Hardy type estimates for Riesz transforms associated with Schr\"{o}dinger operators on the Heisenberg group
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Let $\mathbb{H}^{n}$ be the Heisenberg group and $Q=2n+2$ be its homogeneous dimension. Let $\mathcal{L}=-\Delta_{\mathbb{H}^{n}}+V$ be the Schr\"{o}dinger operator on $\mathbb{H}^{n}$, where $\Delta_{\mathbb{H}^{n}}$ is the sub-Laplacian and the nonnegative potential $V$ belongs to the reverse H\"{o}lder class $B_{q_{1}}$ for $q_{1}\geq Q/2$. Let ${H_{\mathcal{L}}^{p}(\mathbb{H}^{n})}$ be the Hardy space associated with the Schr\"{o}dinger operator $\mathcal{L}$ for $Q/(Q+\delta_{0})<p\leq1$, where $\delta_{0}=\min\{1,2-Q/q_{1}\}$. In this paper, we consider the Hardy type estimates for the operator $T_{\alpha}=V^{\alpha}(-\Delta_{\mathbb{H}^{n}}+V)^{-\alpha}$, and the commutator $[b,T_{\alpha}]$, where $0<\alpha<Q/2$. We prove that $T_{\alpha}$ is bounded from ${H_{\mathcal{L}}^{p}(\mathbb{H}^{n})}$ into $L^{p}(\mathbb{H}^{n})$. Suppose that $b\in BMO_{\mathcal{L}}^{\theta}(\mathbb{H}^{n})$, which is larger than $ BMO(\mathbb{H}^{n})$. We show that the commutator $[b,T_{\alpha}]$ is bounded from $H_{\mathcal{L}}^{1}(\mathbb{H}^{n})$ into weak $L^{1}(\mathbb{H}^{n})$.
Abstract
Let $\mathbb{H}^{n}$ be the Heisenberg group and $Q=2n+2$ be its homogeneous dimension. Let $\mathcal{L}=-\Delta_{\mathbb{H}^{n}}+V$ be the Schr\"{o}dinger operator on $\mathbb{H}^{n}$, where $\Delta_{\mathbb{H}^{n}}$ is the sub-Laplacian and the nonnegative potential $V$ belongs to the reverse H\"{o}lder class $B_{q_{1}}$ for $q_{1}\geq Q/2$. Let ${H_{\mathcal{L}}^{p}(\mathbb{H}^{n})}$ be the Hardy space associated with the Schr\"{o}dinger operator $\mathcal{L}$ for $Q/(Q+\delta_{0})<p\leq1$, where $\delta_{0}=\min\{1,2-Q/q_{1}\}$. In this paper, we consider the Hardy type estimates for the operator $T_{\alpha}=V^{\alpha}(-\Delta_{\mathbb{H}^{n}}+V)^{-\alpha}$, and the commutator $[b,T_{\alpha}]$, where $0<\alpha<Q/2$. We prove that $T_{\alpha}$ is bounded from ${H_{\mathcal{L}}^{p}(\mathbb{H}^{n})}$ into $L^{p}(\mathbb{H}^{n})$. Suppose that $b\in BMO_{\mathcal{L}}^{\theta}(\mathbb{H}^{n})$, which is larger than $ BMO(\mathbb{H}^{n})$. We show that the commutator $[b,T_{\alpha}]$ is bounded from $H_{\mathcal{L}}^{1}(\mathbb{H}^{n})$ into weak $L^{1}(\mathbb{H}^{n})$.
참고문헌 (13)
- B. Bongioanni, "Commutators of Riesz transforms related to Schrödinger operators", J. Fourier Anal. Appl, 17(1), pp.115~134 (2011)[DOI]
- R. R. Coifman, "Factorization theorems for Hardy spaces in several variables", Ann. of Math, 103(3), pp.611~635 (1976)[DOI]
- Y. Hu, "Hardy type estimates for Riesz transforms associated with Schrödinger operators", Anal. Math. Phys, 9(1), pp.275~287 (2019)[DOI]
- P. Li, "A compactness criterion and application to the commutators associated with Schrödinger operators", Math. Nachr, 288(2-3), pp.235~248 (2015)[DOI]
- P. Li, "Lp boundedness of commutator operator associated with Schrödinger operators on Heisenberg group", Acta Math. Sci. Ser. B (Engl. Ed.), 32(2), pp.568~578 (2012)[DOI]
- P. Li, "The boundedness of commutators associated with Schrödinger operators on Herz spaces", J. Inequal. Appl, 2016(172), pp.27~ (2016)[DOI]
- P. Li, "Schrödinger type operators on generalized Morrey spaces", J. Inequal. Appl, 2015(229), pp.21~ (2015)[DOI]
- "Hardy spaces associated with Schrödinger operators on the Heisenberg group"[URL]
- Y. Liu, "Hardy type estimates for Riesz transforms associated with Schrödinger operators on the Heisenberg group", Anal. Theory Appl, 32(1), pp.78~89 (2016)[DOI]
- Y. Liu, "A note for Riesz transforms associated with Schrödinger operators on the Heisenberg group", Anal. Math. Phys, 7(1), pp.31~45 (2017)[DOI]
- G. Lu, "A Fefferman-Phong type inequality for degenerate vector fields and applications", PanAmer. Math. J, 6(4), pp.37~57 (1996)
- G. Pradolini, "Commutators of singular integrals on spaces of homogeneous type", Czechoslovak Math. J, 57(1), pp.75~93 (2007)[DOI]
- Z. Shen, "Lp estimates for Schrödinger operators with certain potentials", Ann. Inst. Fourier (Grenoble), 45(2), pp.513~546 (1995)