Bump Functions with Hölder Derivatives

Gaspari, Thierry

Canadian journal of mathematics, Tome 56 (2004) no. 4, pp. 699-715

Résumé

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.

Détail

DOI : 10.4153/CJM-2004-032-3

Mots-clés : 46T20, 26E15, 26B05, Banach space, bump function, range of the derivative

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