Estimates for Riesz transforms associated with Schr\"odinger type operators
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Let \(\mathcal{L}_2=(-\Delta)^2+V^2 \) be the Schr\"odinger type operator, where nonnegative potential \(V\) belongs to the reverse H\"older class $RH_s$, $s> n/2$. In this paper, we consider the operator $T_{\alpha,\beta}=V^{2\alpha} \mathcal{L}_2^{-\beta}$ and its conjugate $T^*_{\alpha,\beta}$, where $0<\alpha\leq \beta\leq 1$. We establish the $(L^{p},L^{q})$-boundedness of operator $T_{\alpha,\beta}$ and $T^*_{\alpha,\beta}$, respectively, we also show that $T_{\alpha,\beta}$ is bounded from Hardy type space \(H^1_{\mathcal{L}_2}(\mathbb{R}^n)\) into $L^{p_2}(\mathbb{R}^n)$ and $T^*_{\alpha,\beta}$ is bounded from $L^{p_1}(\mathbb{R}^n)$ into $BMO$ type space $BMO_{\mathcal{L}_1}(\mathbb{R}^n)$, where $p_1=\frac{n}{4(\beta-\alpha)}$, $p_2=\frac{n}{n-4(\beta-\alpha)}$.
Abstract
Let \(\mathcal{L}_2=(-\Delta)^2+V^2 \) be the Schr\"odinger type operator, where nonnegative potential \(V\) belongs to the reverse H\"older class $RH_s$, $s> n/2$. In this paper, we consider the operator $T_{\alpha,\beta}=V^{2\alpha} \mathcal{L}_2^{-\beta}$ and its conjugate $T^*_{\alpha,\beta}$, where $0<\alpha\leq \beta\leq 1$. We establish the $(L^{p},L^{q})$-boundedness of operator $T_{\alpha,\beta}$ and $T^*_{\alpha,\beta}$, respectively, we also show that $T_{\alpha,\beta}$ is bounded from Hardy type space \(H^1_{\mathcal{L}_2}(\mathbb{R}^n)\) into $L^{p_2}(\mathbb{R}^n)$ and $T^*_{\alpha,\beta}$ is bounded from $L^{p_1}(\mathbb{R}^n)$ into $BMO$ type space $BMO_{\mathcal{L}_1}(\mathbb{R}^n)$, where $p_1=\frac{n}{4(\beta-\alpha)}$, $p_2=\frac{n}{n-4(\beta-\alpha)}$.
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