Hardy spaces $H^p$ over non-homogeneous metric measure spaces and their applications
Let $({\mathcal X},d,μ)$ be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions. Let $ρ\in (1,\infty)$, $0<p\le1\le q\le\infty$, $p\neq q$, $γ\in[1,\infty)$ and $ε\in(0,\infty)$. In this article, the authors introduce the atomic Hardy space ${\widetilde H_{\mathrm{atb},\,ρ}^{p,\,q,\,γ}(μ)}$ and the molecular Hardy space ${\widetilde H_{\rm{mb},\,ρ}^{p,\,q,\,γ,\,ε}(μ)}$ via the discrete coefficient $\widetilde{K}^{(ρ),\,p}_{B,\,S}$, and prove that the Calderón-Zygmund operator is bounded from ${\widetilde H_{\rm{mb},\,ρ}^{p,\,q,\,γ,\,δ}(μ)}$ (or ${\widetilde H_{\rm{atb},\,ρ}^{p,\,q,\,γ}(μ)}$) into $L^p(μ)$, and from ${\widetilde H_{\rm{atb},\,ρ(ρ+1)}^{p,\,q,\,γ+1}(μ)}$ into ${\widetilde H_{\rm{mb},\,ρ}^{p,\,q,\,γ,\,\frac12(δ-\fracν{p}+ν)}(μ)}$ whose genealized fractional versions are also obtained. The authors also introduce the $ρ$-weakly doubling condition, with $ρ\in (1,\infty)$, of the measure $μ$ and construct a non-doubling measure $μ$ satisfying this condition. If $μ$ is $ρ$-weakly doubling, the authors further introduce the Campanato space ${\mathcal E}^{α,\,q}_{ρ,\,η,\,γ}(μ)$ and show that ${\mathcal E}^{α,\,q}_{ρ,\,η,\,γ}(μ)$ is independent of the choices of $ρ$, $η$, $γ$ and $q$; the authors then introduce the atomic Hardy space $\widehat H_{\rm{atb},\,ρ}^{p,\,q,\,γ}(μ)$ and the molecular Hardy space $\widehat H_{\rm{mb},\,ρ}^{p,\,q,\,γ,\,ε}(μ)$, which coincide with each other; the authors finally prove that $\widehat{H}_{\rm{atb},\,ρ}^{p,\,q,\,γ}(μ)$ is the predual of ${\mathcal E}^{1/p-1,\,1}_{ρ,\,ρ,\,1}(μ)$. Moreover, relations of these Hardy spaces are also discussed.