  In this paper we consider a complete connected noncompact Riemannian manifold
M with Ricci curvature bounded from below, positive injectivity radius and
spectral gap b. We introduce a sequence X^1(M), X^2(M), ... of new Hardy spaces
on M, the sequence Y^1(M/, Y^2(M), ... of their dual spaces, and show that
these spaces may be used to obtain endpoint estimates for purely imaginary
powers of the Laplace-Beltrami operator and for more general spectral
multipliers associated to the Laplace--Beltrami operator L on M. Under the
additional condition that the volume of the geodesic balls of radius r is
controlled by C r^a e^{2\sqrt{b} r} for some real number a and for all large r,
we prove also an endpoint result for first order Riesz transforms D L^{-1/2}.
In particular, these results apply to Riemannian symmetric spaces of the
noncompact type.
