标题： Hilbert变换在Coxeter群和作用在建筑物上的群上 显示英文标题

标题： Hilbert transforms on Coxeter groups and groups acting on buildings

摘要： 在本文中，我们研究Hilbert变换及其在与Coxeter群和作用于建筑物上的群相关的$L_p$-空间上的有界性。我们通过这些群所作用的几何对象建立了Hilbert变换的新模型，并证明这些Hilbert变换满足Cotlar恒等式，该恒等式是在Mei和Ricard以及González-Pérez、Parcet和第二作者的先前工作中开发的，从而得出$L_p$-有界性。我们首先解决了当Coxeter群满足某种嵌套条件时的情况，然后将其扩展到任何有限生成的Coxeter群以及满足嵌套条件的建筑物作用群。我们的结果给出了具有性质(F$\mathbb{R}$)的第一个群的例子，在此可以定义$L_p$-有界的Hilbert变换，并推广了González-Pérez、Parcet和第二作者对于作用于单纯树上的群的工作。 显示英文摘要

摘要： 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.