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

Abstract

Let $(X,d,\mu )$ be a space of homogeneous type, with the upper dimension $\omega$, in the sense of Coifman and Weiss. Assume that $\eta$ is the smoothness index of the wavelets on X constructed by Auscher and Hytönen. 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 functions, the radial maximal functions, the non-tangential maximal functions, the various Littlewood–Paley functions and wavelet functions. This completely answers the question of Coifman and Weiss by showing that no 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.