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논문 기본 정보
- 자료유형
- 학술저널
- 저자정보
- 저널정보
- 대한수학회 대한수학회지 대한수학회지 제59권 제2호
- 발행연도
- 2022.3
- 수록면
- 235 - 254 (20page)
- DOI
- 10.4134/JKMS.j200484
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초록· 키워드
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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 $L^{1}(\mathbb{H}^{n})$.
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