TY  - BOOK
AU  - Heinonen, Juha
AU  - Koskela, Pekka
AU  - Shanmugalingam, Nageswari
AU  - Tyson, Jeremy T.
TI  - Sobolev spaces on metric measure spaces: an approach based on upper gradients 
SN  - 9781107092341
U1  - 515.7 HEI-J 
PY  - 2015///
CY  - Cambridge  University
PB  - Cambridge University Press
KW  - Metric spaces
KW  - Sobolev spaces
KW  - MATHEMATICS -- Calculus
N1  - 	
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
ER  -