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 $\mathrm{\Delta }$ on the homogeneous space $G\mathrm{/}K=S$ by a maximal compact subgroup $K$. We obtain pointwise estimates for the kernel of an oscillating function $\exp (it\sqrt{\mathrm{\mid }x\mathrm{\mid }})\psi (\sqrt{\mathrm{\mid }x\mathrm{\mid }})$ applied to the shifted Laplacian $\mathrm{\Delta }+\mathrm{\mid }\rho {\mathrm{\mid }}^2$. We obtain a polynomial decay in time of the kernel, and of the $L^p-L^q$ norms of the operator, for $1\le p<2<q\le \mathrm{\infty }$. For the related distinguished Laplacian, we obtain bounds for the $L^p-L^p$ norms, $1\le p\le \mathrm{\infty }$, with a slower growth in time than predicted by earlier results.