A note on Hilbert transform over lattices of $\mathrm{PSL}_2(\mathbb{C})$
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913786873511936 |
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| author | García, Jorge Pérez |
| author_facet | García, Jorge Pérez |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08769 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A note on Hilbert transform over lattices of $\mathrm{PSL}_2(\mathbb{C})$ García, Jorge Pérez Functional Analysis Operator Algebras 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. |
| title | A note on Hilbert transform over lattices of $\mathrm{PSL}_2(\mathbb{C})$ |
| topic | Functional Analysis Operator Algebras |
| url | https://arxiv.org/abs/2406.08769 |