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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

References

[1] Azagra, D. and Deville, R., James’ theorem fails for starlike bodies. J. Funct. Anal. 180(2001), 328–346.Google Scholar
[2] Azagra, D., Fabian, M. and Jimenez-Sevilla, M., Exact filling of figures with the derivatives of smooth mappings. Preprint (2001).Google Scholar
[3] Borwein, J. M., Fabian, M., Kortezov, I. and Loewen, P. D., The range of the gradient of a continuously differentiable bump. J. Nonlinear Convex Anal. 2(2001), 1–19.Google Scholar
[4] Borwein, J. M., Fabian, M. and Loewen, P. D., The range of the gradient of a Lipschitz C 1 -smooth bump in infinite dimensions. Israel J. Math. 132(2002), 239–251.Google Scholar
[5] Dieudonné, J., Fundations of Modern Analysis. Academic Press, New York, 1969.Google Scholar
[6] Deville, R., Godefroy, G. and Zizler, V., Smoothness and renormings in Banach spaces. Pitman Monographs and Surveys in Pure and Applied Mathematic 64, Wiley, New York, 1993.Google Scholar
[7] Fabian, M., Kalenda, O. and Kolář, J., Filling analytic sets by the derivatives of C 1 -smooth bumps. Preprint (2002).Google Scholar
[8] Gaspari, T., On the range of the derivative of a real valued function with bounded support. Studia Math. 153(2002), 81–99.Google Scholar
[9] Kolář, J., Kristensen, J., The set of gradients of a bump. Preprint (2002).Google Scholar
[10] Malý, J., The Darboux property for gradients. Real Anal. Exchange 22(1996–1997), 167–173.Google Scholar
[11] Rifford, L., Range of the gradient of a smooth bump function in finite dimensions. Proc. Amer. Math. Soc. 131(2003), 3063–3066.Google Scholar
[12] Saint-Raymond, J., Local inversion for differentiable functions and Darboux property. Preprint (2001).Google Scholar