Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients
Juha Heinonen and Pekka Koskela, "Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients"
English | ISBN: 1107092345 | 2015 | 448 pages | PDF | 2 MB
English | ISBN: 1107092345 | 2015 | 448 pages | PDF | 2 MB
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.