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} .