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Sobolev Spaces on Metric Measure Spaces : An Approach Based on Upper Gradients, Hardback Book


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.


  • Format: Hardback
  • Pages: 448 pages, 4 Line drawings, unspecified
  • Publisher: Cambridge University Press
  • Publication Date:
  • Category: Complex analysis, complex variables
  • ISBN: 9781107092341



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