Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

Lp-BOUNDEDNESS FOR THE COMMUTATORS OF ROUGH OSCILLATORY SINGULAR INTEGRALS WITH NON-CONVOLUTION PHASES

  • Wu, Huoxiong (SCHOOL OF MATHEMATICAL SCIENCES XIAMEN UNIVERSITY)
  • Published : 2009.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, the author studies the k-th commutators of oscillatory singular integral operators with a BMO function and phases more general than polynomials. For 1 < p < $\infty$, the $L^p$-boundedness of such operators are obtained provided their kernels belong to the spaces $L(log+L)^{k+1}(S^{n-1})$. The results of the corresponding maximal operators are also established.

Keywords

References

  1. A. Al-Salman, Rough oscillatory singular integral operators of nonconvolution type, J. Math. Anal. Appl. 299 (2004), no. 1, 72–88 https://doi.org/10.1016/j.jmaa.2004.06.006
  2. A. Al-Salman and A. Al-Jarrah, Rough oscillatory singular integral operators. II, Turkish J. Math. 27 (2003), no. 4, 565–579
  3. Y. Ding, Some problems on oscillatory singular integral and fractional integral with rough kernel, Ph. D. Thesis, Beijing Normal Univ., 1995
  4. Y. Ding and S. Lu, Weighted Lp-boundedness for higher order commutators of oscillatory singular integrals, Tohoku Math. J. (2) 48 (1996), no. 3, 437–449 https://doi.org/10.2748/tmj/1178225342
  5. G. Hu, Weighted norm inequalities for commutators of homogeneous singular integrals, A Chinese summary appears in Acta Math. Sinica 39 (1996), no. 1, 141; Acta Math. Sinica (N.S.) 11 (1995), Special Issue, 77–88
  6. G. Hu, $L^p(R^n)$ boundedness for the commutator of a homogeneous singular integral operator, Studia Math. 154 (2003), no. 1, 13–27
  7. Y. Jiang and S. Lu, Oscillatory singular integrals with rough kernel, Harmonic analysis in China, 135–145, Math. Appl., 327, Kluwer Acad. Publ., Dordrecht, 1995
  8. S. Lu, M. Taibleson, and G. Weiss, Spaces Generated by Blocks, Beijing Normal University Press, Beijing, 1989
  9. S. Lu and H. Wu, Oscillatory singular integrals and commutators with rough kernels, Ann. Sci. Math. Quebec 27 (2003), no. 1, 47–66
  10. S. Lu and Y. Zhang, Criterion on Lp-boundedness for a class of oscillatory singular integrals with rough kernels, Rev. Mat. Iberoamericana 8 (1992), no. 2, 201–219
  11. B. Ma and G. Hu, $L^2(R^n)$ boundedness for commutators of oscillatory singular integral operators, Approx. Theory Appl. (N.S.) 16 (2000), no. 2, 37–44 https://doi.org/10.1007/BF02837391
  12. F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals. I. Oscillatory integrals, J. Funct. Anal. 73 (1987), no. 1, 179–194 https://doi.org/10.1016/0022-1236(87)90064-4
  13. E. M. Stein, Problems in harmonic analysis related to curvature and oscillatory integrals, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 196–221, Amer. Math. Soc., Providence, RI, 1987
  14. E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993
  15. H. Wu, Boundedness of higher order commutators of oscillatory singular integrals with rough kernels, Studia Math. 167 (2005), no. 1, 29–43

Cited by

  1. Non-standard commutators for rough oscillatory singular integrals vol.25, pp.3, 2009, https://doi.org/10.1007/s10496-009-0230-9