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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 Gromov-Hausdorff convergence, and the Keith-Zhong self-improvement theorem for Poincar inequalities.
- Format: Inbunden
- ISBN: 9781107092341
- Språk: Engelska
- Antal sidor: 448
- Utgivningsdatum: 2015-02-05
- Förlag: Cambridge University Press