Shifted wave equation on noncompact symmetric spaces

Let $G$ be a semisimple, connected, and noncompact Lie group with a finite center. We carry out a detailed analysis of oscillating integrals involving the Harish-Chandra $c$-function, in the case of real rank $l\ge 2$. This allows to obtain two main applications. Consider the Laplace-Beltrami operator $\Delta$ on the homogeneous space $G/K=S$ by a maximal compact subgroup $K$. We obtain pointwise estimates for the kernel of an oscillating function $\exp( it\sqrt{|x|}) \psi(\sqrt{|x|})$ applied to the shifted Laplacian $\Delta+|\rho|^2$. We obtain a polynomial decay in time of the kernel, and of the $L^p-L^q$ norms of the operator, for $1\le p<2<q\le \infty$. For the related distinguished Laplacian, we obtain bounds for the $L^p-L^p$ norms, $1\le p\le\infty$, with a slower growth in time than predicted by earlier results.