Acknowledgement
This work was financially supported by Shandong Natural Science Foundation of Chian ZR2017JL008.
References
- B. Bongioanni, E. Harboure, and O. Salinas, Commutators of Riesz transforms related to Schrodinger operators, J. Fourier Anal. Appl. 17 (2011), no. 1, 115-134. https://doi.org/10.1007/s00041-010-9133-6
- R. R. Coifman, R. Rochberg, and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (1976), no. 3, 611-635. https://doi.org/10.2307/1970954
- Y. Hu and Y. Wang, Hardy type estimates for Riesz transforms associated with Schrodinger operators, Anal. Math. Phys. 9 (2019), no. 1, 275-287. https://doi.org/10.1007/s13324-017-0196-2
- P. Li, Y. Mo, and C. Zhang, A compactness criterion and application to the commutators associated with Schrodinger operators, Math. Nachr. 288 (2015), no. 2-3, 235-248. https://doi.org/10.1002/mana.201300286
- P. Li and L. Peng, Lp boundedness of commutator operator associated with Schrodinger operators on Heisenberg group, Acta Math. Sci. Ser. B (Engl. Ed.) 32 (2012), no. 2, 568-578. https://doi.org/10.1016/S0252-9602(12)60039-3
- P. Li and X. Wan, The boundedness of commutators associated with Schrodinger operators on Herz spaces, J. Inequal. Appl. 2016 (2016), Paper No. 172, 27 pp. https://doi.org/10.1186/s13660-016-1118-9
- P. Li, X. Wan, and C. Zhang, Schrodinger type operators on generalized Morrey spaces, J. Inequal. Appl. 2015 (2015), 229, 21 pp. https://doi.org/10.1186/s13660-015-0747-8
- C. Lin, H. Liu, and Y. Liu, Hardy spaces associated with Schrodinger operators on the Heisenberg group, Preprint Available at https://arxiv.org/pdf/1106.4960.pdf
- Y. Liu and G. Tang, Hardy type estimates for Riesz transforms associated with Schrodinger operators on the Heisenberg group, Anal. Theory Appl. 32 (2016), no. 1, 78-89. https://doi.org/10.4208/ata.2016.v32.n1.7
- Y. Liu and G. Tang, A note for Riesz transforms associated with Schrodinger operators on the Heisenberg group, Anal. Math. Phys. 7 (2017), no. 1, 31-45. https://doi.org/10.1007/s13324-016-0128-6
- G. Lu, A Fefferman-Phong type inequality for degenerate vector fields and applications, PanAmer. Math. J. 6 (1996), no. 4, 37-57.
- G. Pradolini and O. Salinas, Commutators of singular integrals on spaces of homogeneous type, Czechoslovak Math. J. 57(132) (2007), no. 1, 75-93. https://doi.org/10.1007/s10587-007-0045-9
- Z. Shen, Lp estimates for Schrodinger operators with certain potentials, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 2, 513-546. https://doi.org/10.5802/aif.1463