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

Let $(\mathcal{X},d,\mu )$ be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions. Let $\rho\in (1,\infty)$,$0<p\le1\le q\le\infty$,$p\neq q$,$\gamma\in[1,\infty)$and$\epsilon\in(0,\infty)$. In this article, the authors introduce the atomic Hardy space ${\tilde{H}}_{\mathrm{a}\mathrm{t}\mathrm{b},\rho }^{p,q,\gamma }(\mu )$ and the molecular Hardy space ${\widetilde H_{\rm{mb},,\rho}^{p,,q,,\gamma,,\epsilon}(\mu)}$ via the discrete coefficient ${\tilde{K}}_{B,S}^{(\rho ),p}$, and prove that the Calder'on-Zygmund operator is bounded from ${\widetilde H_{\rm{mb},,\rho}^{p,,q,,\gamma,,\delta}(\mu)}$(or${\widetilde H_{\rm{atb},,\rho}^{p,,q,,\gamma}(\mu)}$)into$L^p(\mu)$,andfrom${\widetilde H_{\rm{atb},,\rho(\rho+1)}^{p,,q,,\gamma+1}(\mu)}$into${\widetilde H_{\rm{mb},,\rho}^{p,,q,,\gamma,,\frac12(\delta -\frac{\nu}{p}+\nu)}(\mu)}$ whose genealized fractional versions are also obtained. The authors also introduce the $\rho$-weakly doubling condition, with $\rho \in (1,\mathrm{\infty })$, of the measure $\mu$ and construct a non-doubling measure $\mu$ satisfying this condition. If $\mu$ is $\rho$-weakly doubling, the authors further introduce the Campanato space ${\mathcal E}^{\alpha,,q}{\rho,,\eta,,\gamma}(\mu)$andshowthat${\mathcal E}^{\alpha,,q}{\rho,,\eta,,\gamma}(\mu)$isindependentofthechoicesof$\rho$,$\eta$,$\gamma$and$q$; the authors then introduce the atomic Hardy space ${\widehat{H}}_{\mathrm{a}\mathrm{t}\mathrm{b},\rho }^{p,q,\gamma }(\mu )$ and the molecular Hardy space ${\widehat{H}}_{\mathrm{m}\mathrm{b},\rho }^{p,q,\gamma ,ϵ}(\mu )$, which coincide with each other; the authors finally prove that ${\widehat{H}}_{\mathrm{a}\mathrm{t}\mathrm{b},\rho }^{p,q,\gamma }(\mu )$ is the predual of ${\mathcal{E}}_{\rho ,\rho ,1}^{1\mathrm{/}p-1,1}(\mu )$. Moreover, relations of these Hardy spaces are also discussed.