An atomic decomposition of the Hajłasz Sobolev space 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 , defined by Hajłasz, is identified with a Hardy–Sobolev space defined in terms of atoms. Decomposition results are proved for both the homogeneous and the nonhomogeneous spaces.