A note on Hilbert transform over lattices of $\mathrm{PSL}_2(\mathbb{C})$
Jorge Pérez García
TL;DR
The paper analyzes L_p-bounded Fourier multipliers on group von Neumann algebras arising from PSL$_2(\mathbb{C})$, focusing on a symbol $m$ obtained by lifting a hyperbolic partition to PSL$_2(\mathbb{C})$ and restricting to arithmetic lattices $\Gamma_n = \mathrm{PSL}_2(\mathbb{Z}[\sqrt{-n}])$ and Bianchi groups. Building on the non-commutative Cotlar framework, it identifies a kernel set $K_n = \{g\in\Gamma_n: m(g)=0\}$, decomposes it into $K_n^+$ and $K_n^-$ (with extra $L_n^\pm$ pieces in the Bianchi setting), and proves that $m$ is left $(K_n,\chi)$-equivariant with a sign character $\chi$; this structural control enables verification of the Cotlar identity on $\Gamma_n$, yielding L_p-boundedness of the associated multiplier $T_m$ on $L_p(\mathcal{L}\Gamma_n)$ for all $1<p<\infty$ with norm growth $\lesssim\left(\frac{p^2}{p-1}\right)^\beta$ where $\beta=1+\log_2(1+\sqrt{2})$. The work further explores when Cotlar-type control holds on Bianchi groups, showing obstructions for the original $m$ (notably for $n\equiv -1\pmod{4}$) but obtaining Cotlar identity for an alternative multiplier $\widetilde{m}$ induced by a different hyperbolic plane, which holds on $\mathrm{PSL}_2(\mathbb{Z}[i])$ and most $\Gamma_n'$ with $n\neq 3$. Together, these results extend Cotlar-based L_p-boundedness to arithmetic lattices in $\mathrm{PSL}_2(\mathbb{C})$ and provide practical tools for non-commutative harmonic analysis on these groups.
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 $m\colon \mathrm{PSL}_2(\mathbb{C})\to \mathbb{R}$ arising from lifting the indicator function of a partition $\{Σ^+,Σ^+,Σ^-\}$ 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.