Predual Spaces of Hardy Spaces Related to Fractional Schrödinger Operators

Abstract

Let $n\in \mathbb{N}$ and $\hat{I}\pm \in (0,\min \{2,n\})$. For any $a\in [a^{\ast },\infty )$, the fractional Schrödinger operator $L_{\hat{I}}\pm$ is defined by \begin{equation*} L_α:=(-Δ)^{α/2}+a{|x|}^{-α}, \end{equation*} where $a^{\ast }:=-\frac{2^{\hat{I}}\pm \hat{I}“((d+\hat{I}\pm )/4{)}^2}{\hat{I}“((d-\hat{I}\pm )/4{)}^2}$. Let ${\hat{I}}^3\in [0,\frac{\hat{I}}{\pm }n)$. In this paper, we introduce the VMO-type spaces ${\mathrm{VMO}}_{L_{\hat{I}}\pm }^3({\mathbb{R}}^n)$ associated with $L_{\hat{I}}\pm$, and characterize these spaces via some tent spaces. We also prove that, for any given $p\in (\frac{n}{n+\hat{I}\pm },1]$, the space ${\mathrm{VMO}}_{L_{\hat{I}}\pm }^{\frac{1}{p}-1}({\mathbb{R}}^n)$ is the predual space of the Hardy space $H_{L_{\hat{I}}\pm }^p\left({\mathbb{R}}^n\right)$ related to $L_{\hat{I}}\pm$.