Restriction conjecture

Short description: Conjecture about the behaviour of the Fourier transform on curved hypersurfaces

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.

Statement

The restriction conjecture states that $\Vert \widehat{gd\sigma }{\Vert }_{L^q({\mathbb{ℝ}}^n)}\lesssim \Vert g{\Vert }_{L^p(S^{n-1})}$ for certain q and n, where $\Vert f{\Vert }_{L^p}$ represents the L^p norm, or ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x{)}^{p}dx$ and $f\lesssim g$ means that $f\le Cg$ for some constant $C$.

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

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