Maximal characterization of Hardy-Sobolev spaces on manifolds

Let M be a complete non-compact Riemannian manifold with a doubling measure and admitting a Poincare inequality. In the present paper, we identify the Sobolev space M-1(1), introduced by Hajlasz, with a new Hardy-Sobolev space defined by requiring the integrability of a certain maximal function of the gradient. This completes the circle of ideas begun in [4], where M-1(1) was identified with a Hardy-Sobolev space via atomic decomposition.