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

Abstract

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

40 pages, Kyoto J. Math. (to appear)