JournalsrmiVol. 31, No. 3pp. 865–900

Calderón reproducing formulas and applications to Hardy spaces

  • Pascal Auscher

    Université de Paris-Sud, Orsay, France
  • Alan G.R. McIntosh

    Australian National University, Canberra, Australia
  • Andrew J. Morris

    University of Oxford, UK
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We establish new Calderón holomorphic functional calculus whilst the synthesising function interacts with DD through functional calculus based on the Fourier transform. We apply these to prove the embedding HDp(TM)Lp(TM)H^p_D(\wedge T^*M) \subseteq L^p(\wedge T^*M), 1p21 \leq p \leq 2, for the Hardy spaces of differential forms introduced by Auscher, McIntosh and Russ, where D=d+dD=d+d^* is the Hodge–Dirac operator on a complete Riemannian manifold MM that has doubling volume growth. This fills a gap in that work. The new reproducing formulas also allow us to obtain an atomic characterisation of HD1(TM)H^1_D(\wedge T^*M). The embedding HLpLpH^p_L \subseteq L^p, 1p21 \leq p \leq 2, where LL is either a divergence form elliptic operator on Rn\mathbb R^n, or a nonnegative self-adjoint operator that satisfies Davies–Gaffney estimates on a doubling metric measure space, is also established in the case when the semigroup generated by the adjoint L-L^* is ultracontractive.

Cite this article

Pascal Auscher, Alan G.R. McIntosh, Andrew J. Morris, Calderón reproducing formulas and applications to Hardy spaces. Rev. Mat. Iberoam. 31 (2015), no. 3, pp. 865–900

DOI 10.4171/RMI/857