A note on Hilbert transform over lattices of PSL₂(ℂ)

2024/06/13 by Jorge Pérez García, García, Jorge Pérez
Mathematics · Engineering · Computer Science · #Mathematical Analysis and Transform Methods #graph theory and CDMA systems #Advanced Algebra and Logic

paper · pdf · doi:10.48550/arxiv.2406.08769

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 PSL₂(ℂ)→ ℝ arising from lifting the indicator function of a partition \Σ⁺,Σ⁺,Σ^-\ of the hyperbolic space ℍ³ to its isometry group PSL₂(ℂ). The boundedness of T_m on L_p(L PSL₂(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 PSL₂(ℤ[√(-n)]), solving a question left open by the first named authors.

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