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

Let $\mathbb{G}$ be a locally compact quantum group with dual $\widehat{\mathbb{G}}$. Suppose that the left Haar weight $\varphi$ and the dual left Haar weight $\widehat{\varphi}$ are tracial, e.g. $\mathbb{G}$ is a unimodular Kac algebra. We prove that for $1<p\le 2 \le q<\infty$, the Fourier multiplier $m_{x}$ is bounded from $L_p(\widehat{\mathbb{G}},\widehat{\varphi})$ to $L_q(\widehat{\mathbb{G}},\widehat{\varphi})$ whenever the symbol $x$ lies in $L_{r,\infty}(\mathbb{G},\varphi)$, where $1/r=1/p-1/q$. Moreover, we have \begin{equation*} \|m_{x}:L_p(\widehat{\mathbb{G}},\widehat{\varphi})\to L_q(\widehat{\mathbb{G}},\widehat{\varphi})\|\le c_{p,q} \|x\|_{L_{r,\infty}(\mathbb{G},\varphi)}, \end{equation*} where $c_{p,q}$ is a constant depending only on $p$ and $q$. This was first proved by H\"ormander \cite{Hormander1960} for $\mathbb{R}^n$, 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 $L_p$-Fourier multipliers over discrete group von Neumann algebras. A similar result for $\mathcal{S}_p$-$\mathcal{S}_q$ Schur multipliers is also proved.