Estimates for the higher order Riesz transforms related to Schr\"odinger type operators
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We consider the Schr\"odinger type operator \(\mathcal{L}_k=(-\Delta)^k+V^k \) on \(\mathbb{R}^n ( n\geq 2k+1)\), where \(k=1,2\) and the nonnegative potential \(V\) belongs to the reverse H\"older class \(RH_s\) with \( n/2<s<n.\) In this paper, we establish the $(L^{p},L^{q})$-boundedness of the higher order Riesz transform $T_{\alpha,\beta}=V^{2\alpha} \nabla^2 \mathcal{L}_2^{-\beta}(0\leq \alpha\leq 1/2< \beta\leq 1,\beta-\alpha\geq 1/2)$ and its adjoint operator $T^*_{\alpha,\beta}$ respectively. We 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 $BMO_{\mathcal{L}_1}(\mathbb{R}^n)$ when $\beta-\alpha>1/2,$ where $p_1=\frac{n}{4(\beta-\alpha)-2}$, $p_2=\frac{n}{n-4(\beta-\alpha)+2}.$ Moreover, we prove that $T_{\alpha,\beta}$ is bounded from $BMO_{\mathcal{L}_1}(\mathbb{R}^n)$ to itself when $\beta-\alpha=1/2.$
Abstract
We consider the Schr\"odinger type operator \(\mathcal{L}_k=(-\Delta)^k+V^k \) on \(\mathbb{R}^n ( n\geq 2k+1)\), where \(k=1,2\) and the nonnegative potential \(V\) belongs to the reverse H\"older class \(RH_s\) with \( n/2<s<n.\) In this paper, we establish the $(L^{p},L^{q})$-boundedness of the higher order Riesz transform $T_{\alpha,\beta}=V^{2\alpha} \nabla^2 \mathcal{L}_2^{-\beta}(0\leq \alpha\leq 1/2< \beta\leq 1,\beta-\alpha\geq 1/2)$ and its adjoint operator $T^*_{\alpha,\beta}$ respectively. We 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 $BMO_{\mathcal{L}_1}(\mathbb{R}^n)$ when $\beta-\alpha>1/2,$ where $p_1=\frac{n}{4(\beta-\alpha)-2}$, $p_2=\frac{n}{n-4(\beta-\alpha)+2}.$ Moreover, we prove that $T_{\alpha,\beta}$ is bounded from $BMO_{\mathcal{L}_1}(\mathbb{R}^n)$ to itself when $\beta-\alpha=1/2.$
참고문헌 (10)
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