About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 431692, 11 pages
http://dx.doi.org/10.1155/2013/431692
Research Article

Boundedness for a Class of Generalized Commutators of Fractional Hardy Operators with a Rough Kernel

1Department of Mathematics, Shangrao Normal University, Shangrao 334001, China
2School of Mathematical Science, Beijing Normal University, Beijing 100875, China

Received 7 July 2013; Accepted 1 October 2013

Academic Editor: Csaba Varga

Copyright © 2013 Xiao Yu and Shanzhen Lu. 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. A. P. Calderón and A. Zygmund, “On singular integrals,” American Journal of Mathematics, vol. 78, pp. 289–309, 1956. View at MathSciNet
  2. 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 MathSciNet
  3. J. Cohen, “A sharp estimate for a multilinear singular integral in n,” Indiana University Mathematics Journal, vol. 30, no. 5, pp. 693–702, 1981. View at Publisher · View at Google Scholar · View at MathSciNet
  4. J. Cohen and J. Gosselin, “A BMO estimate for multilinear singular integrals,” Illinois Journal of Mathematics, vol. 30, no. 3, pp. 445–464, 1986. View at MathSciNet
  5. S. Lu and Q. Wu, “CBMO estimates for commutators and multilinear singular integrals,” Mathematische Nachrichten, vol. 276, pp. 75–88, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  6. D. Y. Yan, Some problems on multilinear singular integral operators and multilinear oscillatory singular integral operators [Ph.D. thesis], Beijing Normal University, Beijing, China, 2001.
  7. X. Yu and X. X. Tao, “Boundedness for a class of generalized commutators on λ-central Morrey space,” Acta Mathematica Sinica. English Series, vol. 29, no. 10, pp. 1917–1926, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  8. C. Wang and Z. Zhang, “A new proof of Wu's theorem on vortex sheets,” Science China, vol. 55, no. 7, pp. 1449–1462, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  9. S. Wu, “Mathematical analysis of vortex sheets,” Communications on Pure and Applied Mathematics, vol. 59, no. 8, pp. 1065–1206, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  10. G. H. Hardy, “Note on a theorem of Hilbert,” Mathematische Zeitschrift, vol. 6, no. 3-4, pp. 314–317, 1920. View at Publisher · View at Google Scholar · View at MathSciNet
  11. Z.-W. Fu, Z.-G. Liu, S.-Z. Lu, and H.-B. Wang, “Characterization for commutators of n-dimensional fractional Hardy operators,” Science in China A, vol. 50, no. 10, pp. 1418–1426, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  12. M. Christ and L. Grafakos, “Best constants for two nonconvolution inequalities,” Proceedings of the American Mathematical Society, vol. 123, no. 6, pp. 1687–1693, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  13. S. Lu and D. Yang, “The central BMO spaces and Littlewood-Paley operators,” Approximation Theory and its Applications, vol. 11, no. 3, pp. 72–94, 1995. View at MathSciNet
  14. Y. Komori, “Notes on singular integrals on some inhomogeneous Herz spaces,” Taiwanese Journal of Mathematics, vol. 8, no. 3, pp. 547–556, 2004. View at MathSciNet
  15. S. Z. Lu and F. Y. Zhao, “CBMO estimates for multilinear Hardy operators,” Acta Mathematica Sinica. English Series, vol. 26, no. 7, pp. 1245–1254, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  16. G. L. Gao and X. Yu, “Some estimates for the generalized Hardy operators on some function spaces,” Acta Mathematica Sinica. Chinese Series, vol. 55, no. 6, pp. 1101–1110, 2012. View at MathSciNet
  17. X. Yu and S. Lu, “Endpoint estimates for generalized commutators of Hardy operators on H1 space,” Journal of Function Spaces and Applications, vol. 2013, Article ID 410305, 11 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  18. S. Z. Lu and D. C. Yang, “The decomposition of weighted Herz space and its applications,” Science in China A, vol. 38, no. 2, pp. 147–158, 1995. View at MathSciNet
  19. Z. Fu, S. Lu, and F. Zhao, “Commutators of n-dimensional rough Hardy operators,” Science China, vol. 54, no. 1, pp. 95–104, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  20. Y. Ding, “A note on multilinear fractional integrals with rough kernel,” Advances in Mathematics, vol. 30, no. 3, pp. 238–246, 2001. View at MathSciNet
  21. N. Wiener, “Generalized harmonic analysis,” Acta Mathematica, vol. 55, no. 1, pp. 117–258, 1930. View at Publisher · View at Google Scholar · View at MathSciNet
  22. A. Beurling, “Construction and analysis of some convolution algebras,” Annales de l'Institut Fourier, vol. 14, pp. 1–32, 1964. View at MathSciNet
  23. H. Feichtinger, “An elementary approach to Wiener's third Tauberian theorem on Euclidean n-space,” in Proceedings, Conference at Cortona 1984, vol. 29 of Symposia Mathematica, pp. 267–301, Academic Press, New York, NY, USA, 1987.
  24. Y. Z. Chen and K.-S. Lau, “Some new classes of Hardy spaces,” Journal of Functional Analysis, vol. 84, no. 2, pp. 255–278, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  25. J. García-Cuerva, “Hardy spaces and Beurling algebras,” Journal of the London Mathematical Society, vol. 39, no. 3, pp. 499–513, 1989. View at Publisher · View at Google Scholar · View at MathSciNet
  26. J. Alvarez, J. Lakey, and M. Guzmán-Partida, “Spaces of bounded λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures,” Collectanea Mathematica, vol. 51, no. 1, pp. 1–47, 2000. View at MathSciNet
  27. Z. W. Fu, Y. Lin, and S. Z. Lu, “λ-central BMO estimates for commutators of singular integral operators with rough kernels,” Acta Mathematica Sinica. English Series, vol. 24, no. 3, pp. 373–386, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  28. Z. W. Fu, S. Z. Lu, and W. Yuan, “A weighted variant of Riemann-Liouville fractional integrals on Rn,” Abstract and Applied Analysis, vol. 2012, Article ID 780132, 18 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet