Calderon Reproducing Formulas and Applications to Hardy Spaces

TL;DR

This paper develops new Calderón reproducing formulas for self-adjoint operators, enabling advanced analysis of Hardy spaces of differential forms on manifolds and establishing key embeddings and atomic characterizations.

Contribution

It introduces novel Calderón reproducing formulas for operators with finite propagation speed, filling gaps in Hardy space theory on manifolds and for elliptic operators.

Findings

Established embeddings of Hardy spaces into L^p spaces for differential forms.

Provided atomic characterisation of Hardy spaces of differential forms.

Extended embedding results to divergence form elliptic operators and self-adjoint operators with Davies--Gaffney estimates.

Abstract

We establish new Calder\'{o}n 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 $H^p_D(\wedge T^*M) \subseteq L^p(\wedge T^*M)$, $1\leq p\leq 2$, 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 $H^1_D(\wedge T^*M)$. The embedding $H^p_L \subseteq L^p$, $1\leq p\leq 2$, where $L$ is either a divergence form elliptic operator on $\R^n$, 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 $-L^*$ is ultracontractive.