A bilinear approach to cone multipliers I. Restriction estimates (original) (raw)

2000, Geometric And Functional Analysis

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 estimates for the cone, sphere, and paraboloid in R 3 , building upon and unifying previous work in this direction by Bourgain, Wolff, and others. In a subsequent paper [TV] we will give applications of these estimates to some open problems in harmonic analysis and wave equations. T. TAO AND A. VARGAS GAFA respectively; to be consistent with the notation of [TVV] we denote these estimates by R * S (p → 2q) and R * S (p → 2q, α/2) respectively. These estimates are well understood in two dimensions, but there are many open problems remaining in three and higher dimensions, with several applications to harmonic analysis and PDE. In this paper we concentrate on the threedimensional case, although we comment briefly in the four-dimensional case at the end of the paper. Linear restriction estimates have a long history in harmonic analysis and PDE (see for instance [St, Chapter IX and the references therein]), but the systematic and explicit study of bilinear estimates, and their application to the linear problem, has only appeared recently. In [TVV] the bilinear estimates were studied under the assumption that S 1 , S 2 were unit-separated subsets of a graph of an elliptic phase function (see [TVV, Section 2]). In that case the main interest was obtaining new progress on the corresponding linear estimates for the restriction problem, and also for Bochner-Riesz multipliers. One of the basic tools developed was an equivalence between linear restriction estimates and bilinear restriction estimates when the exponents p, q were in the conjectured range for the restriction conjecture. On the other hand the bilinear estimates can also hold for a wider range of exponents, which explains why they can be used to improve upon the linear estimates. The primary purpose of this paper is to extend these results to more general surfaces, in particular subsets of the light cone in R 2+1 , mainly by pursuing the ideas in [Bo4]. Although this paper contains many similar themes to [TVV], we have made an effort to keep it mostly self-contained. The surfaces we shall consider are as follows. Definition 1.1. Suppose that S 1 and S 2 are compact surfaces with boundary in R 3. If ξ ∈ S t , t = 1, 2, we use n t (ξ) ∈ S 2 /± to denote the unit normal to S t at ξ. We say that the pair S 1 and S 2 are of disjoint conic type if the following statements hold: • (Transversality) For all ξ t ∈ S t , t = 1, 2 we have n t (ξ t) ∈ N t , where N 1 and N 2 are small disjoint caps in S 2 /± which are separated by a distance comparable to 1. • (Null direction) The map dn t : T ξ t S t → T ξ t S t has eigenvalue 0 with multiplicity one in the direction w t (ξ t) ∈ S 2 /±. We also assume that the remaining eigenvalue has magnitude ∼ 1. • (Transversality of null directions) For all ξ t ∈ S t , t = 1, 2 we have w t (ξ t) ∈ W t , where W 1 and W 2 are small disjoint caps in S 2 /± which