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Several possible notions of Hardy-Sobolev spaces on a Riemannian manifoldwith a doubling measure are considered. Under the assumption of a Poincar\'einequality, the space $\Mone$, defined by Haj{\l}asz, is identified with aHardy-Sobolev space defined in terms of atoms. Decomposition results are provedfor both the homogeneous and the nonhomogeneous spaces.

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