Let $X$ be a symmetric space of the noncompact type, with Laplace–Beltrami operator $-ℒ$, and let $[b,\infty )$ be the $L^2\left(X\right)$-spectrum of $ℒ$. For $\tau$ in $ℂ$ such that $\mathrm{Re}\tau \ge 0$, let ${𝒫}_{\tau }$ be the operator on $L^2\left(X\right)$ defined formally as $\exp (-\tau {(ℒ-b)}^{1/2})$. In this paper, we obtain $L^p-L^q$ operator norm estimates for ${𝒫}_{\tau }$ for all $\tau$, and show that these are optimal when $\tau$ is small and when $\left|\arg \tau \right|$ is bounded below $\pi /2$.

Let $P$ be a long range metric perturbation of the Euclidean Laplacian on ${ℝ}^d$, $d\ge 2$. We prove local energy decay for the solutions of the wave, Klein-Gordon and Schrödinger equations associated to $P$. The problem is decomposed in a low and high frequency analysis. For the high energy part, we assume a non trapping condition. For low (resp. high) frequencies we obtain a general result about the local energy decay for the group $e^{itf\left(P\right)}$ where $f$ has a suitable development at zero (resp. infinity).