Calderón reproducing formulas and applications to Hardy spaces

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 $H_D^p(\wedge T^{\ast }M)\subseteq L^p(\wedge T^{\ast }M)$, $1\le p\le 2$, for the Hardy spaces of differential forms introduced by Auscher, McIntosh and Russ, where $D=d+d^{\ast }$ 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 $H_D^1(\wedge T^{\ast }M)$. The embedding $H_L^p\subseteq L^p$, $1\le p\le 2$, where $L$ is either a divergence form elliptic operator on ${\mathbb{R}}^n$, 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 $-L^{\ast }$ is ultracontractive.