Localized BMO and BLO Spaces on RD-Spaces and Applications to Schrödinger Operators

Abstract

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 Schrödinger operator. These results apply in a wide range of settings, for instance, to the Schrödinger operator or the degenerate Schrödinger operator on ${\mathbb{R}}^d$, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

Commun. Pure Appl. Anal. (to appear)

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