Description
CAMBRIDGE Sobolev Spaces on Metric Measure Spaces An Approach Based on Upper Gradients 2015 Edition by Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, Jeremy T. Tyson
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.