  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.
