Sobolev spaces on metric measure spaces: An approach based on upper gradients

Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, Jeremy Tyson

Research output: Book/ReportBook

Abstract

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.

Original languageEnglish (US)
PublisherCambridge University Press
Number of pages434
ISBN (Electronic)9781316135914
ISBN (Print)9781107092341
DOIs
StatePublished - Jan 1 2015

Fingerprint

Metric Measure Space
Sobolev Spaces
Gradient
Gromov-Hausdorff Convergence
First-order
Stability Theorem
Landmarks
First-principles
Axiom
Theorem
Metric space

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Sobolev spaces on metric measure spaces : An approach based on upper gradients. / Heinonen, Juha; Koskela, Pekka; Shanmugalingam, Nageswari; Tyson, Jeremy.

Cambridge University Press, 2015. 434 p.

Research output: Book/ReportBook

Heinonen J, Koskela P, Shanmugalingam N, Tyson J. Sobolev spaces on metric measure spaces: An approach based on upper gradients. Cambridge University Press, 2015. 434 p. https://doi.org/10.1017/CBO9781316135914
Heinonen, Juha ; Koskela, Pekka ; Shanmugalingam, Nageswari ; Tyson, Jeremy. / Sobolev spaces on metric measure spaces : An approach based on upper gradients. Cambridge University Press, 2015. 434 p.
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