Fourier restriction problem

Remarks

For $p = 1$ the estimate holds for all $q$, since $\abs{\hat f} \le \norm{f}_{L^1}$ everywhere. For $p = 2$ it fails for every $q$, since $\hat f$ is then an arbitrary $L^2$ function and has no meaningful values on a set of measure zero. When $S$ is a piece of a hyperplane, it fails for every $p > 1$. When $S$ has nonzero Gaussian curvature, the Stein–Tomas restriction theorem gives the estimate for $q = 2$ and $1 \le p \le 2(n+1)/(n+3)$. For the sphere, the conjectured answer is the restriction conjecture; it is known for $n = 2$ and open for $n \ge 3$.