A sharp bilinear restriction estimate for paraboloids

@article{Tao2002ASB,
  title={A sharp bilinear restriction estimate for paraboloids},
  author={Terence Tao},
  journal={Geometric \& Functional Analysis GAFA},
  year={2002},
  volume={13},
  pages={1359-1384}
}
  • T. Tao
  • Published 7 October 2002
  • Mathematics
  • Geometric & Functional Analysis GAFA
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. 
Bilinear restriction estimates for surfaces with curvatures of different signs
Recently, the sharp L 2 -bilinear (adjoint) restriction estimates for the cone and the paraboloid were established by Wolff and Tao, respectively. Their results rely on the fact that for the cone and
An endpoint estimate of the bilinear paraboloid restriction operator
In Fourier restriction problems, a cone and a paraboloid are model surfaces. The sharp bilinear cone restriction estimate was first shown by Wolff, and later the endpoint was obtained by Tao. For a
Restriction theorems for a surface with negative curvature
Abstract.We prove a bilinear restriction theorem for a surface of negative curvature. This is the analogue of the results of T. Wolff [19] and T. Tao [14], [15] for cones and paraboloids. As a
Global restriction estimates for elliptic hyperboloids
We prove global Fourier restriction estimates for elliptic, or two-sheeted, hyperboloids of arbitrary dimension $d \geq 2$, extending recent joint work with Oliveira e Silva and Stovall. Our results
Bilinear restriction estimates for surfaces of codimension bigger than 1
In connection with the restriction problem in $\mathbb R^n$ for hypersurfaces including the sphere and paraboloid, the bilinear (adjoint) restriction estimates have been extensively studied. However,
A trilinear restriction estimate with sharp dependence on transversality
Abstract:We improve the Bennett-Carbery-Tao trilinear restriction estimate for subsets of the paraboloid in three dimensions, giving the sharp bound with respect to the transversality. The main
Extremizers for Adjoint Restriction to a Pair of Reflected Paraboloids
We consider the adjoint restriction inequality associated to the hypersurface {(τ, ξ) : τ = ±|ξ|, ξ ∈ R} at the Stein-Tomas exponent. Extremizers exist in all dimensions and extremizing sequences are
Linear and bilinear restriction to certain rotationally symmetric hypersurfaces
Conditional on Fourier restriction estimates for elliptic hypersurfaces, we prove optimal restriction estimates for polynomial hypersurfaces of revolution for which the defining polynomial has
Optimal bilinear restriction estimates for general hypersurfaces and the role of the shape operator
It is known that under some transversality and curvature assumptions on the hypersurfaces involved, the bilinear restriction estimate holds true with better exponents than what would trivially follow
On the multilinear restriction and Kakeya conjectures
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
...
...

References

SHOWING 1-10 OF 42 REFERENCES
Endpoint bilinear restriction theorems for the cone, and some sharp null form estimates
Abstract. Recently Wolff [25] obtained a nearly sharp $L^2$ bilinear restriction theorem for bounded subsets of the cone in general dimension. We obtain the endpoint of Wolff's estimate and
A bilinear approach to cone multipliers I. Restriction estimates
Abstract. In this paper, we continue the study of three-dimensional bilinear restriction and Kakeya estimates which was initiated in [TVV]. In particular, we give new linear and bilinear restriction
A bilinear approach to cone multipliers II. Applications
Abstract. This paper is a continuation of [TV], in which new bilinear estimates for surfaces in $ {\bold R}^3 $ were proven. We give a concrete improvement to the square function estimate of
Restriction implies Bochner-Riesz for paraboloids
Let Σ ⊆ ℝ n be a (compact) hypersurface with non-vanishing Gaussian curvature, with suitable parameterizations, also called Σ: U → ℝ n ( U open patches in ℝ n−1 ). The restriction problem for Σ is
A Sharp Bilinear Cone Restriction Estimate
The purpose of this paper is to prove an essentially sharp L^2 Fourier restriction estimate for light cones, of the type which is called bilinear in the recent literature.
A sharp bilinear cone restriction estimate
The purpose of this paper is to prove an essentially sharp L2 Fourier restriction estimate for light cones, of the type which is called bilinear in the recent literature. Fix d ≥ 3, denote variables
GLOBAL WELLPOSEDNESS FOR 1 D NON-LINEAR SCHRÖDINGER EQUATION FOR DATA WITH AN INFINITE L 2 NORM
– We prove global wellposedness for the one-dimensional cubic non-linear Schrödinger equation in a space of distributions which is invariant under Galilean transformations and includes L2. This space
A bilinear approach to the restriction and Kakeya conjectures
The purpose of this paper is to investigate bilinear variants of the restriction and Kakeya conjectures, to relate them to the standard formulations of these conjectures, and to give applications of
Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations
A simple duality argument shows these two problems are completely equivalent ifp and q are dual indices, (]/) + (I/q) ]. ]nteresl in Problem A when S is a sphere stems from the work of C. Fefferman
...
...