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Directional upper derivatives and the chain rule formula for locally Lipschitz functions on Banach spaces


Authors: Olga Maleva and David Preiss
Journal: Trans. Amer. Math. Soc. 368 (2016), 4685-4730
MSC (2010): Primary 46G05; Secondary 26B30, 58C20
DOI: https://doi.org/10.1090/tran/6480
Published electronically: November 18, 2015
MathSciNet review: 3456158
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Abstract: Motivated by an attempt to find a general chain rule formula for differentiating the composition $ f\circ g$ of Lipschitz functions $ f$ and $ g$ that would be as close as possible to the standard formula $ (f\circ g)'(x) = f'(g(x))\circ g'(x)$, we show that this formula holds without any artificial assumptions provided derivatives are replaced by complete derivative assignments. The idea behind these assignments is that the derivative of $ f$ at $ y$ is understood as defined only in the direction of a suitable ``tangent space'' $ U(f,y)$ (and so it exists at every point), but these tangent spaces are chosen in such a way that for any $ g$ they contain the range of $ g'(x)$ for almost every $ x$. Showing the existence of such assignments leads us to a detailed study of derived sets and the ways in which they describe pointwise behavior of Lipschitz functions.


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Additional Information

Olga Maleva
Affiliation: School of Mathematics, University of Birmingham, Birmingham, B15 2TT, United Kingdom
Email: O.Maleva@bham.ac.uk

David Preiss
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
Email: D.Preiss@warwick.ac.uk

DOI: https://doi.org/10.1090/tran/6480
Received by editor(s): February 9, 2014
Received by editor(s) in revised form: May 14, 2014
Published electronically: November 18, 2015
Additional Notes: The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.2011-ADG-20110209
Article copyright: © Copyright 2015 American Mathematical Society