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Structure of measures in Lipschitz differentiability spaces


Author: David Bate
Journal: J. Amer. Math. Soc. 28 (2015), 421-482
MSC (2010): Primary 30L99; Secondary 49J52, 53C23
DOI: https://doi.org/10.1090/S0894-0347-2014-00810-9
Published electronically: June 4, 2014
MathSciNet review: 3300699
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Abstract: We prove the equivalence of two seemingly very different ways of generalising Rademacher's theorem to metric measure spaces. One such generalisation is based upon the notion of forming partial derivatives along a very rich structure of Lipschitz curves in a way analogous to the differentiability theory of Euclidean spaces. This approach to differentiability in this generality appears here for the first time and by examining this structure further, we naturally arrive to several descriptions of Lipschitz differentiability spaces.


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Additional Information

David Bate
Affiliation: Department of Mathematics, Mathematics Institute, Zeeman Building, University of Warwick, Coventry, CV4 7AL, UK
Address at time of publication: Department of Mathematics, University of Chicago, 5734 S. University Avenue, Room 208C, Chicago, Illinois 60637
Email: d.s.bate@warwick.ac.uk, bate@math.uchicago.edu

DOI: https://doi.org/10.1090/S0894-0347-2014-00810-9
Received by editor(s): August 10, 2012
Received by editor(s) in revised form: February 6, 2014
Published electronically: June 4, 2014
Additional Notes: I would like to thank David Preiss for his dedicated reading of this manuscript and for our insightful conversations throughout my time as his student. I would also like to thank Guy C. David (UCLA) for spotting a technical error in the first version of Section 6. This work was supported by the EPSRC
Article copyright: © Copyright 2014 American Mathematical Society