Radial Maximal Function Characterizations of Hardy Spaces on RD-Spaces and Their Applications

TL;DR

This paper characterizes Hardy spaces on RD-spaces using radial maximal functions, answering longstanding questions, and extends these results to localized Hardy spaces and manifolds with heat kernel estimates, with broad applications.

Contribution

It provides the first complete characterization of atomic Hardy spaces on RD-spaces via radial maximal functions for the optimal range of p, solving a problem posed in 1977.

Findings

Characterization of Hardy spaces on RD-spaces via radial maximal functions.

Extension of maximal function theory to localized Hardy spaces.

Application to Hardy spaces on manifolds with heat kernel estimates.

Abstract

Let ${\mathcal X}$ be an RD-space with $\mu({\mathcal X})=\infty$, which means that ${\mathcal X}$ is a space of homogeneous type in the sense of Coifman and Weiss and its measure has the reverse doubling property. In this paper, we characterize the atomic Hardy spaces $H^p_{\rm at}(\{\mathcal X})$ of Coifman and Weiss for $p\in(n/(n+1),1]$ via the radial maximal function, where $n$ is the "dimension" of ${\mathcal X}$, and the range of index $p$ is the best possible. This completely answers the question proposed by Ronald R. Coifman and Guido Weiss in 1977 in this setting, and improves on a deep result of Uchiyama in 1980 on an Ahlfors 1-regular space and a recent result of Loukas Grafakos et al in this setting. Moreover, we obtain a maximal function theory of localized Hardy spaces in the sense of Goldberg on RD-spaces by generalizing the above result to localized Hardy spaces and establishing the links between Hardy spaces and localized Hardy spaces. These results have a wide range of applications. In particular, we characterize the Hardy spaces $H^p_{\rm at}(M)$ via the radial maximal function generated by the heat kernel of the Laplace-Beltrami operator $\Delta$ on complete noncompact connected manifolds $M$ having a doubling property and supporting a scaled Poincar\'e inequality for all $p\in(n/(n+\alpha),1]$, where $\alpha$ represents the regularity of the heat kernel. This extends some recent results of Russ and Auscher-McIntosh-Russ.