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1. CJM 2004 (vol 56 pp. 699)
Bump Functions with HÃ¶lder Derivatives 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 set
of the gradients of a $C^{1,1}$-smooth bump function.
Keywords:Banach space, bump function, range of the derivative Categories:46T20, 26E15, 26B05 |
2. CJM 2003 (vol 55 pp. 969)
Lie Groups of Measurable Mappings We describe new construction principles for infinite-dimensional Lie
groups. In particular, given any measure space $(X,\Sigma,\mu)$ and
(possibly infinite-dimensional) Lie group $G$, we construct a Lie
group $L^\infty (X,G)$, which is a Fr\'echet-Lie group if $G$ is so.
We also show that the weak direct product $\prod^*_{i\in I} G_i$ of an
arbitrary family $(G_i)_{i\in I}$ of Lie groups can be made a Lie
group, modelled on the locally convex direct sum $\bigoplus_{i\in I}
L(G_i)$.
Categories:22E65, 46E40, 46E30, 22E67, 46T20, 46T25 |