A sharp bilinear restriction estimate for paraboloids

By Terence Tao. 2002

302 citations recorded by the source corpus.

AbstractRecently Wolff [W3] obtained a sharp L2 bilinear restriction theorem for bounded subsets of the cone in general dimension. Here we adapt the argument of Wolff to also handle subsets of “elliptic surfaces” such as paraboloids. Except for an endpoint, this answers a conjecture of Machedon and Klainerman, and also improves upon the known restriction theory for the paraboloid and sphere.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/5d1d1a7c3f44c47cc82697fd6968bd5c8fe39356

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Bounds on Oscillatory Integral Operators Based on Multilinear Estimates (2010, background): “ce of K1,Kshows that QR≪ Rεfor q> 10 3 . (6.19) Remark. The use of diﬀerent scales in previous analysis (and even more so in §3) is reminiscent of the ‘induction on scales’ approach form [Wo2] and [T1], although the present argument is considerably simpler. In particular, it suﬃces to take K,K1 t”
A Sharp Inequality for the Strichartz Norm (2008, background): “The restriction conjecture asserts that these are sufficient conditions in each case for (1.21) to hold, and so far it has been proved for the range q > (2n + 6)/(n + 1) in both cases, the paraboloid by Tao [ 11 ] and the cone by Wolff [12].”
Improved bounds for Stein's square functions (2010, methodology): “As in the related work by the first author [22], our main tool will be Tao’s bilinear estimate [41] for the adjoint of the Fourier restriction operator.”
Restriction theorems for a surface with negative curvature (2005, methodology): “as in [ 15 ], using the induction hypothesis (10) and (14), by”
Sharp linear and bilinear restriction estimates for paraboloids in the cylindrically symmetric case (2007, methodology): “ot apply the Hardy-Littlewood-Sobolev inequality any more. Instead, we can perform a Whitney-type decomposition to I and thus create some frequency separation for f. Similar arguments can be found in [16], [18], [22], [23]. Proposition 3.5 (q = 4 line). Suppose f ∈ L 1. Then for q = 4, 4 ≤ p ≤ ∞, ε ”
Aspects of Multilinear Harmonic Analysis Related to Transversality (2014, methodology): “and Tao [48] showed that (12) ‖f̂1dσ1f̂2dσ2‖Lq/2(Rd) .”
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.”