Generalized Vanishing Mean Oscillation Spaces Associated with Divergence Form Elliptic Operators

Abstract

Let $L$ be a divergence form elliptic operator with complex bounded measurable coefficients, $ω$ a positive concave function on $(0,\infty)$ of strictly critical lower type $p_ω\in (0, 1]$ and $ρ(t)={t^{-1}}/ω^{-1}(t^{-1})$ for $t\in (0,\infty).$ In this paper, the authors introduce the generalized VMO spaces ${\mathop\mathrm{VMO}_ {ρ, L}({\mathbb R}^n)}$ associated with $L$, and characterize them via tent spaces. As applications, the authors show that $(\mathrm{VMO}_{ρ,L} ({\mathbb R}^n))^\ast=B_{ω,L^\ast}({\mathbb R}^n)$, where $L^\ast$ denotes the adjoint operator of $L$ in $L^2({\mathbb R}^n)$ and $B_{ω,L^\ast}({\mathbb R}^n)$ the Banach completion of the Orlicz-Hardy space $H_{ω,L^\ast}({\mathbb R}^n)$. Notice that $ω(t)=t^p$ for all $t\in (0,\infty)$ and $p\in (0,1]$ is a typical example of positive concave functions satisfying the assumptions. In particular, when $p=1$, then $ρ(t)\equiv 1$ and $({\mathop\mathrm{VMO}_{1, L}({\mathbb R}^n)})^\ast=H_{L^\ast}^1({\mathbb R}^n)$, where $H_{L^\ast}^1({\mathbb R}^n)$ was the Hardy space introduced by Hofmann and Mayboroda.