Estimates for the Riesz transforms associated with Schr\"odinger type operators on the Heisenberg group
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
We consider the Schr\"odinger type operator \(\mathcal{L}=(-\Delta_{\mathbb{H}^n})^2+V^2 \) on the Heisenberg group $\mathbb{H}^n,$ where $\Delta_{\mathbb{H}^n}$ is the sub-Laplacian and the non-negative potential \(V\) belongs to the reverse H\"older class \(RH_s$ for $ s\geq Q/2$ and $Q\geq 6.\) We shall establish the $(L^p,L^q)$ estimates for the Riesz transforms $ T_{\alpha,\beta,j} =V^{2\alpha}\nabla_{\mathbb{H}^n}^j \mathcal{L}^{-\beta},~j=0,1,2,3,$ where $\nabla_{\mathbb{H}^n}$ is the gradient operator on $\mathbb{H}^n,~ 0<\alpha\leq 1-j/4,~ j/4<\beta\leq1,$ and $\beta-\alpha\geq j/4.$
Abstract
We consider the Schr\"odinger type operator \(\mathcal{L}=(-\Delta_{\mathbb{H}^n})^2+V^2 \) on the Heisenberg group $\mathbb{H}^n,$ where $\Delta_{\mathbb{H}^n}$ is the sub-Laplacian and the non-negative potential \(V\) belongs to the reverse H\"older class \(RH_s$ for $ s\geq Q/2$ and $Q\geq 6.\) We shall establish the $(L^p,L^q)$ estimates for the Riesz transforms $ T_{\alpha,\beta,j} =V^{2\alpha}\nabla_{\mathbb{H}^n}^j \mathcal{L}^{-\beta},~j=0,1,2,3,$ where $\nabla_{\mathbb{H}^n}$ is the gradient operator on $\mathbb{H}^n,~ 0<\alpha\leq 1-j/4,~ j/4<\beta\leq1,$ and $\beta-\alpha\geq j/4.$
참고문헌 (15)
- Yue Hu, "Estimates for the commutators of operator V α∇(−∆ + V )−β", Journal of Inequalities and Applications, 2019(1) (2019), Springer Science and Business Media LLC[DOI]
- H. -Q. Li, "Estimations Lp des op´erateurs de Schr¨odinger sur les groupes nilpotents", J. Funct. Anal, 161(1), pp.152~218 (1999)[DOI]
- C. -C. Lin, "BMOL(Hn)spaces and Carleson measures for Schr¨odinger op"erators", Adv. Math, 228(3), pp.1631~1688 (2011)[DOI]
- Y. Liu, "Some estimates of Schr¨odinger type operators on the Heisenberg group", Arch. Math. (Basel), 94(3), pp.255~264 (2010)[DOI]
- Y. Liu, "Riesz transform related to a Schr¨odinger type operator on the Heisenberg group", Georgian Math. J, 22(3), pp.397~407 (2015)[DOI]
- Z. W. Shen, "Lp estimates for Schr¨odinger operators with certain potentials", Ann. Inst. Fourier(Grenoble), 45(2), pp.513~546 (1995)
- E. M. Stein, "Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals" (1993), Princeton University Press
- S. Sugano, "Estimates for the operators V α(-∆ + V)-β and V α∇(-∆ + V)-β with certain nonnegative potentials V", Tokyo J. Math, 21(2), pp.441~452 (1998)[DOI]
- S. Sugano, "Lp estimates for some Schr¨odinger type operators and a Calder´on-Zygmund operator of Schr¨odinger type", Tokyo J. Math, 30(1), pp.179~197 (2007)[DOI]
- Hua Wang, "Semigroup Maximal Functions, Riesz Transforms, and Morrey Spaces Associated with Schrödinger Operators on the Heisenberg Groups", Journal of Function Spaces, 2020, pp.1~22 (2020), Hindawi Limited[DOI]
- Yueshan Wang, "Estimates for Riesz transforms associated with Schr\"odinger type operators", 대한수학회보, 56(5), pp.1117~1127 (2019), 대한수학회[DOI]
- Yanhui Wang, "Estimates for the higher order Riesz transforms related to Schr\"odinger type operators", 대한수학회보, 58(1), pp.235~251 (2021), 대한수학회[DOI]
- Y. Wang, "Some estimates of Riesz transforms related to Schr¨odinger-type operators", Bull. Malays. Math. Sci. Soc, 44(5), pp.3161~3174 (2021)[DOI]
- Zhiyong Wang, "Regularities of Time-Fractional Derivatives of Semigroups Related to Schrodinger Operators with Application to Hardy-Sobolev Spaces on Heisenberg Groups", Journal of Function Spaces, 2020, pp.1~21 (2020), Hindawi Limited[DOI]
- Liu Yu, "Lp ESTIMATES FOR SCHR¨ODINGER TYPE OPERATORS ON THE HEISENBERG GROUP", 대한수학회지, 47(2), pp.425~443 (2010), 대한수학회