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 ra e2b 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.