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Bump Functions with Hölder Derivatives

Published online by Cambridge University Press:  20 November 2018

Thierry Gaspari
Thierry Gaspari*
Affiliation:
Mathématiques Pures de Bordeaux, UMR 5467 CNRS, Université Bordeaux 1 351, cours de la Libération, 33400 Talence, France e-mail: gaspari@math.u-bordeaux.fr
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Abstract

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We study the range of the gradients of a ${{C}^{1,\alpha }}$ -smooth bump function defined on a Banach space. We find that this set must satisfy two geometrical conditions: It can not be too flat and it satisfies a strong compactness condition with respect to an appropriate distance. These notions are defined precisely below. With these results we illustrate the differences with the case of ${{C}^{1}}$-smooth bump functions. Finally, we give a sufficient condition on a subset of ${{X}^{*}}$ so that it is the set of the gradients of a ${{C}^{1,1}}$-smooth bump function. In particular, if $X$ is an infinite dimensional Banach space with a ${{C}^{1,1}}$-smooth bump function, then any convex open bounded subset of ${{X}^{*}}$ containing 0 is the set of the gradients of a ${{C}^{1,1}}$-smooth bump function.

Keywords

46T2026E1526B05Banach spacebump functionrange of the derivative
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 56 , Issue 4 , 01 August 2004 , pp. 699 - 715
Copyright
Copyright © Canadian Mathematical Society 2004

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