Estimates for Riesz Transforms Associated with Schr\"odinger Type Operators
Bull. Korean Math. Soc.
Published online August 2, 2019
Yueshan Wang
Jiaozuo university
Abstract : Let \(\mathcal{L}_k=(-\Delta)^k+V^k (k=1,2)\) be the Schr\"odinger type operator, where nonnegative potential \(V\) belongs to the reverse H\"older class \(RH_s, s\geq n/2.\) In this paper, we consider the operator $T_{\alpha,\beta}=V^{2\alpha} \mathcal{L}_2^{-\beta}$ and its adjoint operator $T^*_{\alpha,\beta}$ for $0<\alpha\leq \beta\leq 1.$ We establish the $(L^{p_1},L^{p_2})$-boundedness of operators $T_{\alpha,\beta}$ and $T^*_{\alpha,\beta},$ respectively. We also show that $T_{\alpha,\beta}$ is bounded from Hardy type space \(H^1_{\mathcal{L}_1}(\mathbb{R}^n)\) 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 $p_1=\frac{n}{4(\beta-\alpha)}, p_2=\frac{n}{n-4(\beta-\alpha)}.$
Keywords : Riesz transform; Schr\"odinger operator; Hardy space,; BMO
MSC numbers : 42B35; 35J10
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