Abstract
We prove d-linear analogues of the classical restriction and Kakeya conjectures in R d. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We conclude by giving some applications to the corresponding variable-coefficient problems and the so-called “joints” problem, as well as presenting some n-linear analogues for n < d.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barceló, J.A., Bennett, J.M., Carbery, A.: A multilinear extension inequality in R n. Bull. London Math. Soc. 36, 407–412 (2004)
Beckner, W., Carbery, A., Semmes, S., Soria, F.: A note on restriction of the Fourier transform to spheres. Bull. London Math. Soc. 21, 394–398 (1989)
Bennett, J.: A trilinear restriction problem for the paraboloid in R 3. Electron. Res. Announc. Amer. Math. Soc. 10, 97–102 (2004)
Bennett, J., Carbery, A., Christ, M., Tao, T.: The Brascamp-Lieb inequalities: finiteness, structure and extremals. To appear in Geom. Funct. Anal.
Bennett, J., Carbery, A., Wright, J.: A non-linear generalisation of the Loomis-Whitney inequality and applications. Math. Res. Lett. 12, 443–457 (2005)
Bergh, J., Löfström, J.: Interpolation spaces. An introduction. Springer, Berlin Heidelberg New York (1976)
Blei, R.C.: Fractional Cartesian products of sets. Ann. Inst. Fourier (Grenoble) 29(2), 79–105 (1979)
Bourgain, J.: Besicovitch type maximal operators and applications to Fourier analysis. Geom. Funct. Anal. 1, 147–187 (1991)
Bourgain, J.: L p-estimates for oscillatory integrals in several variables. Geom. Funct. Anal. 1, 321–374 (1991)
Bourgain, J.: Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I, II. Geom. Funct. Anal. 3, 107–156, 209–262 (1993)
Carlen, E.A., Lieb, E.H., Loss, M.: A sharp analog of Young’s inequality on S N and related entropy inequalities. J. Geom. Anal. 14, 487–520 (2004)
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, 305–323 (1992)
Córdoba, A.: Multipliers of \(\cal F\)(L p), in Euclidean Harmonic Analysis (Proc. Sem., Univ. Maryland, College Park, MD, 1979), Lecture Notes in Math, 779, pp. 162–177. Springer, Berlin Heidelberg New York (1980)
Erdogan, M.B.: A bilinear Fourier extension theorem and applications to the distance set problem. Int. Math. Res. Not. 23, 1411–1425 (2005)
Fefferman, C.: The multiplier problem for the ball. Ann. Math. 94, 330–336 (1971)
Feldman, S., Sharir, M.: An improved bound for joints in arrangements of lines in space. Discrete. Comput. Geom. 33, 307–320 (2005)
Hörmander, L.: Oscillatory integrals and multipliers on FL p. Ark. Mat. 11, 1–11 (1973)
Katz, N.H., Laba, I., Tao, T.: An improved bound on the Minkowski dimension of Besicovitch sets in R 3. Ann. Math. 152, 383–446 (2000)
Klainerman, S., Machedon, M.: Space-time estimates for null forms and the local existence theorem. Comm. Pure Appl. Math. 46, 1221–1268 (1993)
Lee, S.: Improved bounds for Bochner-Riesz and maximal Bochner-Riesz operators. Duke Math. J. 122, 205–232 (2004)
Lieb, E.H.: Gaussian kernels have only Gaussian maximizers. Invent. Math. 102, 179–208 (1990)
Loomis, L.H., Whitney, H.: An inequality related to the isoperimetric inequality. Bull. Amer. Math. Soc. 55, 961–962 (1949)
Moyua, A., Vargas, A., Vega, L.: Restriction theorems and maximal operators related to oscillatory integrals in R 3. Duke Math. J. 96, 547–574 (1999)
Oberlin, D.M., Stein, E.M.: Mapping properties of the Radon transform. Indiana Univ. Math. J. 31, 641–650 (1982)
Sharir, M.: On joints in arrangements of lines in space and related problems. J. Combin. Theory Ser. A 67, 89–99 (1994)
Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, 43. Princeton University Press, Princeton, NJ (1993)
Tao, T.: Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates. Math. Z. 238, 215–268 (2001)
Tao, T.: A sharp bilinear restrictions estimate for paraboloids. Geom. Funct. Anal. 13, 1359–1384 (2003)
Tao, T.: Recent progress on the restriction conjecture. arXiv:math.CA/0311181
Tao, T., Vargas, A., Vega, L.: A bilinear approach to the restriction and Kakeya conjectures. J. Amer. Math. Soc. 11, 967–1000 (1998)
Vargas, A.: Restriction theorems for a surface with negative curvature. Math. Z. 249, 97–111 (2005)
Wisewell, L.: Kakeya sets of curves. Geom. Funct. Anal. 15, 1319–1362 (2005)
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)
Wolff, T.H.: A sharp bilinear cone restriction estimate. Ann. Math. 153, 661–698 (2001)
Wolff, T.H.: Lectures on Harmonic Analysis. University Lecture Series, 29. Amer. Math. Soc., Providence, RI (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bennett, J., Carbery, A. & Tao, T. On the multilinear restriction and Kakeya conjectures. Acta Math 196, 261–302 (2006). https://doi.org/10.1007/s11511-006-0006-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11511-006-0006-4