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
Calderón reproducing formulas and applications to Hardy spaces cover
Download PDF

Abstract

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