Local Hardy Spaces of Differential Forms on Riemannian Manifolds

Abstract

We define local Hardy spaces of differential forms hDᴾ(∧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=D² is the Hodge–Laplacian, then the local geometric Riesz transform D(Δ+aI)⁻¹/² has a bounded extension to hDᴾ for all p∈[1,∞], provided that a>0 is large enough compared to the exponential growth of M. A characterization of h1D 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 HDᴾ(∧T∗M) introduced by Auscher, McIntosh, and Russ