We establish new Calderón holomorphic functional calculus whilst the synthesising function interacts with through functional calculus based on the Fourier transform. We apply these to prove the embedding , , for the Hardy spaces of differential forms introduced by Auscher, McIntosh and Russ, where is the Hodge–Dirac operator on a complete Riemannian manifold 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 . The embedding , , where is either a divergence form elliptic operator on , 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 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. 865–900DOI 10.4171/RMI/857