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

Abstract

Let $\mathbb{G}$ be a locally compact quantum group with dual $\hat{\mathbb{G}}$. Suppose that the left Haar weight $\phi$ and the dual left Haar weight $\hat{\phi }$ 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 $L_p(\hat{\mathbb{G}},\hat{\phi })$ to $L_q(\hat{\mathbb{G}},\hat{\phi })$ whenever the symbol $x$ lies in $L_{r,\infty }(\mathbb{G},\phi )$, where $1/r=1/p-1/q$. Moreover, we have \begin{equation*} \|m_{x}:L_p(\widehat{\mathbb{G}},\widehatφ)\to L_q(\widehat{\mathbb{G}},\widehatφ)\|\le c_{p,q} \|x\|_{L_{r,\infty}(\mathbb{G},φ)}, \end{equation*} where $c_{p,q}$ is a constant depending only on $p$ and $q$. This was first proved by Hörmander \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.

13 pages