Fourier multipliers and group von Neumann algebras

TL;DR

This paper proves boundedness of Fourier multipliers on certain groups using operator algebra techniques, extending classical results like H"ormander's theorem to a broad class of groups.

Contribution

It establishes $L^p$-$L^q$ boundedness of Fourier multipliers on locally compact unimodular groups, generalizing known Euclidean and compact Lie group results.

Findings

Proves $L^p$-$L^q$ boundedness for Fourier multipliers on groups

Develops a Hausdorff-Young-Paley inequality for these groups

Extends classical Fourier multiplier theorems to new group settings

Abstract

In this paper we establish the $L^p$-$L^q$ boundedness of Fourier multipliers on locally compact separable unimodular groups for the range of indices $1<p\leq 2 \leq q<\infty$. Our approach is based on the operator algebras techniques. The result depends on a version of the Hausdorff-Young-Paley inequality that we establish on general locally compact separable unimodular groups. In particular, the obtained result implies the corresponding H\"ormander's Fourier multiplier theorem on $\mathbb{R}^{n}$ and the corresponding known results for Fourier multipliers on compact Lie groups.