Vanishing Mean Oscillation Spaces Associated with Operators Satisfying Davies-Gaffney Estimates

Let $(\mathcal{X}, d, μ)$ be a metric measure space, $L$ a linear operator which has a bounded $H_\infty$ functional calculus and satisfies the Davies-Gaffney estimate, $Φ$ a concave function on $(0,\infty)$ of critical lower type $p_Φ^-\in(0,1]$ and $ρ(t)\equiv t^{-1}/Φ^{-1}(t^{-1})$ for all $t\in(0,\infty)$. In this paper, the authors introduce the generalized VMO space ${\mathrm {VMO}}_{ρ,L}({\mathcal X})$ associated with $L$, and establish its characterization via the tent space. As applications, the authors show that $({\mathrm {VMO}}_{ρ,L}({\mathcal X}))^*=B_{Φ,L^*}({\mathcal X})$, where $L^*$ denotes the adjoint operator of $L$ in $L^2({\mathcal X})$ and $B_{Φ,L^*}({\mathcal X})$ the Banach completion of the Orlicz-Hardy space $H_{Φ,L^*}({\mathcal X})$.

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