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We investigate $L_p$-estimates for balanced averages of Fourier truncationsin group algebras, in terms of differential operators acting on them. Ourresults extend a fundamental inequality of Naor for the hypercube (withprofound consequences in metric geometry) to discrete groups. Differentinequalities are established in terms of directional derivatives which areconstructed 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 onfree Hilbert transforms.

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