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 $H_{\mathcal{D}}^{p,q}\left(\mathcal{X}\right)$ with $p\in (0,1]$ and $q\in [1,\infty ]\cap (p,\infty ]$, the localized Morrey-Campanato space ${\mathcal{E}}_{\mathcal{D}}^{\alpha ,p}\left(\mathcal{X}\right)$, and the localized Morrey-Campanato-BLO (bounded lower oscillation) space ${\tilde{\mathcal{E}}}_{\mathcal{D}}^{\alpha ,p}\left(\mathcal{X}\right)$ with $\alpha \in \mathbb{R}$ and $p\in (0,\infty )$, and they establish their basic properties, including $H_{\mathcal{D}}^{p,q}\left(\mathcal{X}\right)=H_{\mathcal{D}}^{p,\infty }\left(\mathcal{X}\right)$ and several equivalent characterizations for ${\mathcal{E}}_{\mathcal{D}}^{\alpha ,p}\left(\mathcal{X}\right)$ and ${\tilde{\mathcal{E}}}_{\mathcal{D}}^{\alpha ,p}\left(\mathcal{X}\right)$. In particular, the authors prove that when $\alpha >0$ and $p\in [1,\infty )$, then ${\tilde{\mathcal{E}}}_{\mathcal{D}}^{\alpha ,p}\left(\mathcal{X}\right)={\mathcal{E}}_{\mathcal{D}}^{\alpha ,p}\left(\mathcal{X}\right)={Lip}_{\mathcal{D}}(\alpha ;\mathcal{X})$, and when $p\in (0,1]$, then the dual space of $H_{\mathcal{D}}^{p,\infty }\left(\mathcal{X}\right)$ is ${\mathcal{E}}_{\mathcal{D}}^{1/p-1,1}\left(\mathcal{X}\right)$. Let $\rho$ be an admissible function modeled on the known auxiliary function determined by the Schrödinger operator. Denote the spaces ${\mathcal{E}}_{\mathcal{D}}^{\alpha ,p}\left(\mathcal{X}\right)$ and ${\tilde{\mathcal{E}}}_{\mathcal{D}}^{\alpha ,p}\left(\mathcal{X}\right)$, respectively, by ${\mathcal{E}}_{\rho }^{\alpha ,p}\left(\mathcal{X}\right)$ and ${\tilde{\mathcal{E}}}_{\rho }^{\alpha ,p}\left(\mathcal{X}\right)$, when $\mathcal{D}$ is determined by $\rho$. The authors then obtain the boundedness from ${\mathcal{E}}_{\rho }^{\alpha ,p}\left(\mathcal{X}\right)$ to ${\tilde{\mathcal{E}}}_{\rho }^{\alpha ,p}\left(\mathcal{X}\right)$ 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.