A note on Hilbert transform over lattices of ${\mathrm{PSL}}_2(\mathbb{C})$

Abstract

González-Pérez, Parcet and Xia introduced recently a framework to study $L_p$-boundedness of certain families of idempotent multipliers on von Neumann algebras. It includes symbols $m:{\mathrm{PSL}}_2(\mathbb{C})\to \mathbb{R}$ arising from lifting the indicator function of a partition $\{{\Sigma }^{+},{\Sigma }^{+},{\Sigma }^{-}\}$ of the hyperbolic space ${\mathbb{H}}^3$ to its isometry group ${\mathrm{PSL}}_2(\mathbb{C})$. The boundedness of $T_m$ on $L_p(\mathcal{L}{\mathrm{PSL}}_2(\mathbb{C}))$ 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 ${\mathrm{PSL}}_2(\mathbb{Z}[\sqrt{-n}])$, solving a question left open by the first named authors.

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