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 in Rd by (x, xd) with x ∈ Rd−1, and let Γ = {x : xd = |x| and 1 ≤ xd ≤ 2}. Let Γ1 and Γ2 be disjoint conical subsets, i.e. Γi = {x ∈ Γ : x xd ∈ Ωi} where Ωi are disjoint closed subsets of the sphere S d−2. Let f and g be two functions on Γ whose supports are contained in Γ1 and Γ2 respectively. We will prove the following estimate, where σ is surface measure on Γ, and f̂ dσ is the Rd Fourier transform: Theorem 1. If p > 1 + 2 d then