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

Abstract

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 $θ(L)\in (0,\infty].$ Let $ω$ on $(0,\infty)$ be of upper type 1 and of critical lower type $\widetilde p_0(ω)\in (n/(n+θ(L)), 1]$ and $ρ(t)={t^{-1}}/ω^{-1}(t^{-1})$ for $t\in (0,\infty)$. In this paper, the authors first introduce the VMO-type space $\mathrm{VMO}_{ρ,L}({\mathbb R}^n)$ and the tent space $T^{\infty}_{ω,\mathrm v}({\mathbb R}^{n+1}_+)$ and characterize the space $\mathrm{VMO}_{ρ,L}({\mathbb R}^n)$ via the space $T^{\infty}_{ω,\mathrm v}({\mathbb R}^{n+1}_+)$. Let $\widetilde{T}_ω ({\mathbb R}^{n+1}_+)$ be the Banach completion of the tent space $T_ω({\mathbb R}^{n+1}_+)$. The authors then prove that $\widetilde{T}_ω({\mathbb R}^{n+1}_+)$ is the dual space of $T^{\infty}_{ω,\mathrm v}({\mathbb R}^{n+1}_+)$. As an application of this, the authors finally show that the dual space of $\mathrm{VMO}_{ρ,L^\ast}({\mathbb R}^n)$ is the space $B_{ω,L}({\mathbb R}^n)$, where $L^\ast$ denotes the adjoint operator of $L$ in $L^2({\mathbb R}^n)$ and $B_{ω,L}({\mathbb R}^n)$ the Banach completion of the Orlicz-Hardy space $H_{ω,L}({\mathbb R}^n)$. These results generalize the known recent results by particularly taking $ω(t)=t$ for $t\in (0,\infty)$.