On the multilinear restriction and Kakeya conjectures

Acta Mathematica

Volumn 196, Issue 2, 2006, Pages 261-302

  Bennett, Jonathan ^a   Carbery, Anthony ^b   Tao, Terence ^c  

^a UNIVERSITY OF BIRMINGHAM   (United Kingdom)

^b UNIVERSITY OF EDINBURGH   (United Kingdom)

^c UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33750826562     PISSN: 00015962     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11511-006-0006-4     Document Type: Article

References (35)

1. n. Bull. London Math. Soc., 36 (2004), 407-412.
2. BECKNER, W., CARBERY, A., SEMMES, S. & SORIA, F., A note on restriction of the Fourier transform to spheres. Bull. London Math. Soc., 21 (1989), 394-398.
3. 3. Electron. Res. Announc. Amer. Math. Soc., 10 (2004), 97-102.
4. BENNETT, J., CARBERY, A., CHRIST, M. & TAO, T., The Brascamp-Lieb inequalities: finiteness, structure and extremals. To appear in Geom. Funct. Anal.
5. BENNETT, J., CARBERY, A. & WRIGHT, J., A non-linear generalisation of the Loomis-Whitney inequality and applications. Math. Res. Lett., 12 (2005), 443-457.
6. BERGH, J. & LÖFSTRÖM, J., Interpolation Spaces. An Introduction. Springer, Berlin, 1976.
7. BLEI, R. C., Fractional Cartesian products of sets. Ann. Inst. Fourier (Grenoble), 29:2 (1979), 79-105.
8. BOURGAIN, J., Besicovitch type maximal operators and applications to Fourier analysis. Geom. Funct. Anal., 1 (1991), 147-187.
9. p-estimates for oscillatory integrals in several variables. Geom. Funct. Anal., 1 (1991), 321-374.
10. _ Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I, II. Geom. Funct. Anal., 3 (1993), 107-156, 209-262.
11. N and related entropy inequalities. J. Geom. Anal., 14 (2004), 487-520.
12. CHAZELLE, B., EDELSBRUNNER, H., GUIBAS, L. J., POLLACK, R., SEIDEL, R., SHARIR, M. & SNOEYINK, J., Counting and cutting cycles of lines and rods in space. Comput. Geom., 1 (1992), 305-323.
13. p), in Euclidean Harmonic Analysis (Proc. Sem., Univ. Maryland, College Park, MD, 1979), Lecture Notes in Math., 779, pp. 162-177. Springer, Berlin, 1980.
14. ERDOG̃AN, M. B., A bilinear Fourier extension theorem and applications to the distance set problem. Int. Math. Res. Not., 2005:23 (2005), 1411-1425.
15. FEFFERMAN, C., The multiplier problem for the ball. Ann. of Math., 94 (1971), 330-336.
16. FELDMAN, S. & SHARIR, M., An improved bound for joints in arrangements of lines in space. Discrete Comput. Geom., 33 (2005), 307-320.
17. p. Ark. Mat., 11 (1973), 1-11.
18. 3. Ann. of Math., 152 (2000), 383-446.
19. KLAINERMAN, S. & MACHEDON, M., Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math., 46 (1993), 1221-1268.
20. LEE, S., Improved bounds for Bochner-Riesz and maximal Bochner-Riesz operators. Duke Math. J., 122 (2004), 205-232.
21. LIEB, E. H., Gaussian kernels have only Gaussian maximizers. Invent. Math., 102 (1990), 179-208.
22. LOOMIS, L. H. & WHITNEY, H., An inequality related to the isoperimetric inequality. Bull. Amer. Math. Soc, 55 (1949), 961-962.
23. 3. Duke Math. J., 96 (1999), 547-574.
24. OBERLIN, D. M. & STEIN, E. M., Mapping properties of the Radon transform. Indiana Univ. Math. J., 31 (1982), 641-650.
25. SHARIR, M., On joints in arrangements of lines in space and related problems. J. Combin. Theory Ser. A, 67 (1994), 89-99.
26. STEIN, E. M., Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, 43. Princeton University Press, Princeton, NJ, 1993.
27. TAO, T., Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates. Math. Z., 238 (2001), 215-268.
28. _ A sharp bilinear restrictions estimate for paraboloids. Geom. Funct. Anal., 13 (2003), 1359-1384.
29. _ Recent progress on the restriction conjecture. arXiv:math.CA/0311181.
30. TAO, T., VARGAS, A. & VEGA, L., A bilinear approach to the restriction and Kakeya conjectures. J. Amer. Math. Soc., 11 (1998), 967-1000.
31. VARGAS, A., Restriction theorems for a surface with negative curvature. Math. Z., 249 (2005), 97-111.
32. WISEWELL, L., Kakeya sets of curves. Geom. Funct. Anal., 15 (2005), 1319-1362.
33. WOLFF, T. H., Recent work connected with the Kakeya problem, in Prospects in Mathematics (Princeton, NJ, 1996), pp. 129-162. Amer. Math. Soc., Providence, RI, 1999.
34. _ A sharp bilinear cone restriction estimate. Ann. of Math., 153 (2001), 661-698.
35. _ Lectures on Harmonic Analysis. University Lecture Series, 29. Amer. Math. Soc., Providence, RI, 2003.