Directional upper derivatives and the chain rule formula for locally Lipschitz functions on Banach spaces
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- by Olga Maleva and David Preiss PDF
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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.References
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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
- MR Author ID: 141890
- Email: D.Preiss@warwick.ac.uk
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 4685-4730
- MSC (2010): Primary 46G05; Secondary 26B30, 58C20
- DOI: https://doi.org/10.1090/tran/6480
- MathSciNet review: 3456158