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May  2010, 9(3): 779-812. doi: 10.3934/cpaa.2010.9.779

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

Dachun Yang 1, , Dongyong Yang 1, and  Yuan Zhou 1,

1. 

School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex systems, Ministry of Education, Beijing 100875, China, China, China

Received  April 2009 Revised  December 2009 Published  January 2010

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 $BMO_\rho(\mathcal X)$ and $BLO_\rho(\mathcal X)$ and establish their basic properties, including the John-Nirenberg inequality 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 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.
Keywords: Space of homogeneous type, connected and simply connected nilpotent Lie group, Heisenberg group, g-function., admissible function, maximal function, reverse Hölder inequality, localized BMO space, localized BLO space, Schrödinger operator.
Mathematics Subject Classification: Primary: 42B35; Secondary: 42B20, 42B25, 42B3.
Citation: Dachun Yang, Dongyong Yang, Yuan Zhou. Localized BMO and BLO spaces on RD-spaces and applications to Schrödinger operators. Communications on Pure & Applied Analysis, 2010, 9 (3) : 779-812. doi: 10.3934/cpaa.2010.9.779
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