Bump Functions with Hölder Derivatives

[摘要] We study the range of the gradients of a $C^{1,al}$-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 $X^{ast}$ 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 $X^{ast}$ containing $0$ is the setof the gradients of a $C^{1,1}$-smooth bump function.

[学科分类] 数学（综合）