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 Poincar 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 Poincar 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 Poincar Inequalities Under GromovHausdorff Convergence And The KeithZhong Self-Improvement Theorem For Poincar Inequalities.