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.
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Communicated by Fulvio Ricci.
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Carbonaro, A., McIntosh, A. & Morris, A.J. Local Hardy Spaces of Differential Forms on Riemannian Manifolds. J Geom Anal 23, 106–169 (2013). https://doi.org/10.1007/s12220-011-9240-x
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DOI: https://doi.org/10.1007/s12220-011-9240-x
Keywords
- Local Hardy spaces
- Riemannian manifolds
- Differential forms
- Hodge–Dirac operators
- Local Riesz transforms
- Off-diagonal estimates