When is the Haar measure a Pietsch measure for nonlinear mappings? (English) Zbl 1276.28028

Botelho, Geraldo; Pellegrino, Daniel; Rueda, Pilar; Santos, Joedson; Seoane-Sepúlveda, Juan Benigno

Stud. Math. 213, No. 3, 275-287 (2012).

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 \(f\)?

Reviewer: Joe Howard (Portales)

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