$L_p$-$L_q$ Fourier multipliers on locally compact quantum groups

Abstract

Let G be a locally compact quantum group with dual p G. Suppose that the left Haar weight ϕ and the dual left Haar weight p ϕ are tracial, e.g. G is a unimodular Kac algebra. We prove that for 1 ă p ď 2 ď q ă 8, the Fourier multiplier mx is bounded from Lpp p G, p ϕq to Lqp p G, p ϕq whenever the symbol x lies in Lr,8pG, ϕq, where 1{r " 1{p ´1{q. Moreover, we havewhere cp,q is a constant depending only on p and q. This was first proved by Hörmander [Hör60] for R n , and was recently extended to more general groups and quantum groups. Our work covers all these results and the proof is simpler. In particular, this also yields a family of Lp-Fourier multipliers over discrete group von Neumann algebras. A similar result for Sp-Sq Schur multipliers is also proved.