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 Xk(M), introduced in a previous paper of the authors, have an atomic characterization. An atom in Xk(M) is an atom in the Hardy space H1(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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\nabla^{2k} \mathcal{L}^{-k}$\end{document} map Xk(M) into L1(T2kM).