Sobolev spaces on metric measure spaces : an approach based on upper gradients
著者
書誌事項
Sobolev spaces on metric measure spaces : an approach based on upper gradients
（New mathematical monographs, 27）
Cambridge University Press, 2015
- : hardback
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注記
Other authors: Pekka Koskela, Nageswari Shanmugalingam, Jeremy T. Tyson
Includes bibliographical references (p. 412-426) and indexes
内容説明・目次
内容説明
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.
目次
- 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.
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