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

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 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)= eit√ xψ(√ x), our bounds are sharp.