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.
Categories: math.FA