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 $\theta (L)\in (0,\infty ].$ Let $\omega$ on $(0,\infty )$ be of upper type 1 and of critical lower type ${\tilde{p}}_0(\omega )\in (n/(n+\theta (L)),1]$ and $\rho (t)=t^{-1}/{\omega }^{-1}(t^{-1})$ for $t\in (0,\infty )$. In this paper, the authors first introduce the VMO-type space ${\mathrm{VMO}}_{\rho ,L}({\mathbb{R}}^n)$ and the tent space $T_{\omega ,\mathrm{v}}^{\infty }({\mathbb{R}}_{+}^{n+1})$ and characterize the space ${\mathrm{VMO}}_{\rho ,L}({\mathbb{R}}^n)$ via the space $T_{\omega ,\mathrm{v}}^{\infty }({\mathbb{R}}_{+}^{n+1})$. Let ${\tilde{T}}_{\omega }({\mathbb{R}}_{+}^{n+1})$ 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 $T_{\omega ,\mathrm{v}}^{\infty }({\mathbb{R}}_{+}^{n+1})$. 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 $B_{\omega ,L}({\mathbb{R}}^n)$, where $L^{\ast }$ denotes the adjoint operator of $L$ 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 $t\in (0,\infty )$.

arXiv commentJ. Fourier Anal. Appl. (to appear)