Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

A NOTE ON THE FIRST ORDER COMMUTATOR C2

  • Li, Wenjuan (School of Science Northwest Polytechnical University) ;
  • Liu, Suying (School of Science Northwest Polytechnical University)
  • Received : 2018.07.06
  • Accepted : 2018.12.05
  • Published : 2019.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

This paper gives a counterexample to show that the first order commutator $C_2$ is not bounded from $H^1({\mathbb{R}}){\times}H^1({\mathbb{R}})$ into $L^{1/2}({\mathbb{R}})$. Then we introduce the atomic definition of abstract weighted Hardy spaces $H^1_{ato,{\omega}}$$({\mathbb{R}})$ and study its properties. At last, we prove that $C_2$ maps $H^1_{ato,{\omega}}$$({\mathbb{R}}){\times}H^1_{ato,{\omega}}$$({\mathbb{R}})$ into $L^{1/2}_{\omega}$$({\mathbb{R}})$.

Keywords

Acknowledgement

Supported by : National Natural Science Foundation of China, China Postdoctoral Science Foundation, Central Universities

References

  1. F. Bernicot, Use of abstract Hardy spaces, real interpolation and applications to bilinear operators, Math. Z. 265 (2010), no. 2, 365-400. https://doi.org/10.1007/s00209-009-0520-0
  2. F. Bernicot and J. Zhao, New abstract Hardy spaces, J. Funct. Anal. 255 (2008), no. 7, 1761-1796. https://doi.org/10.1016/j.jfa.2008.06.018
  3. A.-P. Calderon, Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092-1099. https://doi.org/10.1073/pnas.53.5.1092
  4. A.-P. Calderon, On commutators of singular integrals, Studia Math. 53 (1975), no. 2, 139-174. https://doi.org/10.4064/sm-53-2-139-174
  5. R. R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. 212 (1975), 315-331. https://doi.org/10.2307/1998628
  6. X. T. Duong, R. M. Gong, L. Grafakos, J. Li, and L. X. Yan, Maximal operator for multilinear singular integrals with non-smooth kernels, Indiana Univ. Math. J. 58 (2009), no. 6, 2517-2541. https://doi.org/10.1512/iumj.2009.58.3803
  7. X. T. Duong, L. Grafakos, and L. Yan, Multilinear operators with non-smooth kernels and commutators of singular integrals, Trans. Amer. Math. Soc. 362 (2010), no. 4, 2089-2113. https://doi.org/10.1090/S0002-9947-09-04867-3
  8. J. Garcia-Cuerva, Weighted $H^p$ spaces, Dissertationes Math. (Rozprawy Mat.) 162 (1979), 63 pp.
  9. G. Hu and S. Lu, Weighted estimates for the multilinear singular integral operators with non-smooth kernels, Sci. China Math. 54 (2011), no. 3, 587-602. https://doi.org/10.1007/s11425-010-4135-z
  10. W. Li, Q. Xue and K. Yabuta, Multilinear Calderon-Zygmund operators on weighted Hardy spaces, Studia Math. 199 (2010), no. 1, 1-16. https://doi.org/10.4064/sm199-1-1
  11. E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993.