Sobolev spaces on metric measure spaces : an approach based on upper gradients

Author(s)

• Heinonen, Juha
• Koskela, Pekka
• Shanmugalingam, Nageswari
• Tyson, Jeremy T.

Bibliographic Information

Sobolev spaces on metric measure spaces : an approach based on upper gradients

Juha Heinonen ... [et al.]

（New mathematical monographs, 27）

Cambridge University Press, 2015

• : hardback

Note

Other authors: Pekka Koskela, Nageswari Shanmugalingam, Jeremy T. Tyson

Includes bibliographical references (p. 412-426) and indexes

Description and Table of Contents

Description

Analysis on metric spaces emerged in the 1990s as an independent research field providing a unified treatment of first-order analysis in diverse and potentially nonsmooth settings. Based on the fundamental concept of upper gradient, the notion of a Sobolev function was formulated in the setting of metric measure spaces supporting a Poincare inequality. This coherent treatment from first principles is an ideal introduction to the subject for graduate students and a useful reference for experts. It presents the foundations of the theory of such first-order Sobolev spaces, then explores geometric implications of the critical Poincare inequality, and indicates numerous examples of spaces satisfying this axiom. A distinguishing feature of the book is its focus on vector-valued Sobolev spaces. The final chapters include proofs of several landmark theorems, including Cheeger's stability theorem for Poincare inequalities under Gromov-Hausdorff convergence, and the Keith-Zhong self-improvement theorem for Poincare inequalities.

Table of Contents

• Preface
• 1. Introduction
• 2. Review of basic functional analysis
• 3. Lebesgue theory of Banach space-valued functions
• 4. Lipschitz functions and embeddings
• 5. Path integrals and modulus
• 6. Upper gradients
• 7. Sobolev spaces
• 8. Poincare inequalities
• 9. Consequences of Poincare inequalities
• 10. Other definitions of Sobolev-type spaces
• 11. Gromov-Hausdorff convergence and Poincare inequalities
• 12. Self-improvement of Poincare inequalities
• 13. An Introduction to Cheeger's differentiation theory
• 14. Examples, applications and further research directions
• References
• Notation index
• Subject index.

by "Nielsen BookData"

Details

• NCID: BB17909271
• ISBN:
  • 9781107092341
• Country Code: uk
• Title Language Code: eng
• Text Language Code: eng
• Place of Publication: Cambridge, U.K.
• Pages/Volumes: xii, 434 p.
• Size: 24 cm
• Classification:
  • LCC : QA611.28
  • DC23 : 515/.7
• Subject Headings:
  • LCSH : Metric spaces
  • LCSH : Sobolev spaces
• Parent Bibliography ID:
  • BA65869793