An atomic decomposition of the Haj{\l}asz Sobolev space $\Mone$ on manifolds

Several possible notions of Hardy-Sobolev spaces on a Riemannian manifold with a doubling measure are considered. Under the assumption of a Poincaré inequality, the space $\Mone$, defined by Haj{\l}asz, is identified with a Hardy-Sobolev space defined in terms of atoms. Decomposition results are proved for both the homogeneous and the nonhomogeneous spaces.