A HAUSDORFF-YOUNG INEQUALITY FOR LOCALLY COMPACT QUANTUM GROUPS

Let G be a locally compact abelian group with dual group (G) over cap. The Hausdorff-Young theorem states that if f is an element of L(p)(G), where 1 <= p <= 2, then its Fourier transform F(p)(f) belongs to L(q)((G) over cap) (where (1/p) + (1/q) = 1) and parallel to F(p)(f)parallel to(q) <= parallel to f parallel to(p). Kunze and Terp extended this to unimodular and locally compact groups, respectively. We further generalize this result to an arbitrary locally compact quantum group G by defining a Fourier transform F(p) : L(p)(G) -> L(q)((G) over cap) and showing that this Fourier transform satisfies the Hausdorff-Young inequality.