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 ‖ g d σ ^ ‖ L q ( R n ) ≲ ‖ g ‖ L p ( S n − 1 ) {\textstyle \|{\widehat {g\,d\sigma }}\|_{L^{q}(\mathbb {R} ^{n})}\lesssim \|g\|_{L^{p}(S^{n-1})}} for certain q and n, where ‖ f ‖ L p {\textstyle \|f\|_{L^{p}}} represents the Lp norm, or ∫ − ∞ ∞ f ( x ) p d x {\textstyle \int _{-\infty }^{\infty }f(x)^{p}\,dx} and f ≲ g {\textstyle f\lesssim g} means that f ≤ C g {\textstyle f\leq Cg} for some constant C {\textstyle C} . The requirements of q and n set by the conjecture are that 1 q < n − 1 2 n {\displaystyle {\frac {1}{q}}<{\frac {n-1}{2n}}} and 1 q ≤ n − 1 n + 1 1 p {\displaystyle {\frac {1}{q}}\leq {\frac {n-1}{n+1}}{\frac {1}{p}}} . The restriction conjecture has been proved for dimension n = 2 {\textstyle n=2} as of 2021.
References