Some Estimates for Riesz Transforms Associated with Schrödinger Operators

Pub Date : 2022-12-23 DOI:10.3103/s1068362322060097

Y. H. Wang

引用次数: 0

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 \(\beta-\alpha=1/2.\)

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与Schrödinger算子相关的Riesz变换的一些估计

AbstractLet \(\mathcal{L}=-\Delta+V\) 是Schrödinger的操作员 \(\mathbb{R}^{n},\) 在哪里 \(n\geq 3,\) 非负电位 \(V\) 属于反向Hölder类 \(RH_{q}\) 有 \(n/2\leq q<n.\) 让 \(H^{p}_{\mathcal{L}}(\mathbb{R}^{n})\) 表示与之相关的Hardy空间 \(\mathcal{L}\) 和 \(BMO_{\mathcal{L}}(\mathbb{R}^{n})\) 表示的对偶空间 \(H^{1}_{\mathcal{L}}(\mathbb{R}^{n}).\) 在本文中，我们证明了这一点 \(T_{\alpha,\beta}=V^{\alpha}\nabla\mathcal{L}^{-\beta}\) 是由 \(H^{p_{1}}_{\mathcal{L}}(\mathbb{R}^{n})\) 进入 \(L^{p_{2}}(\mathbb{R}^{n})\) 为了 \(\dfrac{n}{n+\delta^{\prime}}<p_{1}\leq 1\) 和 \(\dfrac{1}{p_{2}}=\dfrac{1}{p_{1}}-\dfrac{2(\beta-\alpha)}{n},\) 在哪里 \(\delta^{\prime}=\min\{1,2-n/q_{0}\}\) 和 \(q_{0}\) 反向的Hölder指数是 \(V.\) 此外，我们证明 \(T^{*}_{\alpha,\beta}\) 是有界的 \(BMO_{\mathcal{L}}(\mathbb{R}^{n})\) 什么时候 \(\beta-\alpha=1/2.\)

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