Fourier restriction/extension L^p–L^q bounds for the sphere in higher dimensions

Establish L^p→L^q bounds for the Fourier restriction and extension operators associated with the unit sphere S^{n−1} in ℝ^n for n ≥ 3, determining the optimal exponents.

Background

Fourier restriction/extension estimates for curved hypersurfaces such as spheres play a pivotal role in harmonic analysis and PDE. In n=2, the theory is settled, but in n ≥ 3, the optimal Lp–Lq exponents remain unknown.

The paper emphasizes the deep interconnections of restriction/extension bounds with Kakeya and Bochner–Riesz problems, noting that progress on any one of these can yield advances on the others.

References

Three of the most important open questions in euclidean harmonic analysis are: (i) Lp bounds for the Kakeya maximal operator (the corresponding dimensional consequence of these when n=3 has recently been established in spectacular work of Wang and Zahl), (ii) Lp bounds for the Bochner--Riesz operators S\delta, and (iii) Lp -Lq bounds for the Fourier restriction and extension operators for the sphere. These are known to be intimately related, and all of them are resolved when n=2, but all are open in all higher dimensions.

— Littlewood, Paley and Almost-Orthogonality: a theory well ahead of its time  (2511.22605 - Carbery, 27 Nov 2025) in Section 7.3 (Angular decompositions — reverse inequalities and curvature)

The spherical restriction conjecture asserts that R(p,q) holds whenever

1\le q<\rho_n:=\frac{2n}{n+1}, \qquad 1\le p \le \kappa_n(q):=\frac{(n-1)q'}{n+1}.

— Boundary-Weighted Fourier Inequalities for Convex Domains  (2608.19806 - Bampouras et al., 20 Aug 2026) in Section 6, subsection “Consequences of the restriction conjecture”