# Localized Morrey-Campanato Spaces on Metric Measure Spaces and Applications to Schrödinger Operators - Mathematics > Classical Analysis and ODEs

Abstract: Let ${\mathcal X}$ be a space of homogeneous type in the sense of Coifman andWeiss and ${\mathcal D}$ a collection of balls in $\cx$. The authors introducethe localized atomic Hardy space $H^{p, q} {\mathcal D}{\mathcal X}$ with$p\in 0,1$ and $q\in1,\infty\capp,\infty$, the localized Morrey-Campanatospace ${\mathcal E}^{\alpha, p} {\mathcal D}{\mathcal X}$ and the localizedMorrey-Campanato-BLO space $\widetilde{\mathcal E}^{\alpha, p} {\mathcalD}{\mathcal X}$ with $\az\in{\mathbb R}$ and $p\in0, \infty$ and establishtheir basic properties including $H^{p, q} {\mathcal D}{\mathcal X}=H^{p,\infty} {\mathcal D}{\mathcal X}$ and several equivalent characterizationsfor ${\mathcal E}^{\alpha, p} {\mathcal D}{\mathcal X}$ and $\wz{\mathcalE}^{\alpha, p} {\mathcal D}{\mathcal X}$. Especially, the authors prove thatwhen $p\in0,1$, the dual space of $H^{p, \infty} {\mathcal D}{\mathcal X}$is ${\mathcal E}^{1-p-1, 1} {\mathcal D}{\mathcal X}$. Let $ho$ be anadmissible function modeled on the known auxiliary function determined by theSchr\-odinger operator. Denote the spaces ${\mathcal E}^{\alpha, p} {\mathcalD}{\mathcal X}$ and $\widetilde{\mathcal E}^{\alpha, p} {\mathcalD}{\mathcal X}$, respectively, by ${\mathcal E}^{\alpha, p} { ho}{\mathcalX}$ and $\widetilde{\mathcal E}^{\alpha, p} { ho}{\mathcal X}$, when${\mathcal D}$ is determined by $ho$. The authors then obtain the boundednessfrom ${\mathcal E}^{\alpha, p} { ho}{\mathcal X}$ to $\widetilde{\mathcalE}^{\alpha, p} { ho}{\mathcal X}$ of the radial and the Poisson semigroupmaximal functions and the Littlewood-Paley $g$-function which are defined viakernels modeled on the semigroup generated by the Schr\-odinger operator.

Author: Dachun Yang, Dongyong Yang, Yuan Zhou

Source: https://arxiv.org/