Sobolev Spaces on Metric Measure Spaces
An Approach Based on Upper Gradients
Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, Jeremy T. Tyson
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Sobolev Spaces on Metric Measure Spaces
An Approach Based on Upper Gradients
Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, Jeremy T. Tyson
Über dieses Buch
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.