On the multilinear restriction and Kakeya conjectures

By Jonathan Bennett, Anthony Carbery, Terence Tao. 2005

345 citations recorded by the source corpus.

We prove d-linear analogues of the classical restriction and Kakeya conjectures in Rd. 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5e797cd4816e1ad998e77454eb7becd830b5144c

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Improved Fourier restriction estimates in higher dimensions (2018, methodology): “Using this fact alone, one may prove k-broad estimates in the range p > 2kk−1 corresponding to the Bennett–Carbery–Tao multilinear restriction theorem [1].7 • Transverse equidistribution estimates: These inequalities were introduced by Guth [12] and heavily exploit the curvature properties of the pa”
Strichartz estimates for Schrödinger equations on irrational tori (2013, methodology): “More recently, using multilinear restriction theory after [1, 8], Bourgain [7] improved the result (iii) for d 4 and showed that (1.”
Aspects of Multilinear Harmonic Analysis Related to Transversality (2014, methodology): “Theorem 4.1 (B–Carbery–Tao [13]).”
Average decay of the Fourier transform of measures with applications (2015, methodology): “[36] T. Tao, A sharp bilinear restrictions estimate for paraboloids, Geom.”
Fourier Restriction for Hypersurfaces in Three Dimensions and Newton Polyhedra (2016, background): “In particular one can mention milestone papers by Bourgain [3], Wolff [27], Tao [23], Bennett, Carbery, and Tao [2], Bourgain and Guth [4], and Guth [11] which also contain further important references.”
Decouplings for curves and hypersurfaces with nonzero Gaussian curvature (2014, methodology): “References [1] Bennett, J., Carbery, A. and Tao, T.”
POINTWISE CONVERGENCE OF SCHRÖDINGER SOLUTIONS AND MULTILINEAR REFINED STRICHARTZ ESTIMATES (2018, background): “ n+2kfkL2. This closes the induction on radius and completes the proof. 4.2. Proof of Theorem 4.2. One key ingredient in our proof is Bennett-CarberyTao multilinear Kakeya estimates: Theorem 4.4 (see [2] and [10]). Suppose that Sj ⊂ Sm−1,j = 1,··· ,k. Suppose that lj,a are lines in Rm and that the d”
On the Vinogradov mean value (2016, methodology): “writes D˜ dp k (δ−d 1 ,δ1) ≪ D˜dp k (δ−d−1 1 ,δ1) for 1 ≤ d<k (5.9) and follows from wave packet decomposition as explained in (4.1) and multilinear Kakeya type estimates originating from the work [BCT]. Note that in (5.9) and keeping in mind (5.4), we are essentially trading an Lp-norm for an Ld”
An improved bound on the Hausdorff dimension of Besicovitch sets in ℝ³ (2017, background): “This is a consequence of Bennett, Carbery, and Tao’s multilinear Kakeya theorem from [1].”
Sharp L2 estimates of the Schrödinger maximal function in higher dimensions (2018, methodology): “A key tool we are using in the broad case is the following multilinear refined Strichartz estimate from [13], which is proved using l(2) decoupling, induction on scales and multilinear Kakeya estimates (see [1, 17]).”
Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three (2015, background): “These include the one where the boxes are thin tubes, a landmark result due to Bennett, Carbery and Tao [3].”
Heat-flow monotonicity of Strichartz norms (2008, neutral): “Such an approach has proved effective in the context of the general Brascamp–Lieb inequalities, and was central to the approach to the multilinear Kakeya and Strichartz inequalities in [6].”
Moment inequalities for trigonometric polynomials with spectrum in curved hypersurfaces (2011, methodology): “FOR TRIGONOMETRIC POLYNOMIALS WITH SPECTRUM IN CURVED HYPERSURFACES J.Bourgain (0). Summary In this note we develop further the technique from [B-G], based on the multi-linear restriction theory from [B-C-T], to establish some new inequalities on the distribution of trigonometric polynomials on the ”
The proof of the l 2 Decoupling Conjecture (2015, background): “The following multilinear restriction estimate from [3] will play a key role in our proof.”
Sharp estimates for oscillatory integral operators via polynomial partitioning (2017, methodology): “This can be combined with the multilinear estimates of Bennett–Carbery–Tao [4] to prove Theorem 1.1 (this is essentially the argument used in [8]).”