Restriction and kakeya phenomena for finite fields

Duke Mathematical Journal

Volumn 121, Issue 1, 2004, Pages 35-74

  Mockenhaupt, Gerd ^a   Tao, Terence ^b  

^a GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 1642603004     PISSN: 00127094     EISSN: None     Source Type: Journal    
DOI: 10.1215/S0012-7094-04-12112-8     Document Type: Article

References (52)

1. A. BALOG and E. SZEMERÉDI, A statistical theorem of set addition, Combinatorica 14 (1994), 263-268.
2. B. BARCELO TABERNER, On the restriction of the Fourier transform to a conical surface, Trans. Amer. Math. Soc. 292 (1985), 321-333.
3. W. BECKNER, A. CARBERY, S. SEMMES, and F. SORIA, A note on restriction of the Fourier transform to spheres, Bull. London Math. Soc. 21 (1989), 394-398.
4. B. BOLLOBAS, Extremal Graph Theory, London Math. Soc. Monogr. 11, Academic Press, London, 1978
5. J. BOURGAIN, Bounded orthogonal systems and the A(p)-setproblem, Acta Math. 162 (1989), 227-245.
6. J. BOURGAIN, Besicovitch-type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), 147-187.
7. J. BOURGAIN, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I: Schrodinger equations, Geom. Funct. Anal. 3 (1993), 107-156.
8. J. BOURGAIN, “Estimates for cone multipliers” in Geometric Aspects of Functional Analysis (Israel, 1992-1994), Oper. Theory Adv. Appl. 77, Birkhauser, Basel, 1995, 41-60.
9. J. BOURGAIN, “Some new estimates on oscillatory integrals” in Essays on Fourier Analysis in Honor of Elias M. Stein (Princeton, 1991), Princeton Math. Ser. 42, Princeton Univ. Press, Princeton, 1995, 83-112.
10. J. BOURGAIN, On the dimension ofKakeya sets and related maximal inequalities, Geom. Funct. Anal. 9 (1999), 256-282.
11. J. BOURGAIN, “Harmonic analysis and combinatorics: How much may they contribute to each other?” in Mathematics: Frontiers and Perspectives, Amer. Math. Soc., Providence, 2000, 13-32.
12. A. CARBERY, Restriction implies Bochner-Riesz for paraboloids, Math. Proc. Cambridge Philos. Soc. 111 (1992), 525-529.
13. L. CARLESON and P. SJOLIN, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (1972), 287-299.
14. F. M. CHRIST, Restriction of the Fourier transform to submanifolds of low codimension, Ph.D. thesis, University of Chicago, 1982.
15. M. CHRIST, J. DUOANDIKOETXEA, and J. L. RUBIO de FRANCIA, Maximal operators associated to the Radon transform and the Calderén-Zygmund method of rotations, Duke Math. J. 53 (1986), 189-209.
16. p estimates for the X-ray transform, Illinois J. Math. 27 (1983), 125-129.
17. G. ELEKES, On the number of sums and products, Acta Arith. 81 (1997), 365-367.
18. C. FEFFERMAN, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9-36.
19. p norms, Israel J. Math. 18 (1974), 157-166.
20. W. T. GOWERS, A new proof of Szemerédi's theorem for arithmetic progressions of length four, Geom. Funct. Anal. 8 (1998), 529-551.
21. G. H. HARDY and J. E. LITTLEWOOD, Notes on the theory of series XIX: A problem concerning majorants of Fourier series, Quart. J. Math. Oxford Ser. 6 (1935), 304-315.
22. 3, Ann. of Math. (2) 152 (2000), 383-446.
23. N. H. KATZ and T. TAO, Bounds on arithmetic projections, and applications to the Kakeya conjecture, Math Res. Lett. 5-6 (1999), 625-630.
24. N. H. KATZ and T. TAO, Some connections between Falconer’s distance set conjecture and sets of Furstenburg type, New York J. Math. 7 (2001), 149-187.
25. N. H. KATZ and T. TAO, New bounds for Kakeya problems, J. Anal. Math. 87 (2002), 231-263.
26. S. KLAINERMAN and M. MACHEDON, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), 1221-1268.
27. I. LABA and T. TAO, An improved bound for the Minkowski dimension of Besicovitch sets in medium dimension, Geom. Funct. Anal. 11 (2001), 773-806.
28. J. E. LITTLEWOOD, Some Problems in Real and Complex Analysis, D. C. Heath, Lexington, Mass., 1968.
29. G. MOCKENHAUPT, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. (N.S.) 25 (1991), 31-36.
30. G. MOCKENHAUPT, A note on the cone multiplier, Proc. Amer. Math. Soc. 117 (1993), 145-152.
31. G. MOCKENHAUPT, Bounds in Lebesgue spaces of oscillatory integral operators, Habilitationsschrift, Universitat Siegen, Germany, 1996, http://math.gatech.edu/∼gerdm/research/
32. A. MOYUA, A. VARGAS, and L. VEGA, Schrodinger maximal function and restriction properties of the Fourier transform, Internat. Math. Res. Notices 1996, no. 16, 793-815.
33. I. Z. RUZSA, “Sums of finite sets” in Number Theory (New York, 1991 -1995), Springer, New York, 1996, 281-293.
34. P. SJOLIN, Regularity of solutions to the Schrodinger equation, Duke Math. J. 55 (1987), 699-715.
35. E. M. STEIN, “Some problems in harmonic analysis” in Harmonic Analysis in Euclidean Spaces, Part 1 (Williamstown, Mass., 1978), Proc. Sympos. Pure Math. 35, Amer. Math. Soc., Providence, 1979, 3-20.
36. E. M. STEIN, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Monogr. Harmon. Anal. 3, Princeton Univ. Press, Princeton, 1993.
37. R. S. STRICHARTZ, Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), 705-774.
38. E. SZEMEREDI and W. T. TROTTER JR., Extremal problems in discrete geometry, Combinatorica 3 (1983), 381-392.
39. T. TAO, The Bochner-Riesz conjecture implies the restriction conjecture, Duke Math J. 96 (1999), 363-375.
40. 2-functions, and applications to nonlinear dispersive equations, Amer. J. Math. 123 (2001), 839-908.
41. T. TAO, A new bound for finite field Besicovitch sets in four dimensions, to appear in Pacific J. Math., preprint, arXiv:math.CA/0204251
42. T. TAO, Finite field analogues of the Erdos, Falconer, and Furstenberg problems, unpublished, http://math.ucla.edu/∼tao/preprints/kakeya.html
43. T. TAO and A. VARGAS, A bilinear approach to cone multipliers, I: Restriction estimates, Geom. Funct. Anal. 10 (2000), 185-215.
44. T. TAO, A bilinear approach to cone multipliers, II: Applications, Geom. Funct. Anal. 10 (2000), 216-258.
45. T. TAO, A. VARGAS, and L. VEGA, A bilinear approach to the restriction and Kakeya conjectures, J. Amer. Math. Soc. 11 (1998), 967-1000.
46. P. A. TOMAS, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477-478.
47. T. WOLFF, An improved bound for Kakeya type maximal functions, Rev. Mat. Iberoamericana 11 (1995), 651-674.
48. T. WOLFF, A mixed norm estimate for the X-ray transform, Rev. Mat. Iberoamericana 14 (1998), 561-600.
49. T. WOLFF, “Recent work connected with the Kakeya problem” in Prospects in Mathematics (Princeton, 1996), Amer. Math. Soc., Providence, 1999, 129-162.
50. T. WOLFF, A sharp bilinear cone restriction estimate, Ann. of Math. (2) 153 (2001), 661-698.
51. A. ZYGMUND, On Fourier coefficients and transforms of functions of two variables, StudiaMath. 50 (1974), 189-201.
52. A. ZYGMUND, Trigonometric Series, Vols. I, II, Cambridge Math. Lib., Cambridge Univ. Press, Cambridge, 1988.