# 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

## Abstract

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 $H_{D}(∧T_{∗}M)⊆L_{p}(∧T_{∗}M)$, $1≤p≤2$, 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 $H_{D}(∧T_{∗}M)$. The embedding $H_{L}⊆L_{p}$, $1≤p≤2$, where $L$ is either a divergence form elliptic operator on $R_{n}$, 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.

## 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