Hilbert transforms on Coxeter groups and groups acting on buildings

TL;DR

This paper develops a new framework for Hilbert transforms on Coxeter groups and groups acting on buildings, establishing their boundedness on $L_p$-spaces and extending previous results to broader classes of groups.

Contribution

It introduces models for Hilbert transforms on these groups, proves their $L_p$-boundedness under nested conditions, and provides the first examples on groups with property (F$ ext{R}$).

Findings

Established $L_p$-boundedness of Hilbert transforms on Coxeter groups.

Extended results to groups acting on buildings with nested conditions.

Provided the first examples of groups with property (F$ ext{R}$) supporting such transforms.

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 Gonz\'alez-P\'erez, 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 (F$\mathbb{R}$) on which $L_p$-bounded Hilbert transforms can be defined, and generalize the work of Gonz\'alez-P\'erez, Parcet and the second named author for groups acting on simplicial trees.