Predual Spaces of Hardy Spaces Related to Fractional Schrödinger Operators

Let $n\in \mathbb{N}$ and $\alpha \in (0,\min \{2,n\})$. For any $a\in [a^{\ast },\mathrm{\infty })$, the fractional Schrödinger operator $L_{\alpha }$ is defined by \begin{equation*} L_α:=(-Δ)^{α/2}+a{|x|}^{-α}, \end{equation*} where $a^{\ast }:=-\frac{2^{\alpha }\mathrm{\Gamma }((d+\alpha )\mathrm{/}4{)}^2}{\mathrm{\Gamma }((d-\alpha )\mathrm{/}4{)}^2}$. Let $\gamma \in [0,\frac{\alpha }{n})$. In this paper, we introduce the VMO-type spaces ${\mathrm{V}\mathrm{M}\mathrm{O}}_{L_{\alpha }}^{\gamma }({\mathbb{R}}^n)$ 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 ${\mathrm{V}\mathrm{M}\mathrm{O}}_{L_{\alpha }}^{\frac{1}{p}-1}({\mathbb{R}}^n)$ is the predual space of the Hardy space $H_{L_{\alpha }}^p\left({\mathbb{R}}^n\right)$ related to $L_{\alpha }$.