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A structural criterion for the existence of infinite central \(\Lambda(p)\) sets. (English) Zbl 0796.43003

Let \(G\) be a compact group with dual \(\widehat{G}\), let \(p \in (2,\infty)\). \(E \subseteq \widehat{G}\) is a (central) \(\Lambda(p)\) set if \(\| f\|_ p \leq C_ p\| f\|_ 2\) for some \(C_ p \geq 0\) and for all (central) trigonometric polynomials \(f\) with Fourier transform \(\widehat{f}\) supported by \(E\) [cf. E. Hewitt, K. A. Ross: Abstract harmonic analysis. II (1970; Zbl 0213.401)]. For \(E \subset \widehat G\), \(m \in \mathbb{N}\) let \(E^ m\) be the set of representations in \(\widehat G\) contained in an \(n\)-fold tensor product of elements of \(E\). \(G\) is called tall if \(\widehat{G}\) contains at most finitely many elements of each degree.
A main result of the paper under review is the description of compact connected groups admitting infinite central \(\Lambda(4)\) sets (Theorem 5.5). For tall groups (Corollary 5.4) locally central \(\Lambda(4)\) sets are central \(\Lambda(p)\), \(p \in (2,\infty)\). The essential tools is a profound analysis of \(m\)-fold FTR-sets (Figá-Talamanca-Rider sets) which are defined via an explicit representation of the covering group of a compact connected group.
Reviewer: W.Hazod (Dortmund)

MSC:

43A46 Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.)
43A75 Harmonic analysis on specific compact groups
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.

Citations:

Zbl 0213.401
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