Trigonometric chaos and $\mathrm{X}_p$ inequalities I -- Balanced Fourier truncations over discrete groups

TL;DR

This paper extends inequalities related to Fourier truncations and differential operators from the hypercube to discrete groups, utilizing noncommutative $L_p$-space techniques and Riesz transforms.

Contribution

It generalizes Naor's inequality for the hypercube to a broader class of discrete groups using advanced harmonic analysis tools.

Findings

Extended Naor's inequality to discrete groups

Established new inequalities via directional derivatives

Used noncommutative Riesz transforms and free Hilbert transforms

Abstract

We investigate $L_p$-estimates for balanced averages of Fourier truncations in group algebras, in terms of differential operators acting on them. Our results extend a fundamental inequality of Naor for the hypercube (with profound consequences in metric geometry) to discrete groups. Different inequalities are established in terms of directional derivatives which are constructed via affine representations determined by the Fourier truncations. Our proofs rely on the Banach $\mathrm{X}_p$ nature of noncommutative $L_p$-spaces and dimension-free estimates for noncommutative Riesz transforms. In the particular case of free groups we use an alternative approach based on free Hilbert transforms.