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^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 doubling 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 \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^* is ultracontractive.

Locations

• Revista Matemática Iberoamericana - View - PDF
• arXiv (Cornell University) - View - PDF
• HAL (Le Centre pour la Communication Scientifique Directe) - View - PDF
• University of Birmingham Research Portal (University of Birmingham) - View - PDF