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This paper deals with the inequalities devoted to the comparison between thenorm of a function on a compact hypergroup and the norm of its Fouriercoefficients. We prove the classical Paley inequality in the setting of compacthypergroups which further gives the Hardy-Littlewood and Hausdorff-Young-Paley(Pitt) inequalities in the noncommutative context. We establish H\"ormander's$L^p$-$L^q$ Fourier multiplier theorem on compact hypergroups for $1<p \leq 2\leq q<\infty$ as an application of Hausdorff-Young-Paley inequality. Weexamine our results for the hypergroups constructed from the conjugacy classesof compact Lie groups and for a class of countable compact hypergroups.

Hardy-Littlewood inequality and $L^p$-$L^q$ Fourier multipliers on compact hypergroups