Hardy spaces $H^p$ over non-homogeneous metric measure spaces and their applications

Abstract

Let $(\mathcal{X},d,\hat{I}¼)$ be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions. Let $\ddot{I}\in (1,\infty )$, $0<p\le 1\le q\le \infty$, $p\ne q$, ${\hat{I}}^3\in [1,\infty )$ and $\hat{I}\mu \in (0,\infty )$. In this article, the authors introduce the atomic Hardy space ${\tilde{H}}_{\mathrm{atb},\ddot{I}}^{p,q,{\hat{I}}^3}(\hat{I}¼)$ and the molecular Hardy space ${\tilde{H}}_{\mathrm{mb},\ddot{\mathrm{I}}}^{p,q,{\hat{I}}^3,\hat{I}\mu }(\hat{I}¼)$ via the discrete coefficient ${\tilde{K}}_{B,S}^{(\ddot{I}),p}$, and prove that the Calderón-Zygmund operator is bounded from ${\tilde{H}}_{\mathrm{mb},\ddot{\mathrm{I}}}^{p,q,{\hat{I}}^3,\hat{I}´}(\hat{I}¼)$ (or ${\tilde{H}}_{\mathrm{atb},\ddot{\mathrm{I}}}^{p,q,{\hat{I}}^3}(\hat{I}¼)$) into $L^p(\hat{I}¼)$, and from ${\tilde{H}}_{\mathrm{atb},\ddot{\mathrm{I}}(\ddot{\mathrm{I}}+1)}^{p,q,{\hat{I}}^3+1}(\hat{I}¼)$ into ${\tilde{H}}_{\mathrm{mb},\ddot{\mathrm{I}}}^{p,q,{\hat{I}}^3,\frac{1}{2}(\hat{I}´-\frac{\hat{I}}{½}p+\hat{I}½)}(\hat{I}¼)$ whose genealized fractional versions are also obtained. The authors also introduce the $\ddot{I}$-weakly doubling condition, with $\ddot{I}\in (1,\infty )$, of the measure $\hat{I}¼$ and construct a non-doubling measure $\hat{I}¼$ satisfying this condition. If $\hat{I}¼$ is $\ddot{I}$-weakly doubling, the authors further introduce the Campanato space ${\mathcal{E}}_{\ddot{I},\hat{I}\cdot ,{\hat{I}}^3}^{\hat{I}\pm ,q}(\hat{I}¼)$ and show that ${\mathcal{E}}_{\ddot{I},\hat{I}\cdot ,{\hat{I}}^3}^{\hat{I}\pm ,q}(\hat{I}¼)$ is independent of the choices of $\ddot{I}$, $\hat{I}\cdot$, ${\hat{I}}^3$ and $q$; the authors then introduce the atomic Hardy space ${\hat{H}}_{\mathrm{atb},\ddot{\mathrm{I}}}^{p,q,{\hat{I}}^3}(\hat{I}¼)$ and the molecular Hardy space ${\hat{H}}_{\mathrm{mb},\ddot{\mathrm{I}}}^{p,q,{\hat{I}}^3,\hat{I}\mu }(\hat{I}¼)$, which coincide with each other; the authors finally prove that ${\hat{H}}_{\mathrm{atb},\ddot{\mathrm{I}}}^{p,q,{\hat{I}}^3}(\hat{I}¼)$ is the predual of ${\mathcal{E}}_{\ddot{I},\ddot{I},1}^{1/p-1,1}(\hat{I}¼)$. Moreover, relations of these Hardy spaces are also discussed.

82 pages, Sci. China Math. (to appear)