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

TL;DR

This paper extends the boundedness of Fourier multipliers on locally compact quantum groups with tracial Haar weights, generalizing classical results and providing simpler proofs, with applications to discrete group von Neumann algebras and Schur multipliers.

Contribution

It proves new boundedness results for $L_p$-$L_q$ Fourier multipliers on quantum groups with tracial weights, unifying and simplifying previous work.

Findings

Boundedness of Fourier multipliers on quantum groups with tracial Haar weights.

Extension of classical Fourier multiplier results to quantum group setting.

Application to discrete group von Neumann algebras and Schur multipliers.

Abstract

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.