Journal of Operator Theory

Volume 95, Issue 2, Spring 2026 pp. 567-580.

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

Authors: Jorge Perez Garcia
Author institution: Instituto de Ciencias Matematicas (ICMAT), C. Nicolas Cabrera, 13-15, Fuencarral-El Pardo, 28049 Madrid, Spain

Summary: Gonzalez-Perez, Parcet and Xia introduced 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.

DOI: http://dx.doi.org/10.7900/jot.2024oct21.2484
Keywords: Fourier multipliers, noncommutative analysis, von Neumann algebras

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