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

Published Cambridge : Cambridge University Press, 2015 ©2015

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Description 1 online resource (xii, 434 pages) Series New mathematical monographs ; 27 New mathematical monographs ; 27. Contents 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 Summary 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 Bibliography Includes bibliographical references and indexes Notes Print version record Subject Metric spaces. Sobolev spaces. MATHEMATICS -- Calculus. MATHEMATICS -- Mathematical Analysis. Sobolev, Espacios de Metric spaces Sobolev spaces Sobolev-Raum Metrischer Raum Form Electronic book Author Heinonen, Juha, author. Koskela, Pekka, author. Shanmugalingam, Nageswari, author. Tyson, Jeremy T., 1972- author. ISBN 9781316248607 1316248607 9781316250495 1316250490 9781316135914 1316135918

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