soc. |
Total Pages | : 446 |
Release | : 2008 |
Genre | : Mathematics |
ISBN | : 0821844245 |
This volume contains a collection of papers contributed on the occasion of Mazya's 70th birthday by a distinguished group of experts of international stature in the fields of harmonic analysis, partial differential equations, function theory, and spectral analysis, reflecting the state of the art in these areas.
Advanced Courses Of Mathematical Analysis Ii - Proceedings Of The Second International School
Author | : M Victoria Velasco |
Publisher | : World Scientific |
Total Pages | : 227 |
Release | : 2007-03-22 |
Genre | : Mathematics |
ISBN | : 9814478636 |
This volume comprises a collection of articles by leading researchers in mathematical analysis. It provides the reader with an extensive overview of new directions and advances in topics for current and future research in the field.
A Real Variable Method for the Cauchy Transform, and Analytic Capacity
Author | : Takafumi Murai |
Publisher | : Springer |
Total Pages | : 141 |
Release | : 2006-11-15 |
Genre | : Mathematics |
ISBN | : 3540391053 |
This research monograph studies the Cauchy transform on curves with the object of formulating a precise estimate of analytic capacity. The note is divided into three chapters. The first chapter is a review of the Calderón commutator. In the second chapter, a real variable method for the Cauchy transform is given using only the rising sun lemma. The final and principal chapter uses the method of the second chapter to compare analytic capacity with integral-geometric quantities. The prerequisites for reading this book are basic knowledge of singular integrals and function theory. It addresses specialists and graduate students in function theory and in fluid dynamics.