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Conjecture and improved extension theorems for paraboloids in the finite field setting

  • Doowon KohEmail author
Article
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Abstract

We study the extension estimates for paraboloids in d-dimensional vector spaces over finite fields \(\mathbb F_q\) with q elements. We use the connection between \(L^2\) based restriction estimates and \(L^p\rightarrow L^r\) extension estimates for paraboloids. As a consequence, we improve the \(L^2\rightarrow L^r\) extension results obtained by Lewko and Lewko (Proc Am Math Soc 140:2013–2028, 2012) in even dimensions \(d\ge 6\) and odd dimensions \(d=4\ell +3\) for \(\ell \in \mathbb N.\) Our results extend the consequences for 3-D paraboloids due to Lewko (Adv Math 270(1):457–479, 2015) to higher dimensions. We also clarifies conjectures on finite field extension problems for paraboloids.

Keywords

Restriction theorem Extension theorem Paraboloid Finite field 

Mathematics Subject Classification

42B05 

Notes

References

  1. 1.
    Barcelo, B.: On the restriction of the Fourier transform to a conical surface. Trans. Am. Math. Soc. 292, 321–333 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bourgain, J.: Besicovitch-type maximal operators and applications to Fourier analysis. Geom. Funct. Anal. 22, 147–187 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bourgain, J., Demeter, C.: Improved estimates for the discrete Fourier restriction to the higher dimensional sphere. Ill. J. Math. 57(1), 213–227 (2013)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bourgain, J., Demeter, C.: New bounds for the discrete Fourier restriction to the sphere in 4D and 5D. Int. Math. Res. Not. IMRN 11, 3150–3184 (2015)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Carbery, A.: Restriction implies BochnerRiesz for paraboloids. Math. Proc. Camb. Philos. Soc. Ill 3, 525–529 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Guth, L.: A restriction estimate using polynomial partitioning. J. Am. Math. Soc. 29(2), 371–413 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Iosevich, A., Koh, D.: Extension theorems for paraboloids in the finite field setting. Math. Z. 266, 471–487 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Iosevich, A., Koh, D.: Extension theorems for spheres in the finite field setting. Forum. Math. 22(3), 457–483 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Koh, D., Shen, C.: Sharp extension theorems and Falconer distance problems for algebraic curves in two dimensional vector spaces over finite fields. Rev. Mat. Iberoam. 28(1), 157–178 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Koh, D., Shen, C.: Harmonic analysis related to homogeneous varieties in three dimensional vector spaces over finite fields. Canad. J. Math. 64(5), 1036–1057 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lewko, M.: New restriction estimates for the 3-d paraboloid over finite fields. Adv. Math. 270(1), 457–479 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lewko, M.: Finite field restriction estimates based on Kakeya maximal operator estimates. arXiv:1401.8011
  13. 13.
    Lewko, A., Lewko, M.: Endpoint restriction estimates for the paraboloid over finite fields. Proc. Am. Math. Soc. 140, 2013–2028 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mockenhaupt, G., Tao, T.: Restriction and Kakeya phenomena for finite fields. Duke Math. J. 121(1), 35–74 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Stein, E.M.: Some problems in harmonic analysis, harmonic analysis in Euclidean spaces. In: Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass. vol. 1, pp. 3–20 (1978)Google Scholar
  16. 16.
    Stein, E.M.: Harmonic Analysis. Princeton University Press, Princeton (1993)Google Scholar
  17. 17.
    Tao, T.: A sharp bilinear restriction estimate for paraboloids. Geom. Funct. Anal. 13, 1359–1384 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tao, T.: Some recent progress on the restriction conjecture, Fourier analysis and convexity. Appl. Numer. Harmon. Anal., pp. 217–243. Birkhäuser, Boston (2004)CrossRefzbMATHGoogle Scholar
  19. 19.
    Vinh, L.A.: Maximal sets of pairwise orthogonal vectors in finite fields. Can. Math. Bull. 55(2), 418–423 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wolff, T.: A sharp bilinear cone restriction estimate. Ann. Math. 153, 661–698 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Zygmund, A.: On Fourier coefficients and transforms of functions of two variables. Studia Math. 50, 189–201 (1974)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsChungbuk National UniversityCheongjuSouth Korea

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