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arXiv · 2406.08769

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

Jorge Pérez García
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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.

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Jorge Pérez García. 2024-06-13. A note on Hilbert transform over lattices of $\mathrm{PSL}_2(\mathbb{C})$. https://arxiv.org/abs/2406.08769

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