Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and LP- Lq Fourier multipliers on compact homogeneous manifolds

In this paper we prove new inequalities describing the relationship between the "size" of a function on a compact homogeneous manifold and the "size" of its Fourier coefficients. These inequalities can be viewed as noncommutative versions of the Hardy-Littlewood inequalities obtained by Hardy and Littlewood [17] on the circle. For the example case of the group SU(2) we show that the obtained Hardy-Littlewood inequalities are sharp, yielding a criterion for a function to be in LP(SU(2)) in terms of its Fourier coefficients. We also establish Paley and HausdorffYoung-Paley inequalities on general compact homogeneous manifolds. The latter is applied to obtain conditions for the L-P-L-q boundedness of Fourier multipliers for 1 < p <= 2 <= q < infinity on compact homogeneous manifolds as well as the L-P-L-q boundedness of general (non-invariant) operators on compact Lie groups. We also record an abstract version of the Marcinkiewicz interpolation theorem on totally ordered discrete sets, to he used in the proofs with different Plancherel measures on the unitary duals. (C) 2019 Published by Elsevier Inc.