A Complete Real-Variable Theory of Hardy Spaces on Spaces of Homogeneous Type

TL;DR

This paper fully characterizes atomic Hardy spaces on spaces of homogeneous type using various real-variable methods, removing previous geometric restrictions and establishing optimal parameter ranges.

Contribution

It provides the first complete real-variable characterization of Hardy spaces on spaces of homogeneous type without additional geometric conditions.

Findings

Established equivalences of Hardy space norms via maximal and Littlewood-Paley functions.

Proved no extra geometric conditions are needed for radial maximal function characterization.

Derived criteria for boundedness of sublinear operators on Hardy spaces.

Abstract

Let $(X,d,\mu)$ be a space of homogeneous type, with the upper dimension $\omega$, in the sense of R. R. Coifman and G. Weiss. Assume that $\eta$ is the smoothness index of the wavelets on $X$ constructed by P. Auscher and T. Hyt\"onen. In this article, when $p\in(\omega/(\omega+\eta),1]$, for the atomic Hardy spaces $H_{\mathrm{cw}}^p(X)$ introduced by Coifman and Weiss, the authors establish their various real-variable characterizations, respectively, in terms of the grand maximal function, the radial maximal function, the non-tangential maximal functions, the various Littlewood-Paley functions and wavelet functions. This completely answers the question of R. R. Coifman and G. Weiss by showing that no any additional (geometrical) condition is necessary to guarantee the radial maximal function characterization of $H_{\mathrm{cw}}^1(X)$ and even of $H_{\mathrm{cw}}^p(X)$ with $p$ as above. As applications, the authors obtain the finite atomic characterizations of $H^p_{\mathrm{cw}}(X)$, which further induce some criteria for the boundedness of sublinear operators on $H^p_{\mathrm{cw}}(X)$. Compared with the known results, the novelty of this article is that $\mu$ is not assumed to satisfy the reverse doubling condition and $d$ is only a quasi-metric, moreover, the range $p\in(\omega/(\omega+\eta),1]$ is natural and optimal.