A note on $L^p$-$L^q$ boundedness of Fourier multipliers on noncommutative spaces

TL;DR

This paper provides a simplified proof of $L^p$-$L^q$ boundedness for Fourier multipliers on noncommutative spaces, extending classical results to a broad noncommutative setting including unimodular groups.

Contribution

It introduces a straightforward proof technique for $L^p$-$L^q$ estimates of Fourier multipliers on noncommutative spaces associated with semi-finite von Neumann algebras.

Findings

Established $L^p$-$L^q$ bounds for Fourier multipliers in noncommutative spaces

Extended classical Fourier multiplier results to noncommutative and group settings

Simplified the proof of boundedness estimates in noncommutative harmonic analysis

Abstract

In this work, we study Fourier multipliers on noncommutative spaces. In particluar, we show a simple proof of $L^p$-$L^q$ estimate of Fourier multipliers on general noncommutative spaces associated with semi-finite von Neumann algebras. This includes the case of Fourier multipliers on general locally compact unimodular groups.