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Differentiability and Poincaré type inequalities in metric measure spaces

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Bate, David and Li, Sean (2018) Differentiability and Poincaré type inequalities in metric measure spaces. Advances in Mathematics, 333 . pp. 868-930. doi:10.1016/j.aim.2018.06.002 ISSN 0001-8708.

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Official URL: https://doi.org/10.1016/j.aim.2018.06.002

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Abstract

We demonstrate the necessity of a Poincaré type inequality for those metric measure spaces that satisfy Cheeger's generalization of Rademacher's theorem for all Lipschitz functions taking values in a Banach space with the Radon–Nikodym property. This is done by showing the existence of a rich structure of curve fragments that connect nearby points, similar in nature to Semmes's pencil of curves for the standard Poincaré inequality. Using techniques similar to Cheeger–Kleiner [12], we show that our conditions are also sufficient.

We also develop another characterization of RNP Lipschitz differentiability spaces by connecting points by curves that form a rich structure of partial derivatives that were first discussed in [5].

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH): Lipschitz spaces, Poincaré conjecture, Banach spaces, Banach spaces -- Radon-Nikodym property
Journal or Publication Title: Advances in Mathematics
Publisher: Academic Press
ISSN: 0001-8708
Official Date: 31 July 2018
Dates:
DateEvent
31 July 2018Published
18 June 2018Available
27 May 2018Accepted
Volume: 333
Page Range: pp. 868-930
DOI: 10.1016/j.aim.2018.06.002
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Date of first compliant deposit: 25 October 2019
Date of first compliant Open Access: 30 October 2019
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