Restriction conjecture

Remarks

The conditions on $p$ and $q$ are necessary. The condition $p < 2n/(n+1)$ comes from the function $\widehat{d\sigma}$. It decays exactly like $\abs{x}^{-(n-1)/2}$, so it lies in $L^{p'}$ only for $p' > 2n/(n-1)$. By duality, the conjecture requires $\widehat{d\sigma}$ to lie in $L^{p'}$. The condition $q \le \frac{n-1}{n+1}p'$ comes from the Knapp example, a function whose Fourier transform is concentrated on a small cap of the sphere.

Status: the conjecture holds for $n = 2$ and is open for every $n \ge 3$. The Stein–Tomas restriction theorem proves the case $q = 2$, where the range is $p \le 2(n+1)/(n+3)$. The conjecture implies the Kakeya conjecture that every Kakeya set in $\R^n$ has Hausdorff dimension $n$.