Sobolev Spaces on Metric Measure Spaces : An Approach Based on Upper Gradients Hardback
by Juha Heinonen, Pekka (University of Jyvaskyla, Finland) Koskela, Nageswari (University of Cincinnati) Shanmugalingam, Jeremy T. (University of Illinois, Urbana-Champaign) Tyson
Part of the New Mathematical Monographs series
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: 05/02/2015
- Category: Complex analysis, complex variables
- ISBN: 9781107092341