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 onM, 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 an additional geometric condition, we prove also an endpoint result for the first-order Riesz transform ∇L^{−1/2}. In this case, the kernels of the operators L^iu and ∇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.