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 $\mathcal{D}$ a collection of balls in $\cx$. The authors introduce the localized atomic Hardy space $H_{\mathcal{D}}^{p,q}(\mathcal{X})$ with $p\in (0,1]$ and $q\in [1,\infty ]\cap (p,\infty ]$, the localized Morrey-Campanato space ${\mathcal{E}}_{\mathcal{D}}^{\alpha ,p}(\mathcal{X})$ and the localized Morrey-Campanato-BLO space ${\tilde{\mathcal{E}}}_{\mathcal{D}}^{\alpha ,p}(\mathcal{X})$ with $\az\in{\mathbb R}$ and $p\in (0,\infty )$ and establish their basic properties including $H_{\mathcal{D}}^{p,q}(\mathcal{X})=H_{\mathcal{D}}^{p,\infty }(\mathcal{X})$ and several equivalent characterizations for ${\mathcal{E}}_{\mathcal{D}}^{\alpha ,p}(\mathcal{X})$ and $\wz{\mathcal E}^{α, p}_{\mathcal D}({\mathcal X})$. Especially, the authors prove that when $p\in (0,1]$, the dual space of $H_{\mathcal{D}}^{p,\infty }(\mathcal{X})$ is ${\mathcal{E}}_{\mathcal{D}}^{1/p-1,1}(\mathcal{X})$. Let $\rho$ be an admissible function modeled on the known auxiliary function determined by the Schrödinger operator. Denote the spaces ${\mathcal{E}}_{\mathcal{D}}^{\alpha ,p}(\mathcal{X})$ and ${\tilde{\mathcal{E}}}_{\mathcal{D}}^{\alpha ,p}(\mathcal{X})$, respectively, by ${\mathcal{E}}_{\rho }^{\alpha ,p}(\mathcal{X})$ and ${\tilde{\mathcal{E}}}_{\rho }^{\alpha ,p}(\mathcal{X})$, when $\mathcal{D}$ is determined by $\rho$. The authors then obtain the boundedness from ${\mathcal{E}}_{\rho }^{\alpha ,p}(\mathcal{X})$ to ${\tilde{\mathcal{E}}}_{\rho }^{\alpha ,p}(\mathcal{X})$ 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.

Nagoya Math. J. (to appear)

References in corpus (1)

• Localized BMO and BLO Spaces on RD-Spaces and Applications to Schrödinger Operators