Localized BMO and BLO spaces on RD-spacesand applications to Schrödinger operators

Abstract

An RD-space $\mathcal X$ is a space of homogeneoustype in the sense of Coifman and Weiss with the additional propertythat 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$BMO_\rho(\mathcal X)$ and $BLO_\rho(\mathcal X)$and establish their basic properties, including the John-Nirenberginequality for $BMO_\rho(\mathcal X)$,several equivalent characterizations for $BLO_\rho(\mathcal X)$,and some relations between these spaces.Then the authors obtain the boundedness on these localized spacesof several operators including the natural maximal operator,the Hardy-Littlewood maximal operator,the radial maximal functions and their localizedversions associated to $\rho$, and the Littlewood-Paley $g$-functionassociated to $\rho$, where the Littlewood-Paley $g$-functionand some of the radial maximal functions aredefined via kernels which are modeled on the semigroupgenerated by the Schrödinger operator.These results apply in a wide range of settings, for instance,to the Schrödinger operator orthe degenerate Schrödinger operator on $\mathbb R^d$,or the sub-Laplace Schrödinger operator on Heisenberggroups or connected and simply connected nilpotent Lie groups.