Pointwise and uniform bounds for functions of the Laplacian on non-compact symmetric spaces

Abstract

Let $L$ be the distinguished Laplacian on the Iwasawa $AN$ group associated with a semisimple Lie group $G$. Assume $F$ is a Borel function on ${\mathbb{R}}^{+}$. We give a condition on $F$ such that the kernels of the functions $F(L)$ are uniformly bounded. This condition involves the decay of $F$ only and not its derivatives. By a known correspondence, this implies pointwise estimates for a wide range of functions of the Laplace-Beltrami operator on symmetric spaces. In particular, when $G$ is of real rank one and $F(x)={\mathrm{e}}^{it\sqrt{x}}\psi (\sqrt{x})$, our bounds are sharp.