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 
T2  - New mathematical monographs
SN  - 9781316135914 (ebook)
AV  - QA611.28 .H45 2015
U1  - 515/.7 23
PY  - 2015///
CY  - Cambridge
PB  - Cambridge University Press
KW  - Metric spaces
KW  - Sobolev spaces
N1  - Title from publisher's bibliographic system (viewed on 05 Oct 2015); Introduction -- Review of basic functional analysis -- Lebesgue theory of Banach space-valued functions -- Lipschitz functions and embeddings -- Path integrals and modulus -- Upper gradients -- Sobolev spaces -- Poincaré inequalities -- Consequences of Poincaré inequalities -- Other definitions of Sobolev-type spaces -- Gromov-Hausdorff convergence and Poincaré inequalities -- Self-improvement of Poincaré inequalities -- An introduction to Cheeger's differentiation theory -- Examples, applications, and further research directions
N2  - 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
UR  - https://doi.org/10.1017/CBO9781316135914
ER  -