# Revista Matemática Iberoamericana

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**Volume 31, Issue 3, 2015, pp. 865–900**

**DOI: 10.4171/RMI/857**

Published online: 2015-10-29

Calderón reproducing formulas and applications to Hardy spaces

Pascal Auscher^{[1]}, Alan G.R. McIntosh

^{[2]}and Andrew J. Morris

^{[3]}(1) Université de Paris-Sud, Orsay, France

(2) Australian National University, Canberra, Australia

(3) University of Oxford, UK

We establish new Calderón holomorphic functional calculus whilst the synthesising function interacts with $D$ through functional calculus based on the Fourier transform. We apply these to prove the embedding $H^p_D(\wedge T^*M) \subseteq L^p(\wedge T^*M)$, $1 \leq p \leq 2$, for the Hardy spaces of differential forms introduced by Auscher, McIntosh and Russ, where $D=d+d^*$ is the Hodge–Dirac operator on a complete Riemannian manifold $M$ 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 $H^1_D(\wedge T^*M)$. The embedding $H^p_L \subseteq L^p$, $1 \leq p \leq 2$, where $L$ is either a divergence form elliptic operator on $\mathbb R^n$, 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 $-L^*$ is ultracontractive.

*Keywords: *Calderón reproducing formula, Hardy space embedding, self-adjoint operator, finite propagation speed, sectorial operator, off-diagonal estimate, first-order differential operator, Hodge–Dirac operator, divergence form elliptic operator, Riemannian manifold

Auscher Pascal, McIntosh Alan, Morris Andrew: Calderón reproducing formulas and applications to Hardy spaces. *Rev. Mat. Iberoam.* 31 (2015), 865-900. doi: 10.4171/RMI/857