Abstract:
n 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 spectral multipliers associated to the LaplaceBeltrami 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 r^a exp(2\sqrt{b} r) for some real number a and for all large r, we prove also an endpoint result for first order Riesz transforms \nabla L^{1/2}.
Under stronger geometric assumptions on M we prove an atomic characterisation of the spaces X^h(M): we show that an atom in X^h(M) is an atom in the Hardy space H^1(M) introduced by Carbonaro, Mauceri and Meda, satisfying further cancellation conditions.
Keywords:
Riemannian manifold, Hardy space, Laplacian, spectral multipliers, Riesz transforms
MSC:
42, 46, 58
