Journal of Lie Theory 32 (2022), No. 2, 475--498
Copyright Heldermann Verlag 2022
Hardy-Littlewood Inequality and Lp-Lq Fourier Multipliers on Compact Hypergroups
Vishvesh Kumar
Dept. of Mathematics, Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium
vishveshmishra@gmail.com
Michael Ruzhansky
Department of Mathematics, Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium
and: School of Mathematical Sciences, Queen Mary University, London, United Kingdom
michael.ruzhansky@ugent.be
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 Lp-Lq Fourier multiplier theorem on compact hypergroups for 1 < p ≤ 2 ≤ q < ∞ 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.
Keywords: Paley inequality, Hardy-Littlewood inequality, Hausdorff-Paley inequality, compact hypergroups, conjugacy classes of compact Lie groups, Fourier multipliers, Lp-Lq boundedness, compact countable hypergroups.
MSC: 43A62, 43A22; 33C45, 43A90.
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