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