Hardy-type spaces on certain noncompact manifolds and applications

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(alpha) e(2 root br) for some nonnegative real number alpha and for all large r, we prove also an endpoint result for the first-order Riesz transform del L-1/2. In this case, the kernels of the operators L-iu and del L-1/2 are singular both on the diagonal and at infinity. In particular, these results apply to Riemannian symmetric spaces of the noncompact type.