Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

RESTRICTION ESTIMATES FOR ARBITRARY CONVEX CURVES IN R2

  • Received : 2009.12.04
  • Accepted : 2010.04.23
  • Published : 2010.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

We study the restriction estimate of Fourier transform to arbitrary convex curves in $R^2$ with no regularity assumption. Assuming that the convex curve has the lower bound of curvatures, we extend the restriction results from smooth convex curves to arbitrary convex curves. Our work has been motivated by the lecture notes of Terence Tao. The bilinear approach and geometric observations play an important role.

Keywords

References

  1. L. Brandolini, L. Colzani, A. Iosevich, A. Podkorytov and G.Travaglini, Geometry of the Gauss map and lattices points in convex domains, Mathematika. 48 (2001), 107-117. https://doi.org/10.1112/S0025579300014376
  2. J. Bak and D. Mcmichael, Convolution of a measure with itself and a restriction theorem, PAMS. 125 (1997), 463-470. https://doi.org/10.1090/S0002-9939-97-03569-7
  3. L. Brandolini, A. Iosevich and G.Travaglini, Fourier transform, lattices points, and irregularities of distributions, Trans. Amer. Math. Soc. 355 (2003), 3513-3535. https://doi.org/10.1090/S0002-9947-03-03240-9
  4. L. Brandolini, M. Rigoli and G.Travaglini, Average decay of Fourier transforms and geometry of convex sets, Revista Mat. Iber. 14 (1998), 519-560.
  5. C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9-36. https://doi.org/10.1007/BF02394567
  6. B. J. Green, Restriction and Kakeya phenomena, Lecture notes (2003)
  7. E. M. Stein, Harmonic Analysis, Princeton University Press, 1993.
  8. E. Stein, and R. Shakarchi, Fourier Analysis, Princeton University Press, 2003.
  9. E. M. Stein, and G. Weiss, Introduction to Fourier Analysis on Euclidean spaces, Princeton University Press, Princeton, NJ, 1971. MR 46 : 4102.
  10. T. Tao, Two-dimensional restriction theorems, the Kakeya conjecture, Lecture notes, (1999).
  11. T. Tao, Recent progress on the restriction conjecture, (2003), preprint.
  12. A. Zygmund, On Fourier coeffcients and tranforms of functions of two variables, Studia Math. 50, (1974), 189-201. https://doi.org/10.4064/sm-50-2-189-201