A HAUSDORFF–YOUNG INEQUALITY FOR LOCALLY COMPACT QUANTUM GROUPS

Abstract

Let G be a locally compact abelian group with dual group Ĝ. The Hausdorff–Young theorem states that if f ∈ Lp(G), where 1 ≤ p ≤ 2, then its Fourier transform [Formula: see text] belongs to Lq(Ĝ) (where (1/p) + (1/q) = 1) and [Formula: see text]. 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 [Formula: see text] and showing that this Fourier transform satisfies the Hausdorff–Young inequality.