Predual Spaces of Hardy Spaces Related to Fractional Schrödinger Operators

Abstract

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