Hilbert transforms on Coxeter groups and groups acting on buildings

2025/08/25 by Lu, Xiao-Qi, Xia, Runlian
#46B40 #46L05 #47C15 (2010) #FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR) #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2508.18176

Abstract

In this paper, we study Hilbert transforms and their boundedness on L_p-spaces associated with Coxeter groups and groups acting on buildings. We establish new models for Hilbert transforms on these groups through the geometric objects they act on, and we show that these Hilbert transforms satisfy a Cotlar identity which was developed in earlier work of Mei and Ricard, and that of González-Pérez, Parcet and the second named author, thus conclude the L_p-boundedness. We manage to solve the problem firstly for the case when the Coxeter group satisfies a certain nested condition, and then extend it to any finitely generated Coxeter groups and groups that admit actions on buildings satisfying the nested condition. Our results give the first example of groups with property (Fℝ) on which L_p-bounded Hilbert transforms can be defined, and generalize the work of González-Pérez, Parcet and the second named author for groups acting on simplicial trees.

Citations

• A Sobolev estimate for radial L^p-multipliers on a class of semi-simple Lie groups
• Noncommutative Cotlar identities for groups acting on tree-like structures
• Spectral multipliers in group algebras and noncommutative Calderón-Zygmund theory
• Algebraic Calderón-Zygmund theory
• Twisted Hilbert transforms vs Kakeya sets of directions
• Intersections of Apartments
• Kazhdan and Haagerup properties from the median viewpoint
• Pseudo-localization of singular integrals and noncommutative Calderon-Zygmund theory
• Multipliers of the Fourier Algebras of Some Simple Lie Groups and Their Discrete Subgroups

Related

• Around groups in Hilbert Geometry
• A note on Hilbert transform over lattices of PSL₂(ℂ)
• Noncommutative Cotlar identities for groups acting on tree-like structures
• Coxeter group in Hilbert geometry
• Factorial multiparameter Hecke von Neumann algebras and representations of groups acting on right-angled buildings