Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients (New Mathematical Monographs, Series Number 27)

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. Content: 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. Analysis on metric spaces is a field that has expanded dramatically since the 1990s. Written by some of the founders of the theory, this book provides a coherent treatment from first principles. It is an ideal introduction for graduate students and a useful reference for experts in the field.