Abstract and Applied Analysis

Weighted Endpoint Estimates for Commutators of Riesz Transforms Associated with Schrödinger Operators

Yu Liu, Jielai Sheng, and Lijuan Wang

Full-text: Open access

Abstract

Let L = - Δ + V be a Schrödinger operator, where Δ is the laplacian on n and the nonnegative potential V belongs to the reverse Hölder class B s 1 for some s 1 ( n / 2 ) . Assume that ω A 1 ( n ) . Denote by H L 1 ( ω ) the weighted Hardy space related to the Schrödinger operator L = - Δ + V . Let b = [ b , ] be the commutator generated by a function b BMO θ ( n ) and the Riesz transform = ( - Δ + V ) - ( 1 / 2 ) . Firstly, we show that the operator is bounded from L 1 ( ω ) into L weak 1 ( ω ) . Secondly, we obtain the endpoint estimates for the commutator [ b , ] . Namely, it is bounded from the weighted Hardy space H L 1 ( ω ) into L weak 1 ( ω ) .

Article information

Source
Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 281562, 10 pages.

Dates
First available in Project Euclid: 26 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393450330

Digital Object Identifier
doi:10.1155/2013/281562

Mathematical Reviews number (MathSciNet)
MR3139466

Zentralblatt MATH identifier
1293.35063

Citation

Liu, Yu; Sheng, Jielai; Wang, Lijuan. Weighted Endpoint Estimates for Commutators of Riesz Transforms Associated with Schrödinger Operators. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 281562, 10 pages. doi:10.1155/2013/281562. https://projecteuclid.org/euclid.aaa/1393450330


Export citation

References

  • R. R. Coifman, R. Rochberg, and G. Weiss, “Factorization theorems for Hardy spaces in several variables,” Annals of Mathematics, vol. 103, no. 3, pp. 611–635, 1976.
  • Z. Guo, P. Li, and L. Peng, “${L}_{p}$ boundedness of commutators of Riesz transforms associated to Schrödinger operator,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 421–432, 2008.
  • P. Li and L. Peng, “Endpoint estimates for commutators of Riesz transforms associated with Schrödinger operators,” Bulletin of the Australian Mathematical Society, vol. 82, no. 3, pp. 367–389, 2010.
  • B. Bongioanni, E. Harboure, and O. Salinas, “Commutators of Riesz transforms related to Schrödinger operators,” The Journal of Fourier Analysis and Applications, vol. 17, no. 1, pp. 115–134, 2011.
  • B. Bongioanni, E. Harboure, and O. Salinas, “Weighted inequalities for commutators of Schrödinger-Riesz transforms,” Journal of Mathematical Analysis and Applications, vol. 392, no. 1, pp. 6–22, 2012.
  • Y. Liu, “Commutators of BMO functions and degenerate Schrödinger operators with certain nonnegative potentials,” Monatshefte für Mathematik, vol. 165, no. 1, pp. 41–56, 2012.
  • Y. Liu, J. Z. Huang, and J. F. Dong, “Commutators of higher order Riesz transform associated with Schrödinger operators,” Journal of Function Spaces and Applications, vol. 2013, Article ID 842375, 15 pages, 2013.
  • Y. Liu, J. Z. Huang, and J. F. Dong, “Commutators of Calderón-Zygmund operators related to admissible functions on spaces of homogeneous type and applications to Schrödinger operators,” Science China Mathematics, vol. 56, no. 9, pp. 1895–1913, 2013.
  • Z. W. Shen, “${L}_{p}$ estimates for Schrödinger operators with certain potentials,” Annales de l'Institut Fourier, vol. 45, no. 2, pp. 513–546, 1995.
  • J. Dziubański and J. Zienkiewicz, “Hardy space ${H}^{1}$ associated to Schrödinger operator with potential satisfying reverse Hölder inequality,” Revista Matemática Iberoamericana, vol. 15, no. 2, pp. 279–296, 1999.
  • H. Liu, L. Tang, and H. Zhu, “Weighted Hardy spaces and BMO spaces associated with Schrödinger operators,” Mathematische Nachrichten, vol. 285, no. 17-18, pp. 2173–2207, 2012.
  • D. Yang and Y. Zhou, “Localized Hardy spaces ${H}^{1}$ related to admissible functions on RD-spaces and applications to Schrödinger operators,” Transactions of the American Mathematical Society, vol. 363, no. 3, pp. 1197–1239, 2011.
  • E. M. Stein, Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1993.
  • J. Duoandikoetxea, Fourier Analysis, vol. 29 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 2001.
  • C. C. Lin, H. P. Liu, and Y. Liu, “Hardy spaces associated with Schrödinger operators on the čommentComment on ref. [9?]: Please update the information of this reference, if possible.Heisenberg group,” http://arxiv.org/abs/1106.4960.
  • J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, vol. 116 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 1985. \endinput