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181005 ||| eng |
| 020 |
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|a 9781316135914
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| 050 |
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4 |
|a QA611.28
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| 100 |
1 |
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|a Heinonen, Juha
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| 245 |
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|a Sobolev spaces on metric measure spaces
|b an approach based on upper gradients
|c Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, Jeremy T. Tyson
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| 260 |
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|a Cambridge
|b Cambridge University Press
|c 2015
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| 300 |
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|a xii, 434 pages
|b digital
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| 505 |
0 |
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|a 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
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| 653 |
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|a Metric spaces
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| 653 |
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|a Sobolev spaces
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| 700 |
1 |
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|a Koskela, Pekka
|e [author]
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| 700 |
1 |
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|a Shanmugalingam, Nageswari
|e [author]
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| 700 |
1 |
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|a Tyson, Jeremy T.
|e [author]
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| 041 |
0 |
7 |
|a eng
|2 ISO 639-2
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| 989 |
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|b CBO
|a Cambridge Books Online
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| 490 |
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|a New mathematical monographs
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| 028 |
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|a 10.1017/CBO9781316135914
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| 856 |
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|u https://doi.org/10.1017/CBO9781316135914
|x Verlag
|3 Volltext
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| 082 |
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|a 515.7
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| 520 |
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|a 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
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