A bilinear approach to the restriction and Kakeya conjectures

Tao, Terence; Vargas, Ana; Vega, Luis

doi:10.1090/S0894-0347-98-00278-1

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Browser Compatibility Info Help
ISSN 1088-6834(online) ISSN 0894-0347(print)

A bilinear approach to the restriction
and Kakeya conjectures

Authors: Terence Tao, Ana Vargas and Luis Vega
Journal: J. Amer. Math. Soc. 11 (1998), 967-1000
MSC (1991): Primary 42B10, 42B25
MathSciNet review: 1625056
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Bilinear restriction estimates have appeared in work of Bourgain, Klainerman, and Machedon. In this paper we develop the theory of these estimates (together with the analogues for Kakeya estimates). As a consequence we improve the spherical restriction theorem of Wolff from to , and also obtain a sharp spherical restriction theorem for $q> 4 - \frac{5}{27}$ .

1. Michael Beals, Self-spreading and strength of singularities for solutions to semilinear wave equations, Ann. of Math. (2) 118 (1983), no. 1, 187–214. MR 707166 (85c:35057), http://dx.doi.org/10.2307/2006959
2. J. Bourgain, Besicovitch type maximal operators and applications to Fourier analysis, Geom. Funct. Anal. 1 (1991), no. 2, 147–187. MR 1097257 (92g:42010), http://dx.doi.org/10.1007/BF01896376
3. Jean Bourgain, On the restriction and multiplier problems in 𝑅³, Geometric aspects of functional analysis (1989–90), Lecture Notes in Math., vol. 1469, Springer, Berlin, 1991, pp. 179–191. MR 1122623 (92m:42017), http://dx.doi.org/10.1007/BFb0089225
4. J. Bourgain, A remark on Schrödinger operators, Israel J. Math. 77 (1992), no. 1-2, 1–16. MR 1194782 (93k:35071), http://dx.doi.org/10.1007/BF02808007
5. J. Bourgain, Estimates for cone multipliers, Geometric aspects of functional analysis (Israel, 1992–1994) Oper. Theory Adv. Appl., vol. 77, Birkhäuser, Basel, 1995, pp. 41–60. MR 1353448 (96m:42022)
6. Jean Bourgain, Some new estimates on oscillatory integrals, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), Princeton Math. Ser., vol. 42, Princeton Univ. Press, Princeton, NJ, 1995, pp. 83–112. MR 1315543 (96c:42028)
7. Lennart Carleson and Per Sjölin, Oscillatory integrals and a multiplier problem for the disc, Studia Math. 44 (1972), 287–299. (errata insert). Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, III. MR 0361607 (50 #14052)
8. Antonio Cordoba, The Kakeya maximal function and the spherical summation multipliers, Amer. J. Math. 99 (1977), no. 1, 1–22. MR 0447949 (56 #6259)
9. S. W. Drury, 𝐿^{𝑝} estimates for the X-ray transform, Illinois J. Math. 27 (1983), no. 1, 125–129. MR 684547 (85b:44004)
10. Charles Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970), 9–36. MR 0257819 (41 #2468)
11. Charles Fefferman, The multiplier problem for the ball, Ann. of Math. (2) 94 (1971), 330–336. MR 0296602 (45 #5661)
12. Lars Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79–183. MR 0388463 (52 #9299)
13. N. Katz, preprint.
14. S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), no. 9, 1221–1268. MR 1231427 (94h:35137), http://dx.doi.org/10.1002/cpa.3160460902
15. Sergiu Klainerman and Matei Machedon, Remark on Strichartz-type inequalities, Internat. Math. Res. Notices 5 (1996), 201–220. With appendices by Jean Bourgain and Daniel Tataru. MR 1383755 (97g:46037), http://dx.doi.org/10.1155/S1073792896000153
16. Sergiu Klainerman and Matei Machedon, On the regularity properties of a model problem related to wave maps, Duke Math. J. 87 (1997), no. 3, 553–589. MR 1446618 (98e:35118), http://dx.doi.org/10.1215/S0012-7094-97-08718-4
17. A. Moyua, A. Vargas, and L. Vega, Schrödinger maximal function and restriction properties of the Fourier transform, Internat. Math. Res. Notices 16 (1996), 793–815. MR 1413873 (97k:42042), http://dx.doi.org/10.1155/S1073792896000499
18. A. Moyua, A. Vargas, L. Vega, Restriction theorems and Maximal operators related to oscillatory integrals in $\mathbf{R}^3$ , to appear, Duke Math. J.
19. W. Schlag, A generalization of Bourgain’s circular maximal theorem, J. Amer. Math. Soc. 10 (1997), no. 1, 103–122. MR 1388870 (97c:42035), http://dx.doi.org/10.1090/S0894-0347-97-00217-8
20. W. Schlag, A geometric proof of the circular maximal theorem, to appear, Duke Math. J.
21. W. Schlag, A geometric inequality with applications to the Kakeya problem in three dimensions, to appear, Geometric and Functional Analysis.
22. C. D. Sogge, Concerning Nikodym-type sets in 3-dimensional curved space, preprint.
23. Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192 (95c:42002)
24. T. Tao, The Bochner-Riesz conjecture implies the Restriction conjecture, to appear, Duke Math J.
25. T. Tao, A. Vargas, A bilinear approach to cone multipliers and related operators, in preparation.
26. Peter A. Tomas, A restriction theorem for the Fourier transform, Bull. Amer. Math. Soc. 81 (1975), 477–478. MR 0358216 (50 #10681), http://dx.doi.org/10.1090/S0002-9904-1975-13790-6
27. Thomas Wolff, An improved bound for Kakeya type maximal functions, Rev. Mat. Iberoamericana 11 (1995), no. 3, 651–674. MR 1363209 (96m:42034), http://dx.doi.org/10.4171/RMI/188
28. T. H. Wolff, A mixed norm estimate for the x-ray transform, to appear in Revista Mat. Iberoamericana.

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|