Hardy-type spaces on certain noncompact manifolds and applications

Abstract

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 X1(M), X2(M), … of new Hardy spaces on M, the sequence Y1(M), Y2(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 α e 2 b r for some nonnegative real number α and for all large r, we prove also an endpoint result for the first-order Riesz transform ∇ ℒ−1/2. In this case, the kernels of the operators ℒiu 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.