Sobolev Spaces on Metric Measure Spaces

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.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.