Some Estimates for Riesz Transforms Associated with Schrödinger Operators

Abstract

Let \(\mathcal{L}=-\Delta+V\) be the Schrödinger operator on \(\mathbb{R}^{n},\) where \(n\geq 3,\) and nonnegative potential \(V\) belongs to the reverse Hölder class \(RH_{q}\) with \(n/2\leq q<n.\) Let \(H^{p}_{\mathcal{L}}(\mathbb{R}^{n})\) denote the Hardy space related to \(\mathcal{L}\) and \(BMO_{\mathcal{L}}(\mathbb{R}^{n})\) denote the dual space of \(H^{1}_{\mathcal{L}}(\mathbb{R}^{n}).\) In this paper, we show that \(T_{\alpha,\beta}=V^{\alpha}\nabla\mathcal{L}^{-\beta}\) is bounded from \(H^{p_{1}}_{\mathcal{L}}(\mathbb{R}^{n})\) into \(L^{p_{2}}(\mathbb{R}^{n})\) for \(\dfrac{n}{n+\delta^{\prime}}<p_{1}\leq 1\) and \(\dfrac{1}{p_{2}}=\dfrac{1}{p_{1}}-\dfrac{2(\beta-\alpha)}{n},\) where \(\delta^{\prime}=\min\{1,2-n/q_{0}\}\) and \(q_{0}\) is the reverse Hölder index of \(V.\) Moreover, we prove \(T^{*}_{\alpha,\beta}\) is bounded on \(BMO_{\mathcal{L}}(\mathbb{R}^{n})\) when \(\beta-\alpha=1/2.\)