Fourier extension conjecture for the paraboloid in three dimensions

Establish the Fourier extension conjecture for the paraboloid in three dimensions by proving that the Fourier extension operator associated with the paraboloid P^2 ⊂ R^3, defined by (E f)(ξ) = ∫_U e^{-i Φ(x)·ξ} f(x) dx with Φ(x) = (x_1, x_2, x_1^2 + x_2^2) and U ⊂ R^2, is bounded from L^q(U) to L^q(R^3) for all q > 3 (excluding endpoints). The authors conjecture that combining the trilinear bootstrapping argument of Bennett–Carbery–Tao (Proposition 2.1), their Theorem \ref{slide} permitting parallel and repeated tubes, and their earlier main result (Rios–Sawyer, Theorem 3) will yield this proof.

Background

The paper proves the Kakeya maximal operator conjecture in three dimensions through a series of equivalences that connect Kakeya-type estimates with Fourier extension-type square function inequalities on the paraboloid. While their main result resolves the Kakeya maximal operator conjecture, the Fourier extension conjecture for the paraboloid (boundedness of the extension operator from L^q(U) to L^q(R^3) for q > 3) remains a central problem.

In the Introduction, the authors explicitly propose a route to settle the Fourier extension conjecture by adapting a trilinear version of Bourgain’s bootstrapping argument (as used by Bennett, Carbery, and Tao), together with their newly established sliding Gaussian reduction (Theorem \ref{slide}) and a result from their previous paper. This indicates both the open nature of the conjecture and a concrete methodology they believe could resolve it.