Journal of Applied Mathematics

Commutators with Lipschitz Functions and Nonintegral Operators

Peizhu Xie and Ruming Gong

Full-text: Open access

Abstract

Let T be a singular nonintegral operator; that is, it does not have an integral representation by a kernel with size estimates, even rough. In this paper, we consider the boundedness of commutators with T and Lipschitz functions. Applications include spectral multipliers of self-adjoint, positive operators, Riesz transforms of second-order divergence form operators, and fractional power of elliptic operators.

Article information

Source
J. Appl. Math., Volume 2013 (2013), Article ID 178961, 8 pages.

Dates
First available in Project Euclid: 14 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jam/1394808035

Digital Object Identifier
doi:10.1155/2013/178961

Mathematical Reviews number (MathSciNet)
MR3070497

Zentralblatt MATH identifier
1271.35043

Citation

Xie, Peizhu; Gong, Ruming. Commutators with Lipschitz Functions and Nonintegral Operators. J. Appl. Math. 2013 (2013), Article ID 178961, 8 pages. doi:10.1155/2013/178961. https://projecteuclid.org/euclid.jam/1394808035


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.
  • S. Janson, “Mean oscillation and commutators of singular integral operators,” Arkiv för Matematik, vol. 16, no. 2, pp. 263–270, 1978.
  • M. Bramanti and M. C. Cerutti, “Commutators of singularintegrals on homogeneous spaces,” Bollettino dell'Unione Matematica Italiana, vol. 10, no. 4, pp. 843–883, 1996.
  • X. T. Duong and L. X. Yan, “Commutators of BMO functions and singular integral operators with non-smooth kernels,” Bulletin of the Australian Mathematical Society, vol. 67, no. 2, pp. 187–200, 2003.
  • X. T. Duong and A. MacIntosh, “Singular integral operators with non-smooth kernels on irregular domains,” Revista Matemática Iberoamericana, vol. 15, no. 2, pp. 233–265, 1999.
  • G. Hu and D. Yang, “Maximal commutators of BMO functions and singular integral operators with non-smooth kernels on spaces of homogeneous type,” Journal of Mathematical Analysis and Applications, vol. 354, no. 1, pp. 249–262, 2009.
  • P. Auscher and J. M. Martell, “Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: general operator theory and weights,” Advances in Mathematics, vol. 212, no. 1, pp. 225–276, 2007.
  • S. Blunck and P. C. Kunstmann, “Calderón-Zygmund theory for non-integral operators and the ${H}^{\infty }$ functional calculus,” Revista Matemática Iberoamericana, vol. 19, no. 3, pp. 919–942, 2003.
  • P. Auscher and J. M. Martell, “Weighted norm inequalities for fractional operators,” Indiana University Mathematics Journal, vol. 57, no. 4, pp. 1845–1869, 2008.
  • M. Paluszyński, “Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss,” Indiana University Mathematics Journal, vol. 44, no. 1, pp. 1–18, 1995.
  • S. Chanillo, “A note on commutators,” Indiana University Mathematics Journal, vol. 31, no. 1, pp. 7–16, 1982.
  • A. Gogatishvili and V. Kokilashvili, “Criteria of strong type two-weighted inequalities for fractional maximal functions,” Georgian Mathematical Journal, vol. 3, no. 5, pp. 423–446, 1996.
  • X. T. Duong and L. X. Yan, “Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates,” Journal of the Mathematical Society of Japan, vol. 63, no. 1, pp. 295–319, 2011.
  • P. Auscher, S. Hofmann, M. Lacey, A. McIntosh, and Ph. Tchamitchian, “The solution of the Kato square root problem for second order elliptic operators on ${\mathbb{R}}^{n}$,” Annals of Mathematics, vol. 156, no. 2, pp. 633–654, 2002.
  • P. Auscher and J. M. Martell, “Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: harmonic analysis of elliptic operators,” Journal of Functional Analysis, vol. 241, no. 2, pp. 703–746, 2006.