Sobolev Spaces on Metric Measure Spaces
An Approach Based on Upper Gradients

This coherent treatment from first principles is an ideal introduction for graduate students and a useful reference for experts.

Juha Heinonen (Author), Pekka Koskela (Author), Nageswari Shanmugalingam (Author), Jeremy T. Tyson (Author)

9781107092341, Cambridge University Press

Hardback, published 5 February 2015

448 pages, 4 b/w illus.
23.4 x 15.7 x 3.3 cm, 0.65 kg

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 Poincaré 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 Poincaré 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 Poincaré inequalities under Gromov–Hausdorff convergence, and the Keith–Zhong self-improvement theorem for Poincaré inequalities.

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. Poincaré inequalities
9. Consequences of Poincaré inequalities
10. Other definitions of Sobolev-type spaces
11. Gromov–Hausdorff convergence and Poincaré inequalities
12. Self-improvement of Poincaré inequalities
13. An Introduction to Cheeger's differentiation theory
14. Examples, applications and further research directions
References
Notation index
Subject index.

Subject Areas: Functional analysis & transforms [PBKF], Complex analysis, complex variables [PBKD], Calculus & mathematical analysis [PBK], Mathematics [PB]