Local Hardy Spaces of Differential Forms on Riemannian Manifolds

Abstract

We define local Hardy spaces of differential forms $h_{\mathcal{D}}^p(\wedge T^{\ast }M)$ for all $p\in [1,\infty ]$ that are adapted to a class of first order differential operators $\mathcal{D}$ on a complete Riemannian manifold $M$ with at most exponential volume growth. In particular, if $D$ is the Hodge--Dirac operator on $M$ and $\Delta =D^2$ is the Hodge--Laplacian, then the local geometric Riesz transform $D(\Delta +aI{)}^{-1/2}$ has a bounded extension to $h_D^p$ for all $p\in [1,\infty ]$, provided that $a>0$ is large enough compared to the exponential growth of $M$. A characterisation of $h_{\mathcal{D}}^1$ in terms of local molecules is also obtained. These results can be viewed as the localisation of those for the Hardy spaces of differential forms $H_D^p(\wedge T^{\ast }M)$ introduced by Auscher, McIntosh and Russ.

55 pages, 1 figure, minor corrections made for publication

References in corpus (3)

• Conical square function estimates in UMD Banach spaces and applications to H-infinity functional calculi
• H^1 and BMO for certain nondoubling metric measure spaces
• Atomic decomposition of Hardy type spaces on certain noncompact manifolds

Cited by in corpus (1)

• Weighted Local Orlicz-Hardy Spaces on Domains and Their Applications in Inhomogeneous Dirichlet and Neumann Problems