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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 281562, 10 pages
http://dx.doi.org/10.1155/2013/281562
Research Article

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

School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

Received 14 May 2013; Accepted 1 October 2013

Academic Editor: Baoxiang Wang

Copyright © 2013 Yu Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. Z. Guo, P. Li, and L. Peng, “Lp boundedness of commutators of Riesz transforms associated to Schrödinger operator,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 421–432, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. 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. View at Publisher · View at Google Scholar
  8. 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. View at Publisher · View at Google Scholar
  9. Z. W. Shen, “Lp estimates for Schrödinger operators with certain potentials,” Annales de l'Institut Fourier, vol. 45, no. 2, pp. 513–546, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. J. Dziubański and J. Zienkiewicz, “Hardy space H1 associated to Schrödinger operator with potential satisfying reverse Hölder inequality,” Revista Matemática Iberoamericana, vol. 15, no. 2, pp. 279–296, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. D. Yang and Y. Zhou, “Localized Hardy spaces H1 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. E. M. Stein, Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1993. View at MathSciNet
  14. J. Duoandikoetxea, Fourier Analysis, vol. 29 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 2001. View at MathSciNet
  15. C. C. Lin, H. P. Liu, and Y. Liu, “Hardy spaces associated with Schrödinger operators on the Heisenberg group,” http://arxiv.org/abs/1106.4960.
  16. 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. View at MathSciNet