Riesz transform characterization of weighted Hardy spaces associated to Schr\"{o}dinger operators

In this paper, we characterize the weighted local Hardy spaces $h^p_\rho(\omega)$ related to the critical radius function $\rho$ and weights $\omega\in A_{1}^{\rho,\,\infty}(\mathbb{R}^{n})$ by localized Riesz transforms $\widehat{R}_j$, in addition, we give a characterization of weighted Hardy spaces $H^{1}_{\cal L}(\omega)$ via Riesz tranforms associated to Schr\"{o}dinger operator $\cal L$, where $\L=-\Delta+V$ is a Schr\"{o}dinger operator on $\mathbb{R}^{n}$ ($n\ge 3$) and $V$ is a nonnegative function satisfying the reverse H\"older inequality.