Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrödinger operators

Abstract

Let $\mathcal{X}$ be a space of homogeneous type in the sense of Coifman and Weiss, and let $\mathcal{D}$ be a collection of balls in $\mathcal{X}$. The authors introduce the localized atomic Hardy space $H^{p,q}_{\mathcal{D}}({\mathcal{X}})$ with $p\in(0,1]$ and $q\in[1,\infty]\cap(p,\infty]$, the localized Morrey-Campanato space ${\mathcal{E}}^{\alpha,p}_{\mathcal{D}}({\mathcal{X}})$, and the localized Morrey-Campanato-BLO (bounded lower oscillation) space $\widetilde{\mathcal{E}}^{\alpha,p}_{\mathcal{D}}({\mathcal{X}})$ with $\alpha\in{\mathbb{R}}$ and $p\in(0,\infty)$, and they establish their basic properties, including $H^{p,q}_{\mathcal{D}}({\mathcal{X}})=H^{p,\infty}_{\mathcal{D}}({\mathcal{X}})$ and several equivalent characterizations for ${\mathcal{E}}^{\alpha,p}_{\mathcal{D}}({\mathcal{X}})$ and $\widetilde{\mathcal{E}}^{\alpha,p}_{\mathcal{D}}({\mathcal{X}})$. In particular, the authors prove that when $\alpha>0$ and $p\in[1,\infty)$, then $\widetilde{\mathcal{E}}^{\alpha,p}_{\mathcal{D}}(\mathcal{X})=\mathcal{E}^{\alpha,p}_{\mathcal{D}}(\mathcal{X})=\operatorname{Lip}_{\mathcal{D}}(\alpha;\mathcal{X})$, and when $p\in (0,1]$, then the dual space of $H^{p,\infty}_{\mathcal{D}}({\mathcal{X}})$ is ${\mathcal{E}}^{1/p-1,1}_{\mathcal{D}}({\mathcal{X}})$. Let $\rho$ be an admissible function modeled on the known auxiliary function determined by the Schrödinger operator. Denote the spaces ${\mathcal{E}}^{\alpha,p}_{\mathcal{D}}({\mathcal{X}})$ and $\widetilde{\mathcal{E}}^{\alpha,p}_{\mathcal{D}}({\mathcal{X}})$, respectively, by ${\mathcal{E}}^{\alpha,p}_{\rho}({\mathcal{X}})$ and $\widetilde{\mathcal{E}}^{\alpha,p}_{\rho}({\mathcal{X}})$, when $\mathcal{D}$ is determined by $\rho$. The authors then obtain the boundedness from ${\mathcal{E}}^{\alpha,p}_{\rho}({\mathcal{X}})$ to $\widetilde{\mathcal{E}}^{\alpha,p}_{\rho}({\mathcal{X}})$ of the radial and the Poisson semigroup maximal functions and the Littlewood-Paley $g$-function, which are defined via kernels modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, 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.