A note on Hilbert transform over lattices of PSL₂(C)

In a recent work by Gonz\'alez-P\'erez, Parcet and Xia, the boundedness over non-commutative Lₚ-spaces of an analogue of the Hilbert transform was studied. This analogue is defined as a Fourier multiplier with symbol m PSL₂(C)→ R arising from the action by isometries of PSL₂(C) on the $3$-dimensional hyperbolic space H³. More concretely, m is a lifting of the function on H³ that takes values ± 1 in the two regions formed by dividing the space by a hyperbolic plane. The boundedness of Tₘ on Lₚ(L PSL₂(C)) for p≠ 2 was disproved by Parcet, de la Salle and Tablate. Nevertheless, we will show that this Fourier multiplier is bounded when restricted to the arithmetic lattices PSL₂(Z[√-n]), solving a question left open by Gonz\'alez-P\'erez, Parcet and Xia.