Predual Spaces of Hardy Spaces Related to Fractional Schr\"odinger Operators

Abstract

Let $n\in\mathbb{N}$ and ${\alpha}\in(0,\min\{2,n\})$. For any $a\in[a^\ast,\infty)$, the fractional Schr\"odinger operator $L_\alpha$ is defined by \begin{equation*} L_\alpha:=(-\Delta)^{{\alpha}/2}+a{|x|}^{-{\alpha}}, \end{equation*} where $a^*:=-{\frac{2^{\alpha}{\Gamma}((d+{\alpha})/4)^2}{{\Gamma}((d-{\alpha})/4)^2}}$. Let $\gamma\in[0,\frac{\alpha}{n})$. In this paper, we introduce the VMO-type spaces $\mathrm{VMO}_{L_\alpha}^\gamma (\mathbb{R}^{n})$ associated with $L_\alpha$, and characterize these spaces via some tent spaces. We also prove that, for any given $p\in(\frac{n}{n+\alpha},1]$, the space $\mathrm{VMO}_{L_\alpha}^{\frac{1}{p}-1} (\mathbb{R}^{n})$ is the predual space of the Hardy space $H_{L_\alpha}^p\left(\mathbb{R}^{n}\right)$ related to $L_\alpha$.