Weighted local Hardy spaces associated to Schrödinger operators

Abstract

In this paper, we characterize the weighted local Hardy spaces $h^{p}_{\rho}(\omega)$ related to the critical radius function $\rho$ and weights $\omega\in A_{\infty}^{\rho,\infty}(\mathbb{R}^{n})$ which locally behave as Muckenhoupt’s weights and actually include them, by the local vertical maximal function, the local nontangential maximal function and the atomic decomposition. Then, we establish the equivalence of the weighted local Hardy space $h^{1}_{\rho}(\omega)$ and the weighted Hardy space $H^{1}_{\mathcal{L}}(\omega)$ associated to Schrödinger operators $\mathcal{L}$ with $\omega\in A_{1}^{\rho,\infty}(\mathbb{R}^{n})$. By the atomic characterization, we also prove the existence of finite atomic decompositions associated with $h^{p}_{\rho}(\omega)$. Furthermore, we establish boundedness in $h^{p}_{\rho}(\omega)$ of quasi-Banach-valued sublinear operators.