Local Hardy Spaces of Differential Forms on Riemannian Manifolds

Abstract

We define local Hardy spaces of differential forms \(h^{p}_{\mathcal{D}}(\wedge T^{*}M)\) for all p∈[1,∞] that are adapted to a class of first-order differential operators \(\mathcal{D}\) 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 \(h^{p}_{D}\) for all p∈[1,∞], provided that a>0 is large enough compared to the exponential growth of M. A characterization of \(h^{1}_{\mathcal{D}}\) 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 \(H^{p}_{D}(\wedge T^{*}M)\) introduced by Auscher, McIntosh, and Russ.