Atomic Decomposition of Hardy Type Spaces on Certain Noncompact Manifolds

In this paper we consider a complete connected noncompact Riemannian manifold M with bounded geometry and spectral gap. We prove that the Hardy type spaces X ^k (M), introduced in a previous paper of the authors, have an atomic characterization. An atom in X ^k (M) is an atom in the Hardy space H ¹(M) introduced by Carbonaro, Mauceri, and Meda, satisfying an ??infinite dimensional?? cancellation condition. As an application, we prove that the Riesz transforms of even order $nabla^{2k} mathcal{L}^{-k}$ map X ^k (M) into L ¹(T _2k M).