A bilinear approach to the restriction and Kakeya conjectures

Abstract

Bilinear restriction estimates have appeared in work of Bourgain, Klainerman, and Machedon. In this paper we develop the theory of these estimates (together with the analogues for Kakeya estimates). As a consequence we improve the ( L p , L p ) (L^p,L^p) spherical restriction theorem of Wolff from p > 42 / 11 p > 42/11 to p > 34 / 9 p > 34/9 , and also obtain a sharp ( L p , L q ) (L^p,L^q) spherical restriction theorem for q > 4 − 5 27 q> 4 - \frac {5}{27} .

Mass Concentration With Mixed Norm for a Nonelliptic Schrödinger Equation

“…Then the range of p in (3.5) or (3.6) is also extended. In particular, we may substitute 16 13 for the infimum 12 7 of p * in Lemma 3.5. Nevertheless, the conditions in Theorem 3.3 are enough to prove Lemma 3.5.…”

Untitled

“…In 1973, Fefferman showed a connection between the restriction conjecture for the sphere and the Bochner-Riesz conjecture [7]. Recent progress on the restriction conjecture has used a bilinear approach (as in [20]) and recently Lee adapted Fefferman's argument to apply the bilinear results directly to spherical means [10]. This approach applied to the recent bilinear result of Tao [19] proves the Bochner-Riesz conjecture for p > 2 + 4 d for d ≥ 3, the best current range of p. While the case of ρ(ξ) = |ξ| provides an interesting test case, it is much simpler than the question of general ρ.…”

A Fourier restriction estimate for surfaces of positive curvature in $\mathbb{R}^6$

“…Wolff improved Bourgain's result to (2n 2 + n + 6)/(n 2 + n − 1); see [13]. Then in three dimensions Tao, Vargas, and Vega further lowered this exponent, and more importantly they developed the bilinear approach which related this conjecture to restriction estimates for compact, transverse subsets of hypersurfaces; see [10], [11]. The work of Tao in [8], which was a bilinear estimate for compact transverse subsets of paraboloids, through this bilinear method, verified the conjecture for p > 2(n + 2)/n.…”