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

Abstract

Let L be a linear operator in L ²(ℝⁿ) and generate an analytic semigroup {e ^−tL}_t≥0 with kernels satisfying an upper bound of Poisson type, whose decay is measured by θ(L)∈(0,∞]. Let ω on (0,∞) be of upper type 1 and of critical lower type $\widetilde{p}_{0}(\omega)\in(n/(n+\theta(L)),1]$ and ρ(t)=t ⁻¹/ω ⁻¹(t ⁻¹) for t∈(0,∞). In this paper, the authors first introduce the VMO-type space VMO_ρ,L(ℝⁿ) and the tent space $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ and characterize the space VMO_ρ,L(ℝⁿ) via the space $T^{\infty}_{\omega,\mathrm{v}}({{\mathbb{R}}}^{n+1}_{+})$. 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 $\widetilde{T}_{\omega}({\mathbb{R}}^{n+1}_{+})$ is the dual space of $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$. As an application of this, the authors finally show that the dual space of $\mathrm{VMO}_{\rho,L^{\ast}}({\mathbb{R}}^{n})$ is the space B _ω,L(ℝⁿ), where L ^* denotes the adjoint operator of L in L ²(ℝⁿ) and B _ω,L(ℝⁿ) the Banach completion of the Orlicz-Hardy space H _ω,L(ℝⁿ). These results generalize the known recent results by particularly taking ω(t)=t for t∈(0,∞).