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Published in:
Preliminaries Read chapter Read first chapter

2020 | OriginalPaper | Chapter

2. Sobolev Calculus on Metric Measure Spaces

Authors : Nicola Gigli, Enrico Pasqualetto

Published in: Lectures on Nonsmooth Differential Geometry

Publisher: Springer International Publishing

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Abstract

Several different approaches to the theory of weakly differentiable functions over abstract metric measure spaces made their appearance in the literature throughout the last twenty years. Amongst them, we shall mainly follow the one (based upon the concept of test plan) that has been proposed by Ambrosio, Gigli and Savaré. The whole Sect. 2.1 is devoted to the definition of such notion of Sobolev space W 1, 2(X) and to its most important properties.

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previous chapter Preliminaries
next chapter The Theory of Normed Modules
Literature
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Ambrosio, L., Gigli, N., Savaré, G.: Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces. Rev. Mat. Iberoam. 29, 969–996 (2013)MathSciNetCrossRef
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Ambrosio, L., Gigli, N., Savaré, G.: Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Invent. Math. 195, 289–391 (2014)MathSciNetCrossRef
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Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9, 428–517 (1999)MathSciNetCrossRef
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Gigli, N.: Nonsmooth differential geometry - an approach tailored for spaces with Ricci curvature bounded from below. Mem. Am. Math. Soc. (2014, Accepted). arXiv:1407.0809
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Gigli, N.: On the differential structure of metric measure spaces and applications. Mem. Am. Math. Soc. 236, vi+91 (2015). arXiv:1205.6622MathSciNetCrossRef
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Koskela, P., MacManus, P.: Quasiconformal mappings and Sobolev spaces. Studia Math. 131, 1–17 (1998)MathSciNetMATH
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Levi, B.: Sul principio di Dirichlet. Rend. Circ. Mat. Palermo (1906)
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Shanmugalingam, N.: Newtonian spaces: an extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana 16, 243–279 (2000)MathSciNetCrossRef
Metadata
Title
Sobolev Calculus on Metric Measure Spaces
Authors
Nicola Gigli
Enrico Pasqualetto
Copyright Year
2020
Book
Lectures on Nonsmooth Differential Geometry

Print ISBN: 978-3-030-38612-2

Electronic ISBN: 978-3-030-38613-9

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-3-030-38613-9_2

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