Abstract: Let $G$ be a semisimple, connected, and noncompact Lie group with a finite center. We 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 $\Delta+|\rho|^2$, a case not available before. We obtain a polynomial decay in time of the kernel estimate, and of the $L^{p'}-L^p$ norms of the operator, for $2<p<\infty$.
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