Predual Spaces of Banach Completions of Orlicz-Hardy Spaces Associated with Operators

Let $L$ be a linear operator in $L^2({\mathbb{R}}^n)$ and generate an analytic semigroup $\{e^{-tL}{\}}_{t\ge 0}$ with kernels satisfying an upper bound of Poisson type, whose decay is measured by $\theta (L)\in (0,\mathrm{\infty }]\mathrm{.}$ Let $\omega$ on $(0,\mathrm{\infty })$ be of upper type 1 and of critical lower type ${\tilde{p}}_0(\omega )\in (n\mathrm{/}(n+\theta (L)),1]$ and $\rho (t)=t^{-1}\mathrm{/}{\omega }^{-1}(t^{-1})$ for $t\in (0,\mathrm{\infty })$. In this paper, the authors first introduce the VMO-type space $$\mathrm{V}\mathrm{M}\mathrm{O}${\rho,L}({\mathbb R}^n)$andthetentspace$T^{\infty}{\omega,\mathrm v}({\mathbb R}^{n+1}+)$andcharacterizethespace$$\mathrm{V}\mathrm{M}\mathrm{O}${\rho,L}({\mathbb R}^n)$viathespace$T^{\infty}{\omega,\mathrm v}({{\mathbb R}}^{n+1}+)$\mathrm{.}Let$\widetilde{T}{\omega} ({{\mathbb R}}^{n+1}+)$ be the Banach completion of the tent space $T_{\omega }({\mathbb{R}}_{+}^{n+1})$. The authors then prove that ${\tilde{T}}_{\omega }({\mathbb{R}}_{+}^{n+1})$ is the dual space of $T_{\omega ,\mathrm{v}}^{\mathrm{\infty }}({\mathbb{R}}_{+}^{n+1})$. As an application of this, the authors finally show that the dual space of ${\mathrm{V}\mathrm{M}\mathrm{O}}_{\rho ,L^{\ast }}({\mathbb{R}}^n)$ is the space $B_{\omega,L}({\mathbb R}^n)$,where$L^\ast$denotestheadjointoperatorof$L$in$L^2({\mathbb R}^n)$and$B_{\omega,L}({\mathbb R}^n)$ the Banach completion of the Orlicz-Hardy space $H_{\omega ,L}({\mathbb{R}}^n)$. These results generalize the known recent results by particularly taking $\omega (t)=t$ for $t\in (0,\infty)$.