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When is the Haar measure a Pietsch measure for nonlinear mappings? (English) Zbl 1276.28028
Authors’ abstract: “We show that, as in the linear case, the normalized Haar measure on a compact topological group $$G$$ is a Pietsch measure for nonlinear summing mappings on closed translation invariant subspaces of $$C(G)$$. We also show that our result applies to several well-studied classes of nonlinear summing mappings. In the final section, some problems are proposed.”
One of the open problems reads as follows.
Let $$F$$ be a closed translation invariant subspace of $$C(G)$$, let $$X$$ be a metric space and $$f: F\to X$$ be a translation invariant Lipschitz $$p$$-summing mapping. Is the Haar measure a Pietsch measure for $$f$$?

##### MSC:
 28C10 Set functions and measures on topological groups or semigroups, Haar measures, invariant measures 47B10 Linear operators belonging to operator ideals (nuclear, $$p$$-summing, in the Schatten-von Neumann classes, etc.)
##### Keywords:
Haar measure; Pietsch measure; nonlinear mapping
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