rdf-syntax-grammar
mods:
  schema location: http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
  title info:
    title: Analytic Capacity, the Cauchy Transform, and Non-homogeneous Calderón–Zygmund Theory
  name:
    name part: Tolsa, Xavier.
    role:
      role term: creator
    role:
      role term: author.
  name:
    name part: SpringerLink (Online service)
  type of resource: text
  origin info:
    place:
      place term: gw
    date issued: 2014
    issuance: monographic
  language:
    language term: eng
  physical description:
    form: electronic
    form: electronic resource
    reformatting quality: access
    form: remote
    extent: XIII, 396 p. 8 illus. online resource.
  abstract: 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.
  table of contents: Introduction -- Basic notation -- Chapter 1. Analytic capacity -- Chapter 2. Basic Calderón-Zygmund theory with non doubling measures -- Chapter 3. The Cauchy transform and Menger curvature -- Chapter 4. The capacity γ+ -- Chapter 5. A Tb theorem of Nazarov, Treil and Volberg -- Chapter 6. The comparability between γ and γ +, and the semiadditivity of analytic capacity -- Chapter 7. Curvature and rectifiability -- Chapter 8. Principal values for the Cauchy transform and rectifiability -- Chapter 9. RBMO(μ) and H1 atb(μ) -- Bibliography -- Index.
  note: by Xavier Tolsa.
  subject:
    topic: Mathematics
  subject:
    topic: Functions of complex variables
  subject:
    topic: Potential theory (Mathematics)
  subject:
    topic: Calculus of variations
  subject:
    topic: Mathematics
  subject:
    topic: Functions of a Complex Variable
  subject:
    topic: Potential Theory
  subject:
    topic: Calculus of Variations and Optimal Control; Optimization
  classification: QA331-355
  classification: 515.9
  related item:
    title info:
      title: Springer eBooks
  related item:
    title info:
      title: Progress in Mathematics, 307
  identifier: 9783319005966
  identifier: http://dx.doi.org/10.1007/978-3-319-00596-6
  record info:
    record creation date: 131216
    record change date: 20151031091119.0
    record identifier: 978-3-319-00596-6