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Endpoint Estimates of Riesz Transforms Associated with Generalized Schrödinger Operators

Published online by Cambridge University Press:  20 November 2018

Yu Liu
Affiliation:
School of Mathematics and Physics, University of Science and Technology Beijing, 100083, China, e-mail: liuyu75@pku.org.cn, 1005777218@qq.com
Shuai Qi
Affiliation:
School of Mathematics and Physics, University of Science and Technology Beijing, 100083, China, e-mail: liuyu75@pku.org.cn, 1005777218@qq.com
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Abstract

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In this paper we establish the endpoint estimates and Hardy type estimates for the Riesz transform associated with the generalized Schrödinger operator.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Auscher, P. and Ben Ali, B., Maximal inequalities and Riesz transform estimates on LP spaces for Schrb'dinger operators with nonnegative potentials. Ann. Inst. Fourier (Grenoble) 57 (2007), no. 6, 1975-2013. http://dx.doi.Org/10.58O2/aif.232OGoogle Scholar
[2] Cao, J., Liu, Y., and Yang, Da., Hardy spaces associated to Schrodinger type operators . Houston J. Math. 36 (2010), no. 4,1067-1095.Google Scholar
[3] Dziubanski, J. and Zienkiewicz, J., Hardy space H1 associated to Schrodinger operator with potential satisfying reverse Holder inequality. Rev. Mat. Iberoamericana 15 (1999), no. 2, 279-296. http://dx.doi.Org/10.4171/RMI/257Google Scholar
[4] Garcia-Cuerva, J. and de Francia, J. Rubio, Weighted norm inequalities and related topics. North-Holland Mathematics Studies, 116, Notas de Matemâtica, 104, North-Holland Publishing Co., Amsterdam, 1985.Google Scholar
[5] Li, H.-Q., Estimations LP des opérateurs de Schrodinger sur les groupes nilpotents. J. Func. Anal. 161 (1999), no. 1, 152-218. http://dx.doi.Org/10.1006/jfan.1998.3347Google Scholar
[6] Lin, C., Liu, H., and Liu, Y., Hardy spaces associated with Schrodinger operators on the Heisenberg group. arxiv:1106.4960Google Scholar
[7] Liu, Y. and Dong, J., Some estimates of higher order Riesz transform related to Schrodinger operator. Potential. Anal. 32 (2010), no. 1, 41-55. http://dx.doi.Org/10.1007/s11118-009-9143-7Google Scholar
[8] Shen, Z., LP estimates for Schrb'dinger operators with certain potentials. Ann. Inst. Fourier (Grenoble) 45 (1995), 513546. http://dx.doi.org/10.5802/aif.1463Google Scholar
[9] Shen, Z., On fundamental solutions of generalized Schrodinger operators. J. Funct. Anal. 167 (1999), 521564. http://dx.doi.org/10.1006/jfan.1999.3455Google Scholar
[10] Taibleson, M. H. and Weiss, G., The molecular characterization of certain Hardy spaces. In: Representation theorems for Hardy spaces, Astérisque, 77, Soc. Math. France, Paris, 1980, pp. 67149.Google Scholar
[11] Wu, L. and Yan, L., Heat kernels, upper bounds and Hardy spaces associated to the generalized Schrodinger operators. J. Funct. Anal. 270 (2016), no. 10, 3709-3749. http://dx.doi.Org/10.1016/j.jfa.2O15.12.016Google Scholar
[12] Yang, Da., Yang, Do., and Zhou, Y., Endpoint properties of localized Riesz transforms and fractional integrals associated to Schrodinger operators. Potential Anal. 30 (2009), no. 3, 271-300. http://dx.doi.Org/10.1007/s11118-009-9116-xGoogle Scholar
[13] Zhong, J., Harmonic analysis for some Schrodinger type operators. Ph.D., Princeton University, 1993.Google Scholar