Calderon Reproducing Formulas and Applications to Hardy Spaces

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We establish new Calderón reproducing formulas for self-adjoint 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 HᵖD(T*M) Lᵖ(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 polynomial volume growth. This fills a gap in that work. The new reproducing formulas also allow us to obtain an atomic characterisation of H¹D(T*M). The embedding Hᵖₗ Lᵖ, 1≤ p≤ 2, where L is either a divergence form elliptic operator on 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.