The adjoint restriction conjecture for the paraboloid

Let $S$ be a non-empty smooth compact subset of the paraboloid

$$\{(\tau,\xi)\in\mathbb{R}\times\mathbb{R}^n:\tau=|\xi|^2\},$$

where $n\geq1$, and let $\mathrm{d}\sigma$ be the pull-back of $n$-dimensional Lebesgue measure under the projection $(\tau,\xi)\mapsto\xi$. For a Schwartz function $f$, define

$$(f\mathrm{d}\sigma)^{\vee}(t,x)=\int_S f(\tau,\xi)e^{2\pi i(x\cdot\xi+t\tau)}\mathrm{d}\sigma(\xi).$$

For $1\leq p,q\leq\infty$, the adjoint restriction estimate is

$$\|(f\mathrm{d}\sigma)^{\vee}\|_{L^q_{t,x}(\mathbb{R}\times\mathbb{R}^n)}\leq C_{p,q,n,S}\|f\|_{L^p(S;\mathrm{d}\sigma)}.$$

Write $p'$ for the conjugate exponent of $p$. The adjoint restriction conjecture for the paraboloid. The inequality above holds if and only if

$$q>\frac{2(n+1)}{n},\qquad \frac{n+2}{q}\leq\frac{n}{p'}.$$

The displayed conditions are necessary for the estimate, and the conjecture asks whether they are also sufficient. Its status is not determined by the supplied text.