  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
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 $L^p$ on SU(2) in terms of its Fourier coefficients. We also
establish Paley and Hausdorff-Young-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 $1<p\leq 2\leq q<\infty$ 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 be used in the proofs with different Plancherel measures on the
unitary duals.
