Riesz transform characterization of weighted Hardy spaces associated to Schrödinger operators

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

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