Stein’s Fourier restriction conjecture for n ≥ 3

Determine whether the Fourier extension operator Ef(x)=∫_{B(0,1)} f(ω) e^{2πi(x_1 ω_1+⋯+x_{n−1} ω_{n−1}+x_n |ω|^2)} dω satisfies ||Ef||_{L^p(R^n)} ≤ C_{n,p} ||f||_{L^p(R^{n−1})} for all p > 2n/(n−1) in dimensions n ≥ 3.

Background

Bourgain’s 1991 work linked Kakeya phenomena to Fourier restriction through wave packet decompositions, making tube overlap geometry central to Lp bounds for Ef.

Fefferman and Stein proved the conjecture in two dimensions; higher-dimensional cases remain open and are a focal point for techniques from multilinear restriction, polynomial partitioning, and Kakeya theory.

References

In its adjoint form, Stein's Fourier restriction conjecture asserts that

\Vert Ef\Vert_{Lp(Rn)}\leq C_{n,p}\Vert f\Vert_{Lp(R{n-1})},\qquad p>\frac{2n}{n-1}.

The conjecture was proved by Fefferman and Stein for $n=2$, and remains open in dimension $n\geq 3$.

— A Survey of the Kakeya conjecture, 2000-2025  (2512.09397 - Zahl, 10 Dec 2025) in Section 1 (Introduction)

The Fourier restriction conjecture in $\mathbb{R}d$ is proposed by Stein .

— The Szemerédi-Trotter Estimate in Finite Field with its Applications  (2609.35190 - Miao et al., 28 Sep 2026) in Conjecture 1, subsection “Fourier restriction problem,” Section 1

If $p>\frac{2n}{n-1}$, then

|E_Sf|{Lp(\mathbb{R}n)} \lesssim_p |f|{Lp(B{n-1}(0,1))}.

The conjecture is known in dimension two, following the work of Fefferman , but remains open in dimensions $n\geq3$.

— Refinements on the Restriction and Kakeya problems range in $\mathbb{R}^4$  (2610.06181 - Moreira, 5 Oct 2026) in Section 1, introduction; Conjecture (Restriction conjecture)