Localized BMO and BLO Spaces on RD-Spaces and Applications to Schr\"odinger Operators

An RD-space ${\mathcal X}$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling condition holds in ${\mathcal X}$. Let $\rho$ be an admissible function on RD-space ${\mathcal X}$. The authors first introduce the localized spaces $\mathrm{BMO}_\rho({\mathcal X})$ and $\mathrm{BLO}_\rho({\mathcal X})$ and establish their basic properties, including the John-Nirenberg inequality for $\mathrm{BMO}_\rho({\mathcal X})$, several equivalent characterizations for $\mathrm{BLO}_\rho({\mathcal X})$, and some relations between these spaces. Then the authors obtain the boundedness on these localized spaces of several operators including the natural maximal operator, the Hardy-Littlewood maximal operator, the radial maximal functions and their localized versions associated to $\rho$, and the Littlewood-Paley $g$-function associated to $\rho$, where the Littlewood-Paley $g$-function and some of the radial maximal functions are defined via kernels which are modeled on the semigroup generated by the Schr\"odinger operator. These results apply in a wide range of settings, for instance, to the Schr\"odinger operator or the degenerate Schr\"odinger operator on ${{\mathbb R}}^d$, or the sub-Laplace Schr\"odinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.