On the square root of Poisson kernel in $\delta$-hyperbolic spaces and some aspects of boundary representations

TL;DR

This paper establishes a Fatou-type theorem for the square root of the Poisson kernel in hyperbolic spaces, explores boundary representations, and demonstrates their algebraic and decay properties using elementary techniques.

Contribution

It introduces new inequalities for boundary representation matrix coefficients and shows how these lead to a convolution algebra structure and decay estimates.

Findings

Matrix coefficients do not satisfy the weak Harish-Chandra inequality.

A particular inequality related to property RD is satisfied by matrix coefficients.

The Harish-Chandra Schwartz space forms a convolution algebra with decay properties.

Abstract

In this note, we prove a theorem \`a la Fatou for the square root of Poisson Kernel in the context of quasi-convex cocompact discrete groups of isometries of $\delta$-hyperbolic spaces. As a corollary we show that some matrix coefficients of boundary representations cannot satisfy \emph{the weak inequality of Harish-Chandra}. Nevertheless, such matrix coefficients satisfy an inequality which can be viewed as a particular case of the inequality coming from \emph{property RD} for boundary representations. The inequality established in this paper is based on a uniform bound which appears in the proof of the irreducibility of boundary representations. Moreover this uniform bound can be used to prove that the Harish-Chandra's Schwartz space associated with some discrete groups of isometries of $\delta$-hyperbolic spaces carries a natural structure of a convolution algebra. Then in the context of CAT(-1) spaces we show how our elementary techniques enable us to apply an equidisitribution theorem of T. Roblin to obtain informations about the decay of matrix coefficient of boundary representations associated with continuous functions.