Hilbert transforms on Coxeter groups and groups acting on buildings

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 (F$\mathbb{R}$) 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.