Calderón reproducing formulas and applications to Hardy spaces

Abstract

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 embeddingfor 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 (∧T * M ). The embedding H p L ⊆ L p , 1 ≤ p ≤ 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.

Riesz transforms through reverse Hölder and Poincaré inequalities

“…The self-adjointness allows us to work with a better functional calculus than just H ∞ functional calculus, namely functional calculus based on the Fourier transform. As investigated in [9], this calculus interacts nicely with the finite propagation speed, and yields sharper off-diagonal estimates for operators generated by the calculus. propagates at distance at most r. That is, for all f ∈ L 2 (X, µ) with supp f ⊆ E ⊆ X, one has…”

The Hodge–de Rham Laplacian and Lp-boundedness of Riesz transforms on non-compact manifolds

“…can also be obtained by combining the results in [6] and [5] by Auscher, McIntosh, Russ and Morris. Here we give a somewhat direct proof by using the Davies-Gaffney estimates.…”

“…The Hardy spaces H p L associated with operators on metric measured spaces have been studied by several authors, see for example [2,6,5,15,[23][24][25]. The operator L is either acting on functions or differential 1-forms.…”

Hardy and Bmo Spaces on Graphs, Application to Riesz Transform

“…On Riemannian manifolds or in more general contexts, the proof of this fact is much more complicated and is the main result of [2]. Let us emphasize that the proofs of our main results does not go through the E 1…”

“…Let us now prove that the completion of E 1 quad,β ( ) in L 1 exists. To that purpose, it is enough (see Proposition 2.2 in [2]) to check that, for all Cauchy sequences (f n ) n in E 1 quad,β ( ) that converges to 0 in L 1 ( ), f n → 0 for the . H 1 quad,β norm.…”