Calderón reproducing formulas and applications to Hardy spaces

Abstract

We establish new Calderon 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 doubling 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 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.