Proper cocycles, measure equivalence and $L_p$-Fourier multipliers

TL;DR

This paper introduces a new transference method for $L_p$-Fourier multipliers using proper cocycles, extending previous results and enabling applications like convergence of Fourier series, transference from lattices, and a Hilbert transform analogue.

Contribution

It develops a novel transference approach for $L_p$-Fourier multipliers via proper cocycles, broadening applicability beyond the case $p= olinebreak\infty$ and enabling new noncommutative analysis tools.

Findings

Extended transference method for $L_p$-Fourier multipliers.

Simplified proof of pointwise convergence of noncommutative Fourier series.

Constructed a noncommutative Hilbert transform analogue on $SL_2(\mathbb{R})$.

Abstract

We develop a new transference method for completely bounded $L_p$-Fourier multipliers via proper cocycles arising from probability measure-preserving group actions. This method extends earlier results by Haagerup and Jolissaint, which were limited to the case $p = \infty$. Based on this approach, we present a new and simple proof of the main result in [Hong-Wang-Wang, Mem. Amer. Math. Soc. 2024] regarding the pointwise convergence of noncommutative Fourier series on amenable groups, refining the associated estimate for maximal inequalities. In addition, this framework yields a transference principle for $L_p$-Fourier multipliers from lattices in linear Lie groups to their ambient groups, establishing a noncommutative analogue of Jodeit's theorem. As a further application, we construct a natural analogue of the Hilbert transform on $SL_2(\mathbb{R})$.