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

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