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.^[1][2] It was first hypothesized by Elias Stein.^[3] The conjecture states that two necessary conditions needed to solve a problem known as the restriction problem in that scenario are also sufficient.^[2][3]

The restriction conjecture is closely related to the Kakeya conjecture, Bochner-Riesz conjecture and the local smoothing conjecture.^[4][5]

Statement

The restriction conjecture states that ${\textstyle \|{\widehat {g\,d\sigma }}\|_{L^{q}(\mathbb {R} ^{n})}\lesssim \|g\|_{L^{p}(S^{n-1})}}$ for certain q and n, where ${\textstyle \|f\|_{L^{p}}}$ represents the L^p norm, or ${\textstyle \int _{-\infty }^{\infty }f(x)^{p}\,dx}$ and ${\textstyle f\lesssim g}$ means that ${\textstyle f\leq Cg}$ for some constant ${\textstyle C}$ .^{[6][clarification needed]}

The requirements of q and n set by the conjecture are that ${\displaystyle {\frac {1}{q}}<{\frac {n-1}{2n}}}$ and ${\displaystyle {\frac {1}{q}}\leq {\frac {n-1}{n+1}}{\frac {1}{p}}}$ .^[6]

The restriction conjecture has been proved for dimension ${\textstyle n=2}$ as of 2021.^[6]

References

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.
2. Kinnear, George (7 February 2011). "Restriction Theory" (PDF). webhomes.maths.ed.ac.uk.
3. Stedman, Richard James (September 2013). "The Restriction and Kakeya Conjectures" (PDF). University of Birmingham.
4. Tao, Terence (2024-11-17). "Terence Tao (@[e-mail])". 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.
6. Kinnear, George (7 February 2011). "Restriction Theory" (PDF).