The sharp restriction conjecture

Let $n\geq 2$, let $Q=[-1,1]^{n-1}\subset\mathbb{R}^{n-1}$, and let $\Phi:Q\to\mathbb{R}$ be an elliptic phase. For $0<p,q\leq\infty$, let $R^*(p\to q)$ denote the adjoint restriction estimate

$$\|\operatorname{Re}^*f\|_q<_\sim\|f\|_p.$$

The sharp restriction conjecture. $R^*(p\to q)$ holds whenever

$$q>\frac{2n}{n-1}\qquad\text{and}\qquad p'\leq\frac{n-1}{n+1}q.$$

This is a central conjecture for Fourier restriction to elliptic hypersurfaces. It has been verified for $n=2$, but remains open in higher dimensions.