Restriction conjecture

In harmonic analysis, the restriction conjecture, also known as the Fourier restriction conjecture, is a conjecture about the behaviour of the Fourier transform on curved hypersurfaces. It was first hypothesized by Elias Stein. The conjecture states that two necessary conditions needed to solve a problem known as the restriction problem in that scenario are also sufficient.

The restriction conjecture is closely related to the Kakeya conjecture, Bochner-Riesz conjecture and the local smoothing conjecture.

On October 26, 2026, OpenAI made an announcement claiming to have proved the local smoothing conjecture, and thus the restriction conjecture by implication. As of October 26, 2026,^[update] the claim has not been peer reviewed.

Statement

The restriction conjecture states that $‖ \hat{g d σ} ‖_{L^{q} (ℝ^{n})} ≲ ‖ g ‖_{L^{p} (S^{n - 1})}$ for certain q and n, where $‖ f ‖_{L^{p}}$ represents the L^p norm, or $\int_{- \infty}^{\infty} f (x)^{p} d x$ and $f ≲ g$ means that $f \leq C g$ for some constant $C$ .^{[clarification needed]}

The requirements of q and n set by the conjecture are that $\frac{1}{q} < \frac{n - 1}{2 n}$ and $\frac{1}{q} \leq \frac{n - 1}{n + 1} \frac{1}{p}$ .

The restriction conjecture has been proved for dimension $n = 2$ as of 2021.

References

↑ Ansede, Manuel (2025-07-14). "What is the smallest space in which a needle can be rotated to point in the opposite direction? This mathematician has finally solved the Kakeya conjecture". EL PAÍS English. Retrieved 2025-07-20.
1 2 Kinnear, George (7 February 2011). "Restriction Theory" (PDF). webhomes.maths.ed.ac.uk.
1 2 Stedman, Richard James (September 2013). "The Restriction and Kakeya Conjectures" (PDF). University of Birmingham.
↑ Tao, Terence (2024-11-17). "Terence Tao (@tao@mathstodon.xyz)". Mathstodon. Retrieved 2025-07-20.
↑ Cepelewicz, Jordana (2023-09-12). "A Tower of Conjectures That Rests Upon a Needle". Quanta Magazine. Retrieved 2025-07-20.
↑ "Working with academic communities to build scientific understanding". OpenAI. 2026-10-06. Retrieved 2026-10-26.
1 2 3 Kinnear, George (7 February 2011). "Restriction Theory" (PDF).

This mathematical analysis–related article is a stub. You can help Wikipedia by adding missing information.

[1] Ansede, Manuel (2025-07-14). "What is the smallest space in which a needle can be rotated to point in the opposite direction? This mathematician has finally solved the Kakeya conjecture". EL PAÍS English. Retrieved 2025-07-20.

[Kinnear-2] 1 2 Kinnear, George (7 February 2011). "Restriction Theory" (PDF). webhomes.maths.ed.ac.uk.

[Stedman-3] 1 2 Stedman, Richard James (September 2013). "The Restriction and Kakeya Conjectures" (PDF). University of Birmingham.

[4] Tao, Terence (2024-11-17). "Terence Tao (@tao@mathstodon.xyz)". Mathstodon. Retrieved 2025-07-20.

[5] Cepelewicz, Jordana (2023-09-12). "A Tower of Conjectures That Rests Upon a Needle". Quanta Magazine. Retrieved 2025-07-20.

[openai2026claim-6] "Working with academic communities to build scientific understanding". OpenAI. 2026-10-06. Retrieved 2026-10-26.

[:0-7] 1 2 3 Kinnear, George (7 February 2011). "Restriction Theory" (PDF).