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Local Hardy Spaces of Differential Forms on Riemannian Manifolds

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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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Authors and Affiliations

  1. School of Mathematics, University of Birmingham, Watson Building, Edgbaston, Birmingham, B15 2TT, UK

    Andrea Carbonaro

  2. Centre for Mathematics and Its Applications, Mathematical Sciences Institute, Australian National University, Canberra, ACT, 0200, Australia

    Alan McIntosh & Andrew J. Morris

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  1. Andrea Carbonaro
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  2. Alan McIntosh
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  3. Andrew J. Morris
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Correspondence to Andrew J. Morris.

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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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