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(M), X(M), . . . of new Hardy spaces on M , the sequence Y (M), Y (M), . . . of their dual spaces, and show that these spaces may be used to obtain endpoint estimates for spectral multipliers associated to the Laplace–Beltrami operator L on M . These results complement earlier work of J. Cheeger, M. Gromov and M. Taylor and of the authors, and improve a recent result of A. Carbonaro, Mauceri and Meda. Under the additional condition that the volume of the geodesic balls of radius r is controlled by C rα e √ br for some real number α and for all large r, we prove also an endpoint result for first order Riesz transforms ∇L−1/2. Under stronger geometric assumptions on M we prove an atomic characterisation of the spaces Xk(M): we show that an atom in Xk(M) is an atom in the Hardy space H(M) introduced by Carbonaro, Mauceri and Meda, satisfying further cancellation conditions.