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

TL;DR

This paper investigates the boundedness of certain Fourier multipliers over lattices in the complex special linear group, showing boundedness in specific arithmetic lattice cases despite known counterexamples in the general setting.

Contribution

It demonstrates the boundedness of specific Fourier multipliers on arithmetic lattices of _2(4n) _2(4n) _2(4n), addressing an open problem from prior research.

Findings

Boundedness of Fourier multipliers on _2(4n) lattices established.

Counterexamples exist for general _2(\u001b4n) groups.

Extension of boundedness results to specific arithmetic lattices.

Abstract

Gonz\'alez-P\'erez, 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 $m\colon \mathrm{PSL}_2(\mathbb{C})\to \mathbb{R}$ 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.