2013/03/31 by Auscher, Pascal, McIntosh, Alan, Morris, Andrew · 1 citation
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1304.0168
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 embedding HpD(\wedge T^*M) ⊆ Lp(\wedge T^*M), 1≤ p≤ 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 H1D(\wedge T^*M). The embedding HpL ⊆ Lp, 1≤ p≤ 2, where L is either a divergence form elliptic operator on \Rn, 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.