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

In this paper, we characterize the weighted local Hardy spaces $h_{\rho }^p(\omega )$ related to the critical radius function $\rho$ and weights $\omega \in A_1^{\rho ,\mathrm{\infty }}({\mathbb{R}}^n)$ by localized Riesz transforms ${\widehat{R}}_j$, in addition, we give a characterization of weighted Hardy spaces $H_{\mathcal{L}}^1(\omega )$ via Riesz tranforms associated to Schr"{o}dinger operator $\mathcal{L}$, where $\text{L}=-\mathrm{\Delta }+V$ 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.