On the weak (1,1) boundedness of a class of oscillatory singular integrals

Yibiao Pan

Studia Mathematica (1993)

  • Volume: 106, Issue: 3, page 279-287
  • ISSN: 0039-3223

Abstract

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We prove the uniform weak (1,1) boundedness of a class of oscillatory singular integrals under certain conditions on the phase functions. Our conditions allow the phase function to be completely flat. Examples of such phase functions include ϕ ( x ) = e - 1 / x 2 and ϕ ( x ) = x e - 1 / | x | . Some related counterexample is also discussed.

How to cite

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Pan, Yibiao. "On the weak (1,1) boundedness of a class of oscillatory singular integrals." Studia Mathematica 106.3 (1993): 279-287. <http://eudml.org/doc/216017>.

@article{Pan1993,
abstract = {We prove the uniform weak (1,1) boundedness of a class of oscillatory singular integrals under certain conditions on the phase functions. Our conditions allow the phase function to be completely flat. Examples of such phase functions include $ϕ(x) = e^\{-1/x^2\}$ and $ϕ(x) = xe^\{-1/|x|\}$. Some related counterexample is also discussed.},
author = {Pan, Yibiao},
journal = {Studia Mathematica},
keywords = {weak (1,1) boundedness; oscillatory singular integrals; phase functions},
language = {eng},
number = {3},
pages = {279-287},
title = {On the weak (1,1) boundedness of a class of oscillatory singular integrals},
url = {http://eudml.org/doc/216017},
volume = {106},
year = {1993},
}

TY - JOUR
AU - Pan, Yibiao
TI - On the weak (1,1) boundedness of a class of oscillatory singular integrals
JO - Studia Mathematica
PY - 1993
VL - 106
IS - 3
SP - 279
EP - 287
AB - We prove the uniform weak (1,1) boundedness of a class of oscillatory singular integrals under certain conditions on the phase functions. Our conditions allow the phase function to be completely flat. Examples of such phase functions include $ϕ(x) = e^{-1/x^2}$ and $ϕ(x) = xe^{-1/|x|}$. Some related counterexample is also discussed.
LA - eng
KW - weak (1,1) boundedness; oscillatory singular integrals; phase functions
UR - http://eudml.org/doc/216017
ER -

References

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  1. [1] S. Chanillo and M. Christ, Weak (1,1) bounds for oscillatory singular integrals, Duke Math. J. 55 (1987), 141-155. Zbl0667.42007
  2. [2] S. Chanillo, D. Kurtz and G. Sampson, Weighted weak (1,1) and weighted L p estimates for oscillating kernels, Trans. Amer. Math. Soc. 295 (1986), 127-145. Zbl0594.42007
  3. [3] C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9-36. Zbl0188.42601
  4. [4] Y. Hu, Oscillatory singular integrals on weighted Hardy spaces, Studia Math. 102 (1992), 145-156. Zbl0808.42009
  5. [5] Y. Hu and Y. Pan, Boundedness of oscillatory singular integrals on Hardy spaces, Ark. Mat. 30 (1992), 311-320. Zbl0779.42007
  6. [6] A. Nagel, J. Vance, S. Wainger, and D. Weinberg, Hilbert transforms for convex curves, Duke Math. J. 50 (1983), 735-744. Zbl0524.44001
  7. [7] A. Nagel and S. Wainger, Hilbert transforms associated with plane curves, Trans. Amer. Math. Soc. 223 (1976), 235-252. Zbl0341.44005
  8. [8] Y. Pan, Uniform estimates for oscillatory integral operators, J. Funct. Anal. 100 (1991), 207-220. Zbl0735.45010
  9. [9] Y. Pan, Hardy spaces and oscillatory singular integrals, Rev. Mat. Iberoamericana 7 (1991), 55-64. Zbl0728.42013
  10. [10] Y. Pan, Weak (1,1) estimate for oscillatory singular integrals with real-analytic phases, Proc. Amer. Math. Soc., to appear. Zbl0803.42002
  11. [11] D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals and Radon transforms I, Acta Math. 157 (1986), 99-157. Zbl0622.42011
  12. [12] F. Ricci and E. M. Stein, Harmonic analysis on nilpotent groups and singular integrals, I, J. Funct. Anal. 73 (1987), 179-194. Zbl0622.42010
  13. [13] P. Sjölin, Convolution with oscillating kernels on H p spaces, J. London Math. Soc. 23 (1981), 442-454 . Zbl0426.46034
  14. [14] E. M. Stein, Oscillatory integrals in Fourier analysis, in: Beijing Lectures in Harmonic Analysis, Princeton Univ. Press, Princeton 1986, 307-355. 
  15. [15] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton 1970. Zbl0207.13501
  16. [16] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, Cambridge 1959. Zbl0085.05601

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