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Hardy-Littlewood inequality and \(L^p\)-\(L^q\) Fourier multipliers on compact hypergroups. (English) Zbl 1497.43007

Summary: This paper deals with the inequalities comparing the norm of a function on a compact hypergroup and the norm of its Fourier coefficients. We prove the classical Paley inequality in the setting of compact hypergroups which further gives the Hardy-Littlewood and Hausdorff-Young-Paley inequalities in the noncommutative context. We establish Hörmander’s \(L^p\)-\(L^q\) Fourier multiplier theorem on compact hypergroups for \(1<p \leq 2 \leq q < \infty\) as an application of the Hausdorff-Young-Paley inequality. We examine our results for the hypergroups constructed from the conjugacy classes of compact Lie groups and for a class of countable compact hypergroups.

MSC:

43A62 Harmonic analysis on hypergroups
43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
43A90 Harmonic analysis and spherical functions
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