Calderón reproducing formulas and applications to Hardy spaces

We establish new Calderón reproducing formulas for selfadjoint operators D that generate strongly continuous groups with finite propagation speed. These formulas allow the analysing function to interact with D through 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^p(∧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_D¹ (∧T^∗M). The embedding H_L^p ⊆ L^p, 1 ≤ p ≤ 2, where L is either a divergence form elliptic operator on Rⁿ, 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.