Hausdorff-young Inequalities for Nonunimodular Groups

In this paper we deal with a definition of Lp-Fourier transform on locally compact groups. Recall that, for locally compact abelian groups, the Hausdorff-Young inequality reads: Let 1 < p < 2 and q = p/(p − 1). If g ∈ L1(G) ∩ Lp(G), then ĝ ∈ Lq(Ĝ), with ‖ĝ‖q ≤ ‖g‖p. The inequality allows to extend the Fourier transform to a continuous operator Fp : Lp → Lq by continuity. It was generalized to type I unimodular groups by Kunze [13]. Over the years, various authors derived HausdorffYoung inequalities both for concrete groups [14, 1, 8, 7] and for certain classes of groups [16, 17, 18, 10, 2], with the aim of getting a more precise bound in the inequality. The formulation of the results for nonabelian groups requires a certain amount of notation. Given a locally compact group G, we denote by Ĝ its unitary dual, i.e., the set of (equivalence classes of) irreducible unitary representations, endowed with the Mackey Borel structure. The dual space is used to define the operator valued Fourier transform by letting L(G) 3 g 7→ F(g) := (σ(g)) σ∈Ĝ, where σ(g) is defined by the weak operator integral