  We establish sharp Sobolev embedding properties within a broad class of
compact matrix quantum groups of Kac type under the polynomial growth or the
rapid decay property of their duals. Main examples are duals of polynomially
growing discrete quantum groups, duals of free groups and free quantum groups
$O_N^+,S_N^+$. In addition, we generalize sharpend Hausdorff-Young
inequalities, compute degrees of the rapid decay property for
$\widehat{O_N^+},\widehat{S_N^+}$ and prove sharpness of Hardy-Littlewood
inequalities on duals of free groups.
