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Uniform Partitions of unity on locally compact groups

✍ Scribed by Horst Leptin; Detlef Müller

Publisher
Elsevier Science
Year
1991
Tongue
English
Weight
615 KB
Volume
90
Category
Article
ISSN
0001-8708

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✦ Synopsis

Helffer and Nourrigat prove in [2] the following lemma (Lemma 4.52, p. 930): In every connected nilpotent group G there exists a discrete subset M and corresponding to M a non-negative smooth function cp with compact support in G such that 1 cpW)=l for all x E G, ucM i.e., the family ((P~}~~,,, of all translates @: x + cp(px) of cp forms a partition of unity on G. Partitions of this particular form we shall call uniform. The result of Helffer and Nourrigat raises two questions:

  1. Which locally compact groups do admit uniform partitions of unity?

  2. Can one describe the sets M which correspond to uniform partitions of unity?

The first question has a short answer: All. We shall see that the proof of this claim presents no particular difficulties. On the other hand, a comprehensive answer to the second question seems to be beyond the present possibilities. Therefore, we restrict our considerations to the simplest non-trivial case, namely to the group of reals: G = R. Here we can provide a complete solution of the problem: To Mc [w there exists a uniform partition of unity {(P~}~~~, if and only if M is the finite union of cosets of discrete subgroups, i.e., of subsets of the form aZ + /? with a > 0, fi E R.

Throughout the entire paper we shall use the following notations: G always denotes a locally compact group with left-invariant Haar measure dx, X(G) denotes the algebra of continuous real valued functions on G with compact support, XX+(G)= {cp~X(G);cp(x)>O for all XEG}.

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