Local Hardy Spaces of Differential Forms on Riemannian Manifolds

TL;DR

This paper introduces local Hardy spaces for differential forms on Riemannian manifolds, establishing boundedness of certain geometric Riesz transforms and characterizing these spaces via local molecules, extending previous global Hardy space theories.

Contribution

It defines local Hardy spaces of differential forms adapted to first order operators on manifolds with exponential volume growth, and proves boundedness of local Riesz transforms and molecular characterizations.

Findings

Bounded extension of local geometric Riesz transform to $h^p_D$ for all $p$

Characterization of $h^1_{ ext D}$ via local molecules

Extension of global Hardy space theory to localized setting

Abstract

We define local Hardy spaces of differential forms $h^p_{\mathcal D}(\wedge T^*M)$ 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 $h^p_D$ for all $p\in[1,\infty]$, provided that $a>0$ is large enough compared to the exponential growth of $M$. A characterisation of $h^1_{\mathcal D}$ 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 $H^p_D(\wedge T^*M)$ introduced by Auscher, McIntosh and Russ.