2009/03/26 by Yang, Dachun, Yang, Dongyong, Zhou, Yuan
#42B20 #42B25 #42B30 (Secondary) #42B35 (Primary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.0903.4536
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 ρ be an admissible function on RD-space \mathcal X. The authors first introduce the localized spaces BMOρ(\mathcal X) and BLOρ(\mathcal X) and establish their basic properties, including the John-Nirenberg inequality for BMOρ(\mathcal X), several equivalent characterizations for BLOρ(\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 ρ, and the Littlewood-Paley g-function associated to ρ, 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 Rd, or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.