The Lax-Phillips axioms in the context of scattering theory on real hyperbolic spaces, and their quotients by discrete subgroups rely on the so-called Poincaré inequality. It is enough to prove this inequality when the space \(X\) is irreducible. In this case, it is a consequence of the so-called uncertainty principle. In this paper, the author proves the uncertainty princple in full generality (Theorem 3). The proof goes as follows. Via the Radon transform, the author reduces the problem to a purely Euclidean estimate. Then, this is established, using a sharp estimate for the Plancherel measure and a geometric property of irreducible root systems.