Local Hardy spaces of differential forms on Riemannian manifolds

We define local Hardy spaces of differential forms h_D ^p(T*M) for all p∈[1,∞] that are adapted to a class of first-order differential operators D on a complete Riemannian manifold M with at most exponential volume growth. In particular, if D is the Hodge-Dirac operator on M and Δ=D ² is the Hodge-Laplacian, then the local geometric Riesz transform D(Δ+aI)^-1/2 has a bounded extension to h_D^pfor all p∈[1,∞], provided that a>0 is large enough compared to the exponential growth of M. A characterization of h_D¹ in terms of local molecules is also obtained. These results can be viewed as the localization of those for the Hardy spaces of differential forms H_D^p(T*M) introduced by Auscher, McIntosh, and Russ.